Fans in the Theory of Real Semigroups II. Combinatorial Theory

TL;DR

This paper develops the combinatorial theory of ARS-fans in the dual category of abstract real spectra, showing that finite fan structures are fully determined by their specialization partial order.

Contribution

It introduces the combinatorial framework for ARS-fans, demonstrating their classification via specialization order and utilizing ternary semigroups and standard generating systems.

Findings

Finite ARS-fans are classified by their specialization partial order.

Every ARS-fan decomposes into levels with involutions.

The notion of standard generating systems replaces geometric tools.

Abstract

In the paper: Fans in the Theory of Real Semigroups. I. Algebraic Theory (submitted) we introduced the notion of fan in the categories of real semigoups and their dual abstract real spectra and developed the algebraic theory of these structures. In this paper we develop the combinatorial theory of ARS-fans, i.e., fans in the dual category of abstract real spectra. Every ARS is a spectral space and hence carries a natural partial order called the {\it specialization partial order}. Our main result shows that the isomorphism type of a finite fan in the category ARS is entirely determined by its order of specialization. The main tools used to prove this result are: (1) Crucial use of the theory of {\it ternary semigroups}, a class of semigroups underlying that of RSs; (2) Every ARS-fan is a disjoint union of abstract order spaces (called {\it levels}); (3) Every level carries a natural involution of abstract order spaces, and (4) The notion of a {\it standard generating system}, a combinatorial tool replacing, in the context of ARSs, the (absent) tools of combinatorial geometry (matroid theory) employed in the cases of fields and of abstract order spaces.

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Full text

Introduction

In [DP5a] we introduced a notion of fan in each of the dual categories RS and ARS of real semigroups, and of abstract real spectra, dubbed, respectively, RS-fans and ARS-fans. The emphasis in [DP5a] was on the algebraic theory of RS-fans. The present paper, a continuation of [DP5a], is devoted to develop the combinatorial theory of ARS-fans, i.e., fans in the category of abstract real spectra.

The Introduction to [DP5a] gives an account of the role of fans in the theories of preordered fields, of quadratic forms, and in real algebraic and analytic geometry.

Our main result in this paper is Theorem 3.11, showing that the isomorphism type of a finite ARS-fan (in the category ARS) is entirely determined by its order of specialization as a spectral space 111 For a general reference on spectral spaces, see [DST].. The proof of this result relies on a combinatorial machinery that we set up in §§ 1,2. This machinery also gives detailed information on the structure of ARS-fans under their order of specialization.

In Section 1 we introduce the notion of level of an ARS-fan $(X,F)$. Levels are the pieces \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}=\{h\in X\,|\,h^{-1}[0]=I\} of the partition of the character space $X$ of the real semigroup $F$, determined by the ideals $I$ of $F$. Since $F$ is a RS-fan, its ideals are necessarily prime and saturated ([DP5a], Prop. 1.6 (4), Cor. 3.10 (1)), and the family of them is totally ordered under inclusion ([DP5a], Fact 1.4). By Proposition 5.11 of [DP5a], each level is an abstract space of orders and therefore (by results from [D1], [D2] and [Li]) possesses a structure of combinatorial geometry (matroid). Further, there exist canonical AOS-morphisms linking each level to any level determined by a larger ideal (i.e., a “higher” level); Proposition 1.2 (2).

In the next § 2 we exploit the combinatorial geometric structure of the levels to investigate the fine structure of ARS-fans. In Theorem 2.8 we show that multiplication of a character of level L_{\raisebox{-3.0pt}{\scriptstyle I}} by any pair of elements \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in X so that Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}):={\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}}^{-1}[0]\,\mbox{\subseteq}\,I\;(i=1,2) defines an involution of the AOS L_{\raisebox{-3.0pt}{\scriptstyle I}}. These involutions are compatible with the order of specialization between levels induced by inclusion of the determining ideals (2.8 (e)). Further, we prove that these involutions permute certain AOS-subfans of the levels defined by combinatorial conditions (Propositions 2.10 and 2.11). Altogether, the results proved in this section show that the order structure of ARS-fans is subject to strong constraints, illustrated in 2.18.

To prove the isomorphism Theorem 3.11, the combinatorial machinery mentioned above is used together with the notion of a standard generating system introduced in 3.4. This notion is a substitute for the combinatorial geometric notions existing in the context of AOSs, but absent in that of ARSs.

Preliminaries. For easy reference we state, without proof, the following simple facts proved in [DP5a] and frequently used below. The axioms defining the notion of a ternary semigroup (abbreviated TS) appear in [DP5a], Def. 1.1, and [DP1], § 1, p. 100; X_{\raisebox{-3.0pt}{\scriptstyle T}} denotes the set of TS-homomorphisms of a TS, $T$, into the TS ${\bf 3}=\{1,0,-1\}$ (the TS-characters of $T$).

The first Lemma gives several characterizations of the specialization order of the spectral topology on the character set of a ternary semigroup.

Lemma 0.1

Let $T$ be a TS, and let g,h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle T$}}}. The following are equivalent:

$(1)$* g\,\mbox{\rightsquigarrow\,}\,h $($i.e., $h$ is an specialization of g$$).*

$(2)$* h^{-1}[1]\,\mbox{\subseteq}\,g^{-1}[1]  $($equivalently,  h^{-1}[-1]\,\mbox{\subseteq}\,g^{-1}[-1]).*

$(3)$* g^{-1}[\{0,1\}]\,\mbox{\subseteq}\,h^{-1}[\{0,1\}].*

$(4)$* Z(g)\,\mbox{\subseteq}\,Z(h) and \mbox{\forall}\,a\in G\,(a\not\in Z(h)\;\;\mbox{\Rightarrow}\;\;g(a)=h(a)).*

$(5)$* $h=h^{2}g$ $($equivalently, $h^{2}=hg)$. $\Box$*

We also register the following algebraic characterizations of inclusion and equality of zero-sets of elements of X_{\raisebox{-3.0pt}{\scriptstyle T}}.

Lemma 0.2

Let $T$ be a TS, and let u,g,h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle T$}}}. Then:

$(1)$* Z(g)\,\mbox{\subseteq}\,Z(h)\;\mbox{\Leftrightarrow}\;h=hg^{2}.*

$(2)$* Z(g)=Z(h)\;\mbox{\Leftrightarrow}\;g^{2}=h^{2}.*

$(3)$* If  u\,\mbox{\rightsquigarrow\,}\,g,h, then  Z(g)\,\mbox{\subseteq}\,Z(h)  if and only if  g\,\mbox{\rightsquigarrow\,}\,h. $\Box$*

Proposition 0.3

Let $G$ be a RS-fan. Then:

$(1)$* For all elements g,\,h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} such that g\,\mbox{\rightsquigarrow\,}\,h $($hence Z(g)\,\mbox{\subseteq}\,Z(h)) and every ideal $I$ such that Z(g)\,\mbox{\subseteq}\,I\,\mbox{\subseteq}\,Z(h) there is f\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} such that g\,\mbox{\rightsquigarrow\,}f\,\mbox{\rightsquigarrow\,}\,h and $Z(f)=I$.*

$(2)$* For every   g\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} and every ideal  $I\supseteq Z(g)$ there is a $($necessarily unique$)$ f\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} such that g\,\mbox{\rightsquigarrow\,}f and $Z(f)=I$.*

$(3)$* For every ideal $I$ of $F$ there is an f\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} such that $Z(f)=I$. $\Box$*

1 Levels of a ARS-fan

The saturated prime ideals of a real semigroup induce a partition of its character space. The pieces are called levels : the level corresponding to a saturated prime ideal $I$ of $G$ is the set of all g\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} such that $Z(g)=I$. In the case of RS-fans, (proper) ideals —automatically prime and saturated ([DP5a], Prop. 1.6 (4), Cor. 3.10 (1))— are totally ordered under inclusion ([DP5a], Fact 1.4), a fact of much help in studying the relationship between their levels. This notion, together with that of a connected component (2.14), will be the main technical tools employed in the analysis of the fine structure of ARS-fans carried out in this paper.

Proposition 5.11 in [DP5a] shows that the levels of an ARS-fan have a canonical structure of AOS-fans (1.1 (a)), that is, of fans in the category of abstract order spaces (cf. [M], § 3.1, pp. 37 ff.). We prove (Proposition 1.2) that inclusion of ideals induces AOS-morphisms between the corresponding levels (cf. 1.1 (c)). As an application we prove (Corollary 1.10) that the cardinality of a finite RS-fan, $F$, and that of its character space, X_{\raisebox{-3.0pt}{\scriptstyle F}}, are related by the identity {\mathrm{card}}\,(F)=2\cdot{\mathrm{card}}\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}})+1, an analog for RSs of a result known to hold for reduced special groups. $\Box$

Preliminaries and Notation 1.1

(a) Given a real semigroup $G$ and a saturated prime ideal $I$ of $G$, we denote by G_{\raisebox{-3.0pt}{\scriptstyle I}} the RSG $(G/I)\setminus\{\pi(0)\}$, cf. [DP5a], Prop. 5.11. The congruence of $G$ determined by the set of characters \mbox{\mbox{$\cal H$}{\raisebox{-3.0pt}{$\scriptstyle I$}}}=\{h\in\mbox{X{\raisebox{-3.0pt}{$\scriptstyle G$}}}\,|\,Z(h)=I\} is denoted by \sim_{\raisebox{-3.0pt}{\scriptstyle I}}, cf. [DP5a], § 5.C. Every character h\in\mbox{\mbox{$\cal H$}{\raisebox{-3.0pt}{$\scriptstyle I$}}} induces a map  \mbox{\widehat{h}}:\mbox{G{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,\mbox{\longrightarrow}\,\{\,\pm 1\} defined by  \mbox{\widehat{h}}\,\mbox{\circ}\,\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}=h. The correspondence h\,\mapsto\mbox{\widehat{h}} is a bijection between the set  \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(G)=\mbox{\mbox{$\cal H$}{\raisebox{-3.0pt}{$\scriptstyle I$}}}  and the space of orders X_{\raisebox{-3.0pt}{\scriptstyle G{I}}}  of  G_{\raisebox{-3.0pt}{\scriptstyle I}} . (L_{\raisebox{-3.0pt}{\scriptstyle I}} stands for “$I$-th level”; see item (b.ii) below). Thus, we can identify the set \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(G)\;\mbox{\subseteq}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} with the AOS (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G_{I}$}}},\mbox{G_{\raisebox{-3.0pt}{$\scriptstyle I$}}}). We shall systematically use this identification in the sequel, and unambiguously refer to the AOS structure of the set L_{\raisebox{-3.0pt}{\scriptstyle I}}($G$). In case $G$ is a RS-fan, [DP5a], Prop. 5.11, shows that L_{\raisebox{-3.0pt}{\scriptstyle I}}($G$) is an AOS-fan.

(b) Let $F$ be a RS-fan.

(i)   We denote by  \mbox{\mathrm{Spec,}}(F)  the set of all proper ideals of $F$.

(ii)  For I\in\mbox{\mathrm{Spec,}}(F) the set \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F)=\{h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}\,|\,Z(h)=I\} is called the $I$****-th level of X_{\raisebox{-3.0pt}{\scriptstyle F}} .

(iii) For f\in\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}, the depth of $f$, denoted $d(f)$, is the order type of the set \{g\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}\,|\,f\,\mbox{\rightsquigarrow\,}\,g\} under the order of specialization.depthfan!depth of element (Since (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}},\,\mbox{\rightsquigarrow\,}) is a root-system, the order  $\rightsquigarrow$  is total on this set.)

(iv)  For I\in\mbox{\mathrm{Spec,}}(F), the order type of the set \{J\in\mbox{\mathrm{Spec,}}(F)\,|\,J\supseteq\,I\} under the $($total $)$ order of inclusion will be called the depth of $I$, denoted $d(I)$.

(v)  The length of X_{\raisebox{-3.0pt}{\scriptstyle F}}, denoted \ell(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}), is the order type of the $($totally ordered $)$ set  \mbox{\mathrm{Spec,}}(F).

(c) (AOS- and ARS-morphisms; [M], § 2, pp. 23-24, and § 6, p. 103)

(i) Let $(X,G),(Y,H)$ be ARS’s. A map F:X\,\mbox{\longrightarrow}\,Y is an ARS-morphism iff for all  $a\in H$  there is  $b\in G$  so that  \mbox{\widehat{a}}\;\mbox{\circ}\,F=\mbox{\widehat{b}}. Here, for $x\in G$, \mbox{\widehat{x}}:X\,\mbox{\longrightarrow}\,{\bf 3} denotes the map “evaluation at $x$”: \mbox{\widehat{x}}(\mbox{\sigma}):=\mbox{\sigma}(x), for \mbox{\sigma}\in X, and similarly for $H$.

(ii) The definition of an AOS-morphism is similar, with $(X,G),(Y,H)$ AOS’s, and the evaluation maps taking values in $\{\pm 1\}$.

(d) If f:G\,\mbox{\longrightarrow}\,H is a RS-morphism (resp. RSG-morphism), the dual map f^{*}:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle H$}}}\,\mbox{\longrightarrow}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} defined by f^{*}(\mbox{\gamma}):=\mbox{\gamma}\,\mbox{\circ}\,f  for \mbox{\gamma}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle H$}}}, is an ARS-morphism (resp., AOS-morphism). $\Box$

Remarks. (a) Clearly, the union and the intersection of an inclusion chain of (proper) prime ideals in any ternary semigroup is a (proper) prime ideal. In particular, if $F$ is a fan, the totally ordered set (\mbox{\mathrm{Spec,}}(F),\mbox{\subseteq}) is (Dedekind) complete. $\Box$

Proposition 1.2

Let $F$ be a RS-fan and let I\,\mbox{\subseteq}\,J be ideals of $F$. With notation as in 1.1,

$(1)$* The rule $a/J\,\mapsto\,a/I$  $(a\in F\setminus J)$ defines a homomorphism of special groups \mbox{\iota_{\raisebox{-3.0pt}{$\scriptstyle JI$}}}:\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle J$}}}\,\mbox{\longrightarrow}\,\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}}.*

$(2)$* The map \mbox{\mbox{$\kappa$}{\raisebox{-3.0pt}{$\scriptstyle IJ$}}}:\mbox{L{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F)\,\mbox{\longrightarrow}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle J$}}}(F) assigning to each g\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F) the unique element h\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle J$}}}(F) such that g\,\mbox{\rightsquigarrow\,}\,h is an AOS-morphism.*

Proof. (1) (a) \iota_{\raisebox{-3.0pt}{\scriptstyle JI}} is well-defined.

We must show: a,b\,\in F\setminus J\;\mbox{\wedge}\;a\,\mbox{\sim_{\raisebox{-3.0pt}{$\scriptstyle J$}}}\,b\;\,\mbox{\Rightarrow}\;\,a\,\mbox{\sim_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,b. Since I\,\mbox{\subseteq}\,J, this is clear from Lemma 5.10 of [DP5a] which states that, for an ideal $K$ of $F$ and a,b\in F\setminus K,\;\,a\,\mbox{\sim_{\raisebox{-3.0pt}{$\scriptstyle J$}}}\,b\;\mbox{\Leftrightarrow}\;\mbox{\exists}z\not\in K\,(az=bz).

Clearly, we have:

(b) \iota_{\raisebox{-3.0pt}{\scriptstyle JI}} is a group homomorphism sending $-1/J$ to $-1/I$.

Since F_{\raisebox{-3.0pt}{\scriptstyle J}} is a RSG-fan, \iota_{\raisebox{-3.0pt}{\scriptstyle JI}} is automatically a homomorphism of special groups.

(2) By (1) and 1.1 (d), the map \mbox{\iota^{}{\raisebox{-3.0pt}{$\scriptstyle JI$}}}:\mbox{X{\raisebox{-3.0pt}{$\scriptstyle F_{I}$}}}\,\mbox{\longrightarrow}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F_{J}$}}} dual to \iota_{\raisebox{-3.0pt}{\scriptstyle JI}} is an AOS-morphism. The map \mbox{\kappa}_{\raisebox{-3.0pt}{\scriptstyle IJ}} is   \mbox{\mbox{$\kappa$}{\raisebox{-3.0pt}{$\scriptstyle IJ$}}}=({\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}})^{-1}\circ\mbox{\iota^{}{\raisebox{-3.0pt}{$\scriptstyle JI$}}}\circ\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!I$}}}, where \raisebox{1.72218pt}{\mbox{\varphi}}_{\raisebox{-3.0pt}{\scriptstyle!I}} denotes the bijection g\mapsto\mbox{\widehat{g}} (g\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F)), identifying \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F) with X_{\raisebox{-3.0pt}{\scriptstyle F_{I}}} (1.1 (a)), and similarly for L_{\raisebox{-3.0pt}{\scriptstyle J}}($F$). It only remains to prove g\,\mbox{\rightsquigarrow\,}\,\mbox{\mbox{$\kappa$}{\raisebox{-3.0pt}{$\scriptstyle IJ$}}}(g), for g\in\mbox{L{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F). To ease notation, write h=\mbox{\mbox{$\kappa$}_{\raisebox{-3.0pt}{$\scriptstyle IJ$}}}(g). According to Lemma 0.1 (4) we must show Z(g)\,\mbox{\subseteq}\,Z(h) and a\not\in Z(h)\;\mbox{\Rightarrow}\;g(a)=h(a). The inclusion of zero-sets is I\,\mbox{\subseteq}\,J. Let $a\not\in Z(h)=J$. Since:

\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(h)=\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(\mbox{\mbox{$\kappa$}{\raisebox{-3.0pt}{$\scriptstyle IJ$}}}(g))=\mbox{\iota^{*}{\raisebox{-3.0pt}{$\scriptstyle JI$}}}(\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!I$}}}(g))=\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!I$}}}(g)\,\mbox{\circ}\,\mbox{\iota_{\raisebox{-3.0pt}{$\scriptstyle JI$}}} ,   \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!I$}}}(g)(a/I)=g(a)  and  \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(h)(a/J)=h(a),

(cf. 1.1), we get,

h(a)=(\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!I$}}}(g))(\mbox{\iota{\raisebox{-3.0pt}{$\scriptstyle JI$}}}(a/J))=\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}_{\raisebox{-3.0pt}{$\scriptstyle\!I$}}}(g)(a/I)=g(a),

as required. $\Box$

Next we prove that the depth of an ideal in a fan is the same as the depth of any element in the corresponding level; in particular, elements of the same depth belong to the same level.

Proposition 1.3

Let $F$ be a RS-fan. For f\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}} we have $d(f)=d(Z(f))$; equivalently, the sets \{g\in\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}\,|\,f\,\mbox{\rightsquigarrow\,}\,g\} $($ordered under specialization $)$ and $\{J\in$ Spec$(F)\,|\,J\supseteq Z(f)\}$ $($ordered under inclusion $)$ are order-isomorphic.

Proof. To ease notation, set f\!\uparrow\;=\{\,g\in\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}\,|\,f\,\mbox{\rightsquigarrow\,}\,g\} and $I\!\uparrow\;=\{J\in\textrm{Spec}(F)\,|\,J\supseteq I\}$ $(I\in\textrm{Spec}(F))$. The required order isomorphism is the map Z:f\!\uparrow\,\mbox{\longrightarrow}\;Z(f)\!\uparrow  assigning to each  $g\in f\!\uparrow$  its zero-set. That

(a)  $Z$ is increasing,    and    (b)  $Z$ is surjective,

is clear, from g\,\mbox{\rightsquigarrow\,}\,h\;\,\mbox{\Rightarrow}\;\,Z(g)\,\mbox{\subseteq}\,Z(h) and Proposition 0.3 (2), respectively. That

(c) $Z$ is injective.

follows from 0.2 (3). $\Box$

A trivial variant of the proof of 1.3 gives:

Proposition 1.4

Let $F$ be a RS-fan. Given  f_{1},f_{2}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}} such that f_{1}\,\mbox{\rightsquigarrow\,}\,f_{2}, the intervals \{\,g\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}\,|\,f_{1}\,\mbox{\rightsquigarrow\,}\,g\,\mbox{\rightsquigarrow\,}\,f_{2}\} $($ordered under specialization $)$ and \{J\in Spec(F)\,|\,Z(f_{1})\,\mbox{\subseteq}\,J\,\mbox{\subseteq}\,Z(f_{2})\} $($ordered under inclusion $)$ are order-isomorphic. $\Box$

The results in the next two Lemmas will be frequently used in this and in subsequent sections.

Lemma 1.5

Let $G$ be a RS and let \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}},h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} be so that  \mbox{\bigcup}_{i=1}^{r}Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,\mbox{\subseteq}\,Z(h). For $i=1,\dots,r$, let \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} be such that g_{\raisebox{-3.0pt}{\scriptstyle i}} $\rightsquigarrow$  f_{\raisebox{-3.0pt}{\scriptstyle i}} and  Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,\mbox{\subseteq}\,Z(\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,\mbox{\subseteq}\,Z(h). Then,

$(*)$*     h\cdot\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\dots\,\cdot\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}=h\cdot\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\dots\,\cdot\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle r$}}}.*

Note. The products in (*) may not be in X_{\raisebox{-3.0pt}{\scriptstyle G}}.

Proof. Obviously, () holds whenever $x\in Z(h)$. If $x\not\in Z(h)$, from the assumptions we get x\not\in\,\mbox{\bigcup}_{i=1}^{r}Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}})  and  x\not\in\;\mbox{\bigcup}_{i=1}^{r}Z(\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle i$}}}). Since g_{\raisebox{-3.0pt}{\scriptstyle i}} $\rightsquigarrow$ f_{\raisebox{-3.0pt}{\scriptstyle i}}, we get \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}(x)=\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle i$}}}(x) for $i=1,\dots,\,r$ (Lemma 0.1 (4)), and () follows. $\Box$

Lemma 1.6

Let $F$ be a RS-fan. Then,

$(a)$*  For  $i=1,\dots,\,r$, with $r$ odd, let   \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}} be such that  g_{\raisebox{-3.0pt}{\scriptstyle i}} $\rightsquigarrow$  h_{\raisebox{-3.0pt}{\scriptstyle i}}. Then,  \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\dots\,\cdot\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\,\,\mbox{\rightsquigarrow\,}   \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\dots\,\cdot\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}.*

$(b)$*  Let \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},f,g,k\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}} be such that f,\,g\,\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\;k\,\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}, and Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\,\mbox{\subseteq}\,Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}). Then,fg\,k\,\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}.*

Note. Here the products are in X_{\raisebox{-3.0pt}{\scriptstyle!F}} as the number of factors is odd.

Proof. (a) For  $i=1,\dots,\,r$ we have \mbox{h^{2}{\raisebox{-3.0pt}{$\scriptstyle i$}}}=\mbox{h{\raisebox{-3.0pt}{$\scriptstyle i$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}} (Lemma 0.1 (5)). Multiplying these equalities termwise gives (\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\dots\,\cdot\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}})^{2}=(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\dots\,\cdot\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}})(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\dots\,\cdot\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}), which proves the assertion.

(b) By Lemma 0.1 we must prove \mbox{h^{2}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(fg\,k). Obviously, this equality holds at every x\in Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}). If x\not\in Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), then x\not\in Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}), and f,\,g\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}} implies \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(x)=f(x)=g(x)\neq 0; also k\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}} implies \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(x)=k(x)\neq 0, whence $f(x)g(x)=1$ and \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(x)k(x)=1. This yields  (\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}fg\,k)(x)=(f(x)g(x))(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(x)k(x))=1. On the other hand, (\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(x))^{2}=1, proving that the required identity holds at x\not\in Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})  as well. $\Box$

Our last result in this section, Corollary 1.10, shows that if $F$ is a finite RS-fan and X_{\raisebox{-3.0pt}{\scriptstyle F}} its character space, then {\mathrm{card}}\,(F)=2\cdot{\mathrm{card}}\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}})+1. This identity is the analog of a well-known result relating the cardinalities of a finite RSG-fan and its space of orders ([ABR], p. 75). The result follows from a more general observation, valid for RS-fans of arbitrary cardinality.

Proposition 1.7

Let $I\subset J$ be consecutive ideals of a RS-fan $F$ $($with, possibly, $J=F)$. Then,

$($i $)$   Under product induced by $F$, $J\setminus I$ is a group of exponent $2$ with unit $x^{2}$ for any $x\in J\setminus I$ $($and distinguished element $-1=-x^{2})$.

$($ii $)$ The restriction of the quotient map \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,\lceil\,(J\setminus I):J\setminus I\,\mbox{\longrightarrow}\,\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}}=F/I\setminus\{\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(0)\} is a group isomorphism preseving the distinguished element $-1$.

Proof. (i) Since $I$ is prime, $J\setminus I$ is closed under product. Given $x,y\in J\setminus I$, we must prove $x^{2}=y^{2}$ (which implies $x^{2}y=y^{3}=y$). By the separation theorem for TSs ([DP1], Thm. 1.9, pp. 103-104) it suffices to show that $h(x^{2})=h(y^{2})$ for all h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}. If J\,\mbox{\subseteq}\,Z(h), then $h(x^{2})=h(y^{2})=0$. If Z(h)\,\mbox{\subseteq}\,I, then $h(x),h(y)\neq 0$, whence $h(x^{2})=h(y^{2})=1$.

(ii) Clearly, \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(x)\neq\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(0), i.e., \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(x)\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}}, for all $x\in J\setminus I$, and \pi_{\raisebox{-3.0pt}{\scriptstyle I}} preserves product.

— \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,\lceil\,(J\setminus I) is injective.

Suppose \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(x)=\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(y), i.e., x\>\mbox{\sim_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,y, with $x,y\in J\setminus I$. By [DP5a], Lemma 5.10 (cf. proof of 1.2), $xz=yz$ for some $z\not\in I$. To prove $x=y$, let h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}. If J\,\mbox{\subseteq}\,Z(h), then $h(x)=h(y)=0$. If Z(h)\,\mbox{\subseteq}\,I, then $h(z)\neq 0$, and we get $h(x)=h(y)$.

— \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(x^{2})=\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(1), for $x\in J\setminus I$.

Clear, for $Z(h)=I$ implies $h(x^{2})=1$. In particular, \pi_{\raisebox{-3.0pt}{\scriptstyle I}} preserves $-1$.

— \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,\lceil\,(J\setminus I) is onto F_{\raisebox{-3.0pt}{\scriptstyle I}}.

Let p\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}} ; then, p=\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(q) with $q\not\in I$. Taking $z\in J\setminus I$, we have $qz^{2}\in J\setminus I$, whence \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(qz^{2})=\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(q)\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(z^{2})=\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(q)\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(1)=\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(q)=p. $\Box$

Notation 1.8

Let $F$ be a finite RS-fan, and let

\{0\}=\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle n$}}}\subset\,\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle n-1$}}}\subset\cdots\subset\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\subset\,\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\subset\,F=\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}

be the set of all its ideals; thus, for $1\leq d\leq n$, I_{\raisebox{-3.0pt}{\scriptstyle d}} is the ideal of depth $d$. We set \mbox{F_{\raisebox{-3.0pt}{$\scriptstyle d$}}}=\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I_{d}$}}}=(F/\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle d$}}})\setminus\{\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle d$}}}(0)\}, where \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle d$}}}:F\,\mbox{\longrightarrow}\;F/\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle d$}}} denotes the canonical quotient map. We also write L_{\raisebox{-3.0pt}{\scriptstyle d}} for L_{\raisebox{-3.0pt}{\scriptstyle I_{d}}}; cf. 1.1 (b). $\Box$

Clearly, F\setminus\{0\}=\mbox{\bigcup}_{d=1}^{n}(\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle d-1$}}}\setminus\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle d$}}}) (disjoint union), whence, by 1.7 we have \mbox{\mathrm{card}}\,(F)=\sum_{d=1}^{n}\mbox{\mathrm{card}}\,(\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle d-1$}}}\setminus\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle d$}}})+1=\sum_{d=1}^{n}\mbox{\mathrm{card}}\,(\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle d$}}})+1. Since the levels partition X_{\raisebox{-3.0pt}{\scriptstyle F}}, 1.1 yields:

Proposition 1.9

For any finite RS-fan $F$, $\mathrm{card}$ (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}})=\sum_{d=1}^{n}\mbox{\mathrm{card}}\,(\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle d$}}})=\sum_{d=1}^{n}{\mathrm{card}}\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F_{d}$}}}).$\Box$

Corollary 1.10

For a finite RS-fan, $F$,   {\mathrm{card}}\,(F)=2\cdot{\mathrm{card}}\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}})+1.

Proof. Since the F_{\raisebox{-3.0pt}{\scriptstyle d}} are finite RSG-fans ([DP5a], Prop. 5.11), we know that {\mathrm{card}}\,(\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle d$}}})=$$2\cdot{\mathrm{card}}\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F_{d}$}}}) for $1\leq\,d\,\leq\,n$ (see [ABR], p. 75). The result follows, then, from Proposition 1.7 and the preceding cardinality identities. $\Box$

2 Involutions of ARS-fans

Notation 2.1

In addition to the notation introduced in Definition 1.1, for J\,\mbox{\subseteq}\,I in Spec($F$) we define the sets:

\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}=\{\,h\in\mbox{L{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,|\,\mbox{\exists}g\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}}\,(\,g\,\mbox{\rightsquigarrow\,}\,h\;\mbox{\wedge}\;Z(g)=J\,)\}.

\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}=\{\,h\in\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}\,|\,\mbox{\forall}g^{\prime}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}}\,(\,g^{\prime}\,\mbox{\rightsquigarrow\,}\,h\;\mbox{\Rightarrow}\;J\,\mbox{\subseteq}\,Z(g^{\prime}))\}.

That is,  S^{I}_{\raisebox{-3.0pt}{\scriptstyle!J}} consists of those elements of level $I$ having predecessors of level $J$ or lower in the specialization partial order;  C^{I}_{\raisebox{-3.0pt}{\scriptstyle!J}} is the set of elements in L_{\raisebox{-3.0pt}{\scriptstyle I}} having predecessors at level $J$ but not lower. $\Box$

Remarks 2.2

(i) For $I\in$ Spec($F$), \mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\{0\}$}}}=\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\{0\}$}}}, and S^{I}_{\raisebox{-3.0pt}{\scriptstyle I}} = L_{\raisebox{-3.0pt}{\scriptstyle I}}. (Recall that $\{0\}$ is the smallest member of Spec($F$), i.e., the zero-set of the lowest level of X_{\raisebox{-3.0pt}{\scriptstyle!F}}.)

(ii)   For J\,\mbox{\subseteq}\,I in Spec($F$),  \mbox{S^{I}_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}\neq\mbox{\emptyset}.

Proof. Let g\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}} be such that $Z(g)=J$ (exists by Proposition 0.3 (3)). If $h$ is the unique  $\rightsquigarrow$  -successor of $g$ of level $I$ (Proposition 0.3 (2)), then h\in\mbox{S^{I}_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}.

(iii)  For J\,\mbox{\subseteq}\,I in Spec($F$),  \mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}={\mathrm{Im}}\,(\mbox{\mbox{$\kappa$}{\raisebox{-3.0pt}{$\scriptstyle JI$}}}), where \mbox{\mbox{$\kappa$}{\raisebox{-3.0pt}{$\scriptstyle JI$}}}:\mbox{L{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(F)\,\mbox{\longrightarrow}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F) is the AOS-morphism defined in Proposition 1.2 (2).

(iv)  For J\,\mbox{\subseteq}\,I in Spec($F$),  \mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}\supseteq\ \mbox{\bigcup}\>\{\>\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J^{\prime}$}}}\,|\,J^{\prime}\in\,\mbox{Spec}(\mbox{F})\;\,\mbox{and}\;\,J^{\prime}\,\mbox{\subseteq}\,J\>\}. (Note that C^{I}_{\raisebox{-3.0pt}{\scriptstyle!J^{\prime}}} may be empty for some J^{\prime}\,\mbox{\subseteq}\,J.)

(v)   For J\,\mbox{\subseteq}\,I in Spec($F$),  \mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}=\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}\setminus\mbox{\bigcup}\>\{\,\mbox{S^{I}_{\raisebox{-3.0pt}{$\scriptstyle\!J^{\prime}$}}}\,|\,J^{\prime}\in\,\mbox{Spec}(\mbox{F})\;\,\mbox{and}\;\,J^{\prime}\,\subset\,J\>\}.

(vi)  For J,J^{\prime}\,\mbox{\subseteq}\,I in Spec($F$), $J\neq J^{\prime}$, we have \mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle J$}}}\,\cap\,\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J^{\prime}$}}}=\mbox{\emptyset}. $\Box$

In order to render later arguments as transparent as possible, we recall the following simple (and well-known) facts about fans in the categories RSG and AOS.

Lemma 2.3

Let  g:H\,\mbox{\longrightarrow}\,G  be a SG-homomorphism between RSG-fans, and let g^{*}:(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}},G) \mbox{\longrightarrow}\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle H$}}},H) denote the AOS-morphism dual to $g$ $($cf. 1.1 $(d))$. Then,

$(1)$* With representation induced by that of $H$, ${\mathrm{Im}}(g)$ is a RSG-fan, and $G$ is isomorphic to the extension of  ${\mathrm{Im}}(g)$ by the exponent-two group \mbox{\Delta}=G/\,{\mathrm{Im}}(g).*

$(2)$* $({\mathrm{Im}}(g^{*}),\,H/\,ker(g))$ is an AOS-fan.*

Remarks 2.4

(a) For the definition of extension of a SG by a group of exponent two, see [DM1], Ex. 1.10, p. 10.

(b) By the duality between RSGs and AOSs ([DM1], Ch. 3), the dual statement holds as well: given an AOS-morphism of (AOS-)fans, \mbox{\kappa}:(X,G)\,\mbox{\longrightarrow}\,(Y,H), the assertions (1) and (2) hold with $g$ := \mbox{\kappa}^{*} (the SG-morphism dual to $\kappa$), and with g^{*}=\mbox{\kappa}. $\Box$

Sketch of proof of 2.3. (1) The first assertion is easily checked. For the second, ${\mathrm{Im}}(g)$ is a direct summand of the group $G$. Let pr:\;G\,\mbox{\longrightarrow}\;{\mathrm{Im}}(g) be the projection onto the factor ${\mathrm{Im}}(g)$; $pr$ is a SG-morphism ($G$ and ${\mathrm{Im}}(g)$ are fans), and is the identity on ${\mathrm{Im}}(g)$. The isomorphism between $G$ and {\mathrm{Im}}(g)[\mbox{\Delta}] is  f(a)=\mbox{\langle,pr(a),,a/{\mathrm{Im}}(g),\rangle}, for $a\in\,G$.

(2) Recall that $g^{*}$ is defined by composition,  g^{*}(\mbox{\sigma})=\mbox{\sigma}\,\circ\,g\ (\mbox{\sigma}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}), see 1.1 (d),  and that  {\mathrm{Im}}(g^{*})^{\mbox{\bot}}=\mbox{\bigcap}\,\{ker(\mbox{\gamma})\,|\,\mbox{\gamma}\in\,{\mathrm{Im}}(g^{*})\}=\mbox{\bigcap}\,\{ker(\mbox{\sigma}\,\mbox{\circ}\>g)\,|\,\mbox{\sigma}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}\}. Routine checking from these definitions proves that ${\mathrm{Im}}(g^{*}$) is closed under product of any three members (since X_{\raisebox{-3.0pt}{\scriptstyle G}} is), and that  {\mathrm{Im}}(g^{*})^{\mbox{\bot}}=ker(g)  (since \mbox{\bigcap}\,\{ker(\mbox{\sigma})\,|\,\mbox{\sigma}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}\}=\{1\}), whence {\mathrm{Im}}(g^{*})\,\mbox{\subseteq}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle H/ker(g)$}}}.

Clearly, the map \mbox{\overline{g}}:H/ker(g)\,\mbox{\longrightarrow}\,{\mathrm{Im}}(g) induced by $g$ is an SG-isomorphism. Thus, we have a commutative diagram of SG-morphisms:

[TABLE]

It only remains to show that {\mathrm{Im}}(g^{*})\,\supseteq\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle H/ker(g)$}}}. Any SG-character \mbox{\gamma}:\,H/ker(g)\,\mbox{\longrightarrow} \mbox{\mathbb{Z}}_{\raisebox{-3.0pt}{\scriptstyle 2}} can be lifted to a map \mbox{\sigma}:\,G\,\mbox{\longrightarrow}\,\mbox{\mbox{\mbox{\mathbb{Z}}_{\raisebox{-3.0pt}{\scriptstyle 2}}}}, via the identification of  $G$ with  {\mathrm{Im}}(g)[\mbox{\Delta}], as follows: for each $a\in G$ there is $b\in H$ such that $pr(a)=g(b)$. We set \mbox{\sigma}(a)=\mbox{\gamma}(b/ker(g))=\mbox{\gamma}(\pi(b)). In terms of the diagram above, we have: \mbox{\sigma}=\mbox{\gamma}\,\mbox{\circ}\,(\mbox{\overline{g}})^{-1}\,\mbox{\circ}\,pr. It follows that  $\sigma$  is a well-defined SG-morphism, i.e., \mbox{\sigma}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}, and (since pr\,\mbox{\circ}\,g=g and (\mbox{\overline{g}})^{-1}\,\mbox{\circ}\;g=\pi), g^{*}(\mbox{\sigma})=\mbox{\sigma}\,\mbox{\circ}\,g=\mbox{\gamma}\,\mbox{\circ}\,\pi. $\Box$

Lemma 2.3, together with 2.2 (iii) and 1.2 (2), gives:

Corollary 2.5

Let $F$ be a RS-fan, and let J\,\mbox{\subseteq}\,I be in Spec(\mbox{F}). The set S^{I}_{\raisebox{-3.0pt}{\scriptstyle!J}} is an AOS-fan. Indeed, it is a sub-fan of \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{F}), when the latter is endowed with its structure of AOS-fan, as indicated in 1.1. More generally, if  \mbox{\cal F}\,\mbox{\subseteq}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(\mbox{F}) is an AOS-fan, the set  \mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(\mbox{\cal F})= \{\,h\in\mbox{L{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,|\, \mbox{\exists}g\in\mbox{\cal F}\,(\,g\,\mbox{\rightsquigarrow\,}\,h)\} is an AOS-subfan of \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{F}).

Proof. The first assertion is a special case of the second (with $\cal F$  = L_{\raisebox{-3.0pt}{\scriptstyle!J}}($F$)). For the latter, observe that \mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(\mbox{\cal F})=\mbox{\mbox{$\kappa$}{\raisebox{-3.0pt}{$\scriptstyle JI$}}}[\mbox{\cal F}]={\mathrm{Im}}(\mbox{\mbox{$\kappa$}_{\raisebox{-3.0pt}{$\scriptstyle JI$}}}\lceil\mbox{\cal F}) and use Remark 2.4 (b). $\Box$

The following definition will have a crucial role in the sequel:

Definition 2.6

Let $F$ be a RS-fan, let \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}}, and fix $I\in{\mathrm{Spec}}(F)$ so that Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}), Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})\,\mbox{\subseteq} $I$. We define a map  \mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}:\mbox{L{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F)\,\mbox{\longrightarrow}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F) as follows: for  h\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F),

\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}(h)=h\,\mbox{g{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}. $\Box$

Note. Since Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,\mbox{\subseteq}\;I=Z(h)\;(i=1,2), we have Z(h\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})=I, whence h\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}.

Fact 2.7

With notation as in Definition 2.6, let $J\in$ Spec$(F)$ be such that Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\cup Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})\,\mbox{\subseteq}\,J\,\mbox{\subseteq}\;I, and for  $i=1,2,$ let g^{\prime}_{\raisebox{-3.0pt}{\scriptstyle i}} be the unique  $\rightsquigarrow$ -successor of  g_{\raisebox{-3.0pt}{\scriptstyle i}} of level  $J$. Then, \mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}=\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g^{\prime}{1},g^{\prime}{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}. Thus, in 2.6 we may assume Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})=Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}).

Proof. Lemma 1.5 shows that h\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=h\,\mbox{g^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}, for h\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}. $\Box$

Theorem 2.8

With notation as in Definition 2.6, we have:

$(a)$* {\raisebox{1.72218pt}{\mbox{\varphi}}}^{\,g_{1},g_{2}}_{\raisebox{-3.0pt}{\scriptstyle I}} is an AOS-automorphism of  L_{\raisebox{-3.0pt}{\scriptstyle I}}.*

$(b)$* {\raisebox{1.72218pt}{\mbox{\varphi}}}^{\,g_{1},g_{2}}_{\raisebox{-3.0pt}{\scriptstyle I}} is an involution: for h\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}} ,  \mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g{1},g_{2}}_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(h))=h.*

$(c)$* For $i=1,2,$ let h_{\raisebox{-3.0pt}{\scriptstyle i}} be the unique  $\rightsquigarrow$ -successor of  g_{\raisebox{-3.0pt}{\scriptstyle i}} in L_{\raisebox{-3.0pt}{\scriptstyle I}}. Then,  \mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}})=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}.*

In particular,

$(d)$* If g_{\raisebox{-3.0pt}{\scriptstyle 1}}, g_{\raisebox{-3.0pt}{\scriptstyle 2}}, have a common specialization $h$ at some level I\supseteq Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}),Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), then  $h$  is a*

fixed point of  {\raisebox{1.72218pt}{\mbox{\varphi}}}^{\,g_{1},g_{2}}_{\raisebox{-3.0pt}{\scriptstyle I}}.

$(e)$* Let J\,\mbox{\subseteq}\,I be in $Spec(F)$. Assume Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}),Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})\,\mbox{\subseteq}\;J, and let \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}},\;\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}. Then,*

\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\;\;\;\mbox{\Rightarrow}\;\;\;\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\,\mbox{\rightsquigarrow\,}\,\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 2$}}}).

For the proof of this Theorem we will need an improvement on 1.1 (a), valid for fans but not for arbitrary RSs; namely :

Fact 2.9

Let $F$ be a RS-fan, and  $I$ be an ideal of $F$. Any  g\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}} such that Z(g)\,\mbox{\subseteq}\,I induces a SG-character   \mbox{\widehat{g}}:\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,\mbox{\longrightarrow}\,\mbox{\mbox{\mbox{\mathbb{Z}}_{\raisebox{-3.0pt}{\scriptstyle 2}}}}, by setting  \mbox{\widehat{g}}\;\mbox{\circ}\,\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle I$}}}=g.

Proof. The only delicate point is well-definedness: for  $a\in F\setminus I$,   a\;\mbox{\sim_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,1\;\,\mbox{\Rightarrow}\;\,g(a)=1. By [DP5a], Lemma 5.10,  a\;\mbox{\sim_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,1 means $az=z$ for some $z\not\in\,I$ (see proof of 1.2); then $g(z)\neq 0$, and taking images under $g$ in this equality yields $g(a)=1$. $\Box$

Proof of Theorem 2.8. We begin by proving:

(b) For  h\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F),\;\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}}(h))=h\,\mbox{g^{2}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{g^{2}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}. But  h\,\mbox{g^{2}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{g^{2}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=h; this is clear if $h(x)=0$ $(x\in F)$; if $h(x)\neq 0$, then \mbox{g{\raisebox{-3.0pt}{$\scriptstyle i$}}}(x)\neq 0 (since Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,\mbox{\subseteq}\,Z(h)), and hence \mbox{g^{2}_{\raisebox{-3.0pt}{$\scriptstyle i$}}}(x)=1\;(i=1,2), proving the stated identity, and item (b).

(a)  i) {\raisebox{1.72218pt}{\mbox{\varphi}}}^{\,g_{1},g_{2}}_{\raisebox{-3.0pt}{\scriptstyle I}} is an AOS-morphism.

Since F_{\raisebox{-3.0pt}{\scriptstyle I}} is the RSG-fan dual to L_{\raisebox{-3.0pt}{\scriptstyle I}}($F$), we must show (see 1.1 (c)):

(*)   For every \mbox{\alpha}\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}} there is \mbox{\beta}\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}} such that \mbox{\widehat{\mbox{$\alpha$}}}\;\mbox{\circ}\;\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}_{\raisebox{-3.0pt}{$\scriptstyle I$}}}=\,\mbox{\widehat{\mbox{$\beta$}}},

where \mbox{\widehat{\mbox{$\alpha$}}}:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F_{I}$}}}\,\mbox{\longrightarrow}\,\mbox{\mbox{\mbox{\mathbb{Z}}_{\raisebox{-3.0pt}{\scriptstyle 2}}}} denotes evaluation at $\alpha$. We claim that \mbox{\beta}=\mbox{\alpha}\>\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\alpha})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\alpha}) does the job. By Fact 2.9, \mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle i$}}}(\mbox{\alpha})\in\mbox{\mbox{\mbox{\mathbb{Z}}_{\raisebox{-3.0pt}{\scriptstyle 2}}}}\;(i=1,2), whence \mbox{\beta}\in\mbox{F{\raisebox{-3.0pt}{$\scriptstyle I$}}}. For h\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(F) we have:

(\mbox{\widehat{\mbox{$\alpha$}}}\;\mbox{\circ}\;\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}})(h)=\mbox{\widehat{\mbox{$\alpha$}}}\,(h\,\mbox{g{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})=h(\mbox{\alpha})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\alpha})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\alpha})=h(\mbox{\alpha}\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\alpha})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\alpha}))=h(\mbox{\beta})=\mbox{\widehat{\mbox{$\beta$}}}\,(h),

as required. Note that (b) implies

ii)  {\raisebox{1.72218pt}{\mbox{\varphi}}}^{\,g_{1},g_{2}}_{\raisebox{-3.0pt}{\scriptstyle I}} is bijective.

iii) The dual map (\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}})^{*}:\mbox{F{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,\mbox{\longrightarrow}\,\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}} is also bijective.

Item (i) proves that, for \mbox{\alpha}\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,,\;(\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}{\raisebox{-3.0pt}{$\scriptstyle I$}}})^{*}(\mbox{\alpha})=\mbox{\alpha}\>\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\alpha})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\alpha}). For injectivity, assume \mbox{\alpha}\>\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\alpha})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\alpha})=1; if \mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\alpha})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\alpha})=-1, then \mbox{\alpha}=-1, whence (as \mbox{\widehat{g}}_{\raisebox{-3.0pt}{\scriptstyle i}} is a SG-character), \mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle i$}}}(\mbox{\alpha})=-1\;(i=1,2), and \mbox{\alpha}=1, contradiction. Thus, \mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\alpha})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\alpha})=1, which entails  \mbox{\alpha}=1. For surjectivity, given \mbox{\beta}\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle I$}}}, set  \mbox{\alpha}=\mbox{\beta}\>\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\beta})\,\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\beta}). Then, \mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\alpha})=\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\beta}) and \mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(\mbox{\alpha})=\mbox{\mbox{$\widehat{g}$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\mbox{\beta}), whence (\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{,g_{1},g_{2}}_{\raisebox{-3.0pt}{$\scriptstyle I$}}})^{*}(\mbox{\alpha})=\mbox{\beta}.

(c) We must prove \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}. This clearly holds at any x\in Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})=Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}). If x\not\in Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}})  $(i=1,2)$, then x\not\in Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}); since g_{\raisebox{-3.0pt}{\scriptstyle i}} $\rightsquigarrow$  h_{\raisebox{-3.0pt}{\scriptstyle i}}, it follows \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}(x)=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}(x)\neq 0 (Lemma 0.1 (4)), and \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}(x)\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}(x)=1; hence, \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(x)=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(x)=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(x).

(e) Lemma 1.6 (a) immediately implies  \mbox{h^{2}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\;\,\mbox{\Rightarrow}\;\,(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})^{2}=(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}). $\Box$

By use of these involutions we obtain a number of regularity results concerning the order structure of ARS-fans.

Proposition 2.10

Let $F$ be a RS-fan. For J\,\mbox{\subseteq}\,\mbox{J_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\subseteq}\,\mbox{J_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,\mbox{\subseteq}\,I in  Spec$(F)$, and h\in\mbox{S^{I}_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}} set:

B^{J_{1},J_{2}}(h)=\{\,g\in\mbox{S^{J_{2}}{\raisebox{-3.0pt}{$\scriptstyle J_{1}$}}}\;|\;g\,\mbox{\rightsquigarrow\,}\,h\,\}*,   and   A^{J_{1},J_{2}}(h)=\{\,g\in\mbox{C^{J{2}}_{\raisebox{-3.0pt}{$\scriptstyle J_{1}$}}}\;|\;g\,\mbox{\rightsquigarrow\,}\,h\,\}.*

Then,

$(a)$* For  \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\,\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}} , we have   {\mathrm{card}}\,(B^{J_{1},J_{2}}(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}}))={\mathrm{card}}\,(B^{J_{1},J_{2}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})).*

$(b)$* For  \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\,\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}} , we have   {\mathrm{card}}\,(A^{J_{1},J_{2}}(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}}))={\mathrm{card}}\,(A^{J_{1},J_{2}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})).*

Remark. The assumptions of the Proposition guarantee that the sets $B^{J_{1},J_{2}}(h)$ are non-empty. In fact, given h\in\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}, there is u\,\mbox{\rightsquigarrow\,}\,h so that Z(u)\,\mbox{\subseteq}\,J; set $J^{\prime}=Z(u)$. Since J^{\prime}\,\mbox{\subseteq}\,J\,\mbox{\subseteq}\,\mbox{J{\raisebox{-3.0pt}{$\scriptstyle 2$}}}, $u$ has a unique  $\rightsquigarrow$  - successor $g$ in L_{\raisebox{-3.0pt}{\scriptstyle J_{2}}}. But u\,\mbox{\rightsquigarrow\,}\,g,\,h  and  J_{2}=Z(g)\,\mbox{\subseteq}\;I=Z(h) imply g\,\mbox{\rightsquigarrow\,}\,h (Lemma 0.2 (3)). Since J^{\prime}\,\mbox{\subseteq}\,J\,\mbox{\subseteq}\,\mbox{J_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}, we conclude that g\in\mbox{S^{J_{2}}_{\raisebox{-3.0pt}{$\scriptstyle J_{1}$}}}, i.e., $g\in B^{J_{1},J_{2}}(h)$.

The sets $A^{J_{1},J_{2}}(h)$ may be empty for some choices of  $h$  and the J_{\raisebox{-3.0pt}{\scriptstyle i}}’s. However, if h\in\,\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}} and \mbox{J{\raisebox{-3.0pt}{$\scriptstyle 1$}}}=J, we have A^{J_{1},J_{2}}(h)\neq\mbox{\emptyset}. Indeed, in this case the element g\in\mbox{S^{J_{2}}{\raisebox{-3.0pt}{$\scriptstyle J$}}} constructed above is in C^{J_{2}}_{\raisebox{-3.0pt}{\scriptstyle J}}, for if g\in\mbox{S^{J{2}}{\raisebox{-3.0pt}{$\scriptstyle J^{\prime}$}}} for some $J^{\prime}\subset J$, then g\,\mbox{\rightsquigarrow\,}\,h would imply h\in\,\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J^{\prime}$}}}, contrary to the assumption h\in\,\mbox{C^{I}_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}. $\Box$

Proof of Proposition 2.10. (a) With J_{\raisebox{-3.0pt}{\scriptstyle 1}}, J_{\raisebox{-3.0pt}{\scriptstyle 2}} as in the statement, write B_{\raisebox{-3.0pt}{\scriptstyle i}} for B^{J_{1},J_{2}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\;(i=1,2). The assumption \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\in\,\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}} implies the existence of elements \mbox{u{\raisebox{-3.0pt}{$\scriptstyle i$}}}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}} so that u_{\raisebox{-3.0pt}{\scriptstyle i}} $\rightsquigarrow$  h_{\raisebox{-3.0pt}{\scriptstyle i}}  and   Z(\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,\mbox{\subseteq}\>J. Replacing u_{\raisebox{-3.0pt}{\scriptstyle i}} by its unique successor of level $J$ we may assume Z(\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle i$}}})=J (see 2.7). We fix u_{\raisebox{-3.0pt}{\scriptstyle i}}’s with these properties throughout the proof, and for J\,\mbox{\subseteq}\>J^{\prime}\,\mbox{\subseteq}\>I we denote by \raisebox{1.72218pt}{\mbox{\varphi}}_{\raisebox{-3.0pt}{\scriptstyle!!J^{\prime}}} the involution {\raisebox{1.72218pt}{\mbox{\varphi}}}^{u_{1},u_{2}}_{\raisebox{-3.0pt}{\scriptstyle!J^{\prime}}} of L_{\raisebox{-3.0pt}{\scriptstyle!J^{\prime}}} defined in 2.6.

Since the maps \raisebox{1.72218pt}{\mbox{\varphi}}_{\raisebox{-3.0pt}{\scriptstyle!J^{\prime}}} are bijective, it is enough to prove \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}[\mbox{B{\raisebox{-3.0pt}{$\scriptstyle 1$}}}]=\mbox{B_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}. Further, since \raisebox{1.72218pt}{\mbox{\varphi}}_{\raisebox{-3.0pt}{\scriptstyle!!J_{2}}} is an involution it suffices just to prove the inclusion $\subseteq$, i.e.,

(*)       g\in\mbox{S^{J_{2}}{\raisebox{-3.0pt}{$\scriptstyle\!J_{1}$}}}  and  g\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\;\,\mbox{\Rightarrow}\;\,\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g)\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 2$}}}  and  \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g)\in\mbox{S^{J{2}}_{\raisebox{-3.0pt}{$\scriptstyle\!J_{1}$}}}.

(i)  \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g)=g\,\mbox{u{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}.

Immediate consequence of Lemma 1.6 (b), since g,\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}} and u_{\raisebox{-3.0pt}{\scriptstyle 2}} $\rightsquigarrow$  h_{\raisebox{-3.0pt}{\scriptstyle 2}}.

(ii) \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g)\in\,\mbox{S^{J{2}}_{\raisebox{-3.0pt}{$\scriptstyle\!J_{1}$}}}.

Since g\in\,\mbox{S^{J_{2}}{\raisebox{-3.0pt}{$\scriptstyle J_{1}$}}}, there is v\,\mbox{\rightsquigarrow\,}\,g so that Z(v)=\mbox{J{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\supseteq J=Z(\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\;(i=1,2); thus, $v$ is in the domain of \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!Z(v)$}}}=\,\mbox{{\raisebox{1.72218pt}{\mbox{$\varphi$}}}^{u{1},u_{2}}{\raisebox{-3.0pt}{$\scriptstyle Z(v)$}}}, and Theorem 2.8 (e) gives \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!Z(v)$}}}(v)\,\mbox{\rightsquigarrow\,}\,\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}_{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g), proving (ii) and item (a).

(b) Write A_{\raisebox{-3.0pt}{\scriptstyle i}} for A^{J_{1},J_{2}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\;(i=1,2). As above, it suffices to prove the analogue of (*):

(**)       g\in\mbox{C^{J_{2}}{\raisebox{-3.0pt}{$\scriptstyle J_{1}$}}}  and  g\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\;\,\mbox{\Rightarrow}\;\,\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g)\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 2$}}}  and  \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g)\in\mbox{C^{J{2}}_{\raisebox{-3.0pt}{$\scriptstyle\!J_{1}$}}} ,

where h_{\raisebox{-3.0pt}{\scriptstyle 1}}, h_{\raisebox{-3.0pt}{\scriptstyle 2}} are now assumed to be in  C^{I}_{\raisebox{-3.0pt}{\scriptstyle!J}}. In fact, by (*) it only remains to show:

(iii) There is no  w\in\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}} such that  Z(w)\subset\mbox{J_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}  and  w\,\mbox{\rightsquigarrow\,}\,\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}_{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g).

Otherwise, we would have w\,\mbox{\rightsquigarrow\,}\,\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g)\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 2$}}} (the last relation holding by (*)). Since \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\,\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}, we get J\,\mbox{\subseteq}\,Z(w), and since Z(\mbox{u{\raisebox{-3.0pt}{$\scriptstyle i$}}})=J, \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!Z(w)$}}}(w) is defined. Theorem 2.8 (e) applied to the first of the preceding inequalities yields: \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!Z(w)$}}}(w)\,\mbox{\rightsquigarrow\,}\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J_{2}$}}}(g))=g. This, together with \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!Z(w)$}}}(w)\in\mbox{L{\raisebox{-3.0pt}{$\scriptstyle Z(w)$}}}  and Z(w)\subset\mbox{J_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}, contradicts the assumption g\in\mbox{C^{J_{2}}_{\raisebox{-3.0pt}{$\scriptstyle\!J_{1}$}}}, proving (iii), and item (b). $\Box$

A slight variant of the argument proving Proposition 2.10 yields:

Proposition 2.11

Let $F$ be a RS-fan and let J\,\mbox{\subseteq}\,I be in \mbox{\mathrm{Spec,}}(F). For  \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}} such that  Z(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,\mbox{\subseteq}\,J   $(i=1,2)$, the map  {\raisebox{1.72218pt}{\mbox{\varphi}}}^{\,g_{1},g_{2}}_{\raisebox{-3.0pt}{\scriptstyle I}} is a permutation of  S^{I}_{\raisebox{-3.0pt}{\scriptstyle!J}}  and of  C^{I}_{\raisebox{-3.0pt}{\scriptstyle!J}}. $\Box$

For a RS-fan, $F$, and h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}}, we denote by P_{\raisebox{-3.0pt}{\scriptstyle h}} = \{\,g\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle\!F$}}}\,|\;g\,\mbox{\rightsquigarrow\,}\,h\,\} the root-system of predecessors of $h$ under specialization. We begin by proving:

Proposition 2.12

$(1)$* P_{\raisebox{-3.0pt}{\scriptstyle h}} is an ARS-fan. In particular,*

$(2)$* Any connected component of an ARS-fan is an ARS-fan.*

Proof. (1) Lemma 0.1 (2) shows that  g\,\mbox{\rightsquigarrow\,}\,h  iff  T=h^{-1}[1]\,\mbox{\subseteq}\;g^{-1}[1]. With notation as in [M], § 6.3, p. 110, and § 6.6, p. 126, the latter condition just means $g\in U(T)$, i.e., P_{\raisebox{-3.0pt}{\scriptstyle h}} is the saturated set $U(T)(=W(T)\,\cap\,U(T^{2}))$. [M], Cor. 6.6.8, p. 126, proves that sets of this form are ARSs. Lemma 1.6 (a) shows that it is closed under products of three elements, hence a fan by the results of [DP5a], § 3.

(2) Follows from (1) by taking $h$ to be the (unique)  $\rightsquigarrow$ -top element of the given connected component. $\Box$

Continuing the analysis of (ARS-)fans of the form P_{\raisebox{-3.0pt}{\scriptstyle h}}, we show:

Theorem 2.13

Let $F$ be a RS-fan and let J\,\mbox{\subseteq}\,I be in  Spec$(F)$. Let \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\in\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}},\;\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}. For $i=1,2,$ we write P_{\raisebox{-3.0pt}{\scriptstyle i}} for P_{\raisebox{-3.0pt}{\scriptstyle h{i}}}. Then,

$(1)$* There is an ARS-embedding  $\varphi$  of  P_{\raisebox{-3.0pt}{\scriptstyle 1}} into P_{\raisebox{-3.0pt}{\scriptstyle 2}}. Further, \raisebox{1.72218pt}{\mbox{\varphi}}[\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}]=\{\,u\in\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,|\;J\,\mbox{\subseteq}\;Z(u)\>\}. In particular, $\varphi$ is an order-embedding of  (\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\,\mbox{\rightsquigarrow\,}) into (\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\,\mbox{\rightsquigarrow\,}).*

$(2)$* If, in addition, \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\mbox{C^{I}_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}, then  $\varphi$ is an isomorphism of ARSs.*

Proof. Our assumption on the h_{\raisebox{-3.0pt}{\scriptstyle i}}’s guarantees the existence of \mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}} so that \mbox{u_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\;(i=1,2). For J\,\mbox{\subseteq}\,J^{\prime}\,\mbox{\subseteq}\,I in $\textrm{Spec}(F)$ let \raisebox{1.72218pt}{\mbox{\varphi}}_{\raisebox{-3.0pt}{\scriptstyle!!J^{\prime}}} denote the involution \raisebox{1.72218pt}{\mbox{\varphi}}^{u_{1},u_{2}}_{\raisebox{-3.0pt}{\scriptstyle!!J^{\prime}}} of L_{\raisebox{-3.0pt}{\scriptstyle J^{\prime}}} (Definition 2.6).

(1) We construct \raisebox{1.72218pt}{\mbox{\varphi}}:\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\longrightarrow}\,\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 2$}}} by “collecting together” all the relevant maps \raisebox{1.72218pt}{\mbox{\varphi}}_{\raisebox{-3.0pt}{\scriptstyle!!J^{\prime}}}  (J\,\mbox{\subseteq}\,J^{\prime}\,\mbox{\subseteq}\,I): given g\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle J^{\prime}$}}}\,,\,g\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}, we set

\raisebox{1.72218pt}{\mbox{\varphi}}(g)=\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}_{\raisebox{-3.0pt}{$\scriptstyle\!\!J^{\prime}$}}}(g).

Since the levels L_{\raisebox{-3.0pt}{\scriptstyle!J^{\prime}}} are pairwise disjoint,  $\varphi$ is well-defined.

i)   \raisebox{1.72218pt}{\mbox{\varphi}}[\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}]\,\mbox{\subseteq}\,\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}.

By Theorem 2.8 (e),  g\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}} implies \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J^{\prime}$}}}(g)\,\mbox{\rightsquigarrow\,}\,\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!I$}}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}). Since h_{\raisebox{-3.0pt}{\scriptstyle i}} is the unique successor of u_{\raisebox{-3.0pt}{\scriptstyle i}} at level $I$, 2.8 (c) yields \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!I$}}}(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}})=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}, whence \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J^{\prime}$}}}(g)\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 2$}}}, as required. Note this also gives J\,\mbox{\subseteq}\;J^{\prime}=Z(\raisebox{1.72218pt}{\mbox{\varphi}}(g)).

ii)  \{\,u\in\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,|\;J\,\mbox{\subseteq}\;Z(u)\>\}\,\mbox{\subseteq}\;\raisebox{1.72218pt}{\mbox{\varphi}}[\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}].

Let $u$ be in the left-hand side, with $J^{\prime}=Z(u)$, say. Set v=\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J^{\prime}$}}}(u); then, \mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J^{\prime}$}}}(v)=u (2.8 (b)). By 1.6 (b), u_{\raisebox{-3.0pt}{\scriptstyle 1}} $\rightsquigarrow$  h_{\raisebox{-3.0pt}{\scriptstyle 1}} and u,\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}} imply u\,\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=\mbox{\raisebox{1.72218pt}{\mbox{$\varphi$}}{\raisebox{-3.0pt}{$\scriptstyle\!\!J^{\prime}$}}}(u)=v\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}}, i.e., v\in\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}. Hence \raisebox{1.72218pt}{\mbox{\varphi}}(v)=u\in\,\raisebox{1.72218pt}{\mbox{\varphi}}[\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}].

iii) $\varphi$  is injective.

This is clear using 2.8 (b), since Z(\raisebox{1.72218pt}{\mbox{\varphi}}(g))=Z(g) for g\in\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}.

iv)  $\varphi$  is an ARS-morphism.

The proof is similar to that of item (a) in Theorem 2.8. The statement to be proved is:

(†) For every $a\in F$ there is $b\in F$ such that (\mbox{\widehat{a/\mbox{T_{\raisebox{-3.0pt}{\scriptstyle 2}}}}})\,\mbox{\circ}\,\raisebox{1.72218pt}{\mbox{\varphi}}=\mbox{\widehat{b/\mbox{T_{\raisebox{-3.0pt}{\scriptstyle 1}}}}},

where, for $i=1,2$, \mbox{T_{\raisebox{-3.0pt}{$\scriptstyle i$}}}=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}^{-1}[1],\>\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle i$}}}=U(\mbox{T_{\raisebox{-3.0pt}{$\scriptstyle i$}}}),\>\mbox{\widehat{a/\mbox{T_{\raisebox{-3.0pt}{\scriptstyle 2}}}}}:\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,\mbox{\longrightarrow}\;{\bf 3}  is the evaluation map: \mbox{\widehat{a/\mbox{T_{\raisebox{-3.0pt}{\scriptstyle 2}}}}}\,(g)=\mbox{\widehat{g}}\,(a/\mbox{T_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})=g(a), for g\in\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}, and similarly for \mbox{\widehat{b/\mbox{T_{\raisebox{-3.0pt}{\scriptstyle 1}}}}}:\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\longrightarrow}\;{\bf 3}. (Note that g\in\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=U(\mbox{T_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}) ensures that \widehat{a/\mbox{T_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}} depends only on the congruence class of $a$ modulo T_{\raisebox{-3.0pt}{\scriptstyle 2}}.)

The conclusion of (†) can equivalently be written as  \mbox{\widehat{\raisebox{1.72218pt}{\mbox{$\varphi$}}(g)}}(a/\mbox{T_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})=\mbox{\widehat{g}}\,(b/\mbox{T_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}), i.e.,  (\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,g)(a) $=g(b)$. Since \mbox{u_{\raisebox{-3.0pt}{$\scriptstyle i$}}}(a)\in\{0,1,-1\}\;(i=1,2), it is clear that the element b=a\,\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(a)\mbox{u_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(a)\in F verifies (†); see 2.8 (a).

(2) Since \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in\mbox{C^{I}_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}, the preceding construction can be performed with the roles of h_{\raisebox{-3.0pt}{\scriptstyle 1}} and h_{\raisebox{-3.0pt}{\scriptstyle 2}} reversed. Routine verification using 2.8(b) shows that the map obtained is  \raisebox{1.72218pt}{\mbox{\varphi}}^{-1}, which then is an ARS-morphism, proving that  $\varphi$ is an ARS-isomorphism. $\Box$

Proposition 2.10 and Theorem 2.13 provide significant information on the structure of the connected components of ARS-fans.

Definition and Remarks 2.14

(a) Let $(X,\preceq\,)$ be a root-system and let \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in X. Define:

g_{\raisebox{-3.0pt}{\scriptstyle 1}} \equiv_{\raisebox{-3.0pt}{\scriptstyle C}} g_{\raisebox{-3.0pt}{\scriptstyle 2}}   iff   g_{\raisebox{-3.0pt}{\scriptstyle 1}}, g_{\raisebox{-3.0pt}{\scriptstyle 2}} have a common $\preceq$ - upper bound.

\equiv_{\raisebox{-3.0pt}{\scriptstyle C}}  is an equivalence relation; its classes are called connected components of  $(X,\preceq\,)$.

(b) The  $\rightsquigarrow$  - top elements of the connected components of an ARS-fan $(X,F)$ have all the same level, namely the level determined by the maximal ideal $M$ of $F$; cf. Proposition 0.3 (3).

(c) Since every connected component of an ARS-fan is itself an ARS-fan, 2.12 (2), the zero-sets of its elements attain a lowest level, which can be explicitly determined, cf. Proposition 2.15 below. However, different components may have different lowest levels, see Corollary 2.17. * $\Box$*

Notation. The sets L_{\raisebox{-3.0pt}{\scriptstyle I}}, S^{I}_{\raisebox{-3.0pt}{\scriptstyle!J}} and C^{I}_{\raisebox{-3.0pt}{\scriptstyle!J}} defined in 1.1 and 2.1 relativize in an obvious way to the connected components of a fan $(X,F)$; if $K$ is such a component and J\,\mbox{\subseteq}\,I are in Spec($F$) we set:

\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(K)=\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,\cap\,K,     \mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(K)=\mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}\,\cap\,K,    and    \mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(K)=\mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}\,\cap\,K.

Note that some (or all) of these sets may be empty, depending on $I$, $J$ and the component $K$. Clearly, if h_{\raisebox{-3.0pt}{\scriptstyle 0}} is the  $\rightsquigarrow$  - top element of $K$, we have \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(K)=\{\,g\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}\,|\,g\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\,\}, and similarly for \mbox{S^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(K)  and  \mbox{C^{I}{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(K). \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(K)\neq\mbox{\emptyset} just means that $K$ “reaches at least” the $I$-th level of $X$ (possibly lower). $\Box$

Proposition 2.15

Let $K$ be a connected component of an ARS-fan $(X,F)$. Let h_{\raisebox{-3.0pt}{\scriptstyle 0}} be the  $\rightsquigarrow$  - top element of $K$, and let T=\mbox{h^{-1}{\raisebox{-3.0pt}{$\scriptstyle 0$}}}[1]. Then, the lowest level of $K$ $($i.e., the smallest ideal $I$ of $F$ such that \mbox{L{\raisebox{-3.0pt}{$\scriptstyle I$}}}(K)\neq\mbox{\emptyset}) is  I=\mbox{\Gamma}\,\cap\,-\mbox{\Gamma}, where $\Gamma$ is the saturated subsemigroup of  $F$ generated by  ${\mathrm{Id}}(F)\cdot T$.

Note. The subsemigroup Id$(F)\cdot T$ may not be saturated, since ${\mathrm{Id}}(F)\cdot T\,\cap\,-({\mathrm{Id}}(F)\cdot T)$ is not, in general, an ideal; see [DP5a], Cor. 3.10 (2).

Proof. With notation as in 2.12, we have K=\mbox{P_{\raisebox{-3.0pt}{$\scriptstyle h_{0}$}}}=U(T)=\{g\in X\,|\,g\,\lceil\,T=1\}= the ARS X_{\raisebox{-3.0pt}{\scriptstyle F/T}} (where F/T=F/{\mbox{\sim_{\raisebox{-3.0pt}{$\scriptstyle K$}}}}, with \sim_{\raisebox{-3.0pt}{\scriptstyle K}} denoting the congruence on $F$ induced by $K$). Let \mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle T$}}}:F\,\mbox{\longrightarrow}\ F/T be the quotient map. The lowest level of X_{\raisebox{-3.0pt}{\scriptstyle F/T}} is $\{0\}$; with $K$ identified to a subset of $X$ via the map g\,\mapsto\mbox{\widehat{g}}\;\;(\mbox{\widehat{g}}\;\mbox{\circ}\,\mbox{\pi_{\raisebox{-3.0pt}{$\scriptstyle T$}}}=g), the corresponding ideal of $F$ is \mbox{\pi^{-1}{\raisebox{-3.0pt}{$\scriptstyle T$}}}[0]=\{a\in F\,|\,a\,\mbox{\sim{\raisebox{-3.0pt}{$\scriptstyle K$}}}\,0\}. Then, with the ideal $I$ defined in the statement, we must prove, for $a\in F$ :

a\;\mbox{\sim_{\raisebox{-3.0pt}{$\scriptstyle K$}}}\,0\;\;\Longleftrightarrow\;\;a\in I.

($\Leftarrow$) This follows from I\,\mbox{\subseteq}\,Z(g) for all $g\in K$. Since $g\,\lceil\,T=1$, we get {\mathrm{Id}}\cdot T\;\mbox{\subseteq}\;P(g)=g^{-1}[0,1]; since $P(g)$ is a saturated subsemigroup, it comes \mbox{\Gamma}\,\mbox{\subseteq}\,P(g). Hence, x\in I=\mbox{\Gamma}\,\cap\,-\mbox{\Gamma} implies $g(x)=0$.

($\Rightarrow$) Assume $a\not\in I$. In order to prove a\;\mbox{\not\sim_{\raisebox{-3.0pt}{$\scriptstyle K$}}}\,0 we construct a character $g\in X$ such that $g\,\lceil\,T=1$ and $g(a)\neq 0$ (i.e., $g(a^{2})=1$). The ideal $I$ is prime and saturated ([DP5a], 3.10 (1)). Since I=\mbox{\Gamma}\,\cap\,-\mbox{\Gamma}, there is a saturated subsemigroup $S$ of $F$ such that \mbox{\Gamma}\,\mbox{\subseteq}\,S and $S$ maximal with $S\,\cap\,-S=I$. By [DP1], Lemma 3.5, p. 114, $S\,\cup\,-S=F$, and $S$ defines a character $g\in X$ with $Z(g)=I$, by setting $g\,\lceil\,(S\setminus-S)=1,\;g\,\lceil\,(-S\setminus S)=-1$ and $g\,\lceil\,I=0$. Note that we have,

(${\dagger}$)   I\,\cap\,a^{2}T=\mbox{\emptyset}.

Otherwise, there is $t\in T$ such that $a^{2}t\in I$; since $I$ is prime and $a\not\in I$, we get $t\in I$, contradicting T\,\cap\,I=\mbox{h^{-1}{\raisebox{-3.0pt}{$\scriptstyle 0$}}}[1]\,\cap\,Z(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}})=\mbox{\emptyset}.

Since a^{2}T\,\mbox{\subseteq}\,S, (†) implies -S\,\cap\,a^{2}T=\mbox{\emptyset}, whence $g\,\lceil\,a^{2}T=1$ by the definition of $g$. $\Box$

Proposition 2.10 implies:

Corollary 2.16

Let  $(X,F)$ be an ARS-fan and let K_{\raisebox{-3.0pt}{\scriptstyle 1}}, K_{\raisebox{-3.0pt}{\scriptstyle 2}} be connected components of  $(X,F)$. Then,

$(1)$* Let $I\in$ Spec$(F)$; if  \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\neq\mbox{\emptyset} for  $i=1,2$, then  {\mathrm{card}}\,(\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}))={\mathrm{card}}\,(\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})).*

$(2)$* Let J\,\mbox{\subseteq}\;J^{\prime} be in  Spec$(F)$, and assume \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\neq\mbox{\emptyset}\;\;(i=1,2). Then,  {\mathrm{card}}\,(\mbox{S^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle 1$}}}))={\mathrm{card}}\,(\mbox{S^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle 2$}}})).*

Proof. (1) follows from (2), as \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}=\mbox{S^{I}_{\raisebox{-3.0pt}{$\scriptstyle I$}}}.

(2) Fix $i\in\{1,2\}$. Let h_{\raisebox{-3.0pt}{\scriptstyle i}} be the  $\rightsquigarrow$  - top element of K_{\raisebox{-3.0pt}{\scriptstyle i}}. The assumption \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle\!J$}}}(\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\neq\mbox{\emptyset} implies that the sets \mbox{S^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle i$}}})=\{\,g\in\mbox{S^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}\,|\,g\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle i$}}}\,\} are non-empty. Now, applying Proposition 2.10(a) with $I$ = $M$ (= the maximal ideal of $F$), \mbox{J_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}=J, \mbox{J_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=J^{\prime} we have B^{J,J^{\prime}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}})=\{\,g\in\mbox{S^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}\;|\;g\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle i$}}}\,\}=\mbox{S^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle i$}}}), and the result follows. $\Box$

Remark. Assertion (2) of Corollary 2.16 fails, in general, if the sets \mbox{S^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle i$}}}) are replaced by \mbox{C^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle i$}}}), even if both sets \mbox{C^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle i$}}}),\;i=1,2,  are assumed non-empty. The snag is that \mbox{C^{J^{\prime}}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle i$}}})\neq\mbox{\emptyset}  does not imply that the  $\rightsquigarrow$  - top element h_{\raisebox{-3.0pt}{\scriptstyle i}} of K_{\raisebox{-3.0pt}{\scriptstyle i}} belongs to \mbox{C^{M}{\raisebox{-3.0pt}{$\scriptstyle J$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle i$}}}), a condition required for Proposition 2.10(b) to apply. It is easy to construct counterexamples. $\Box$

Theorem 2.13 gives:

Corollary 2.17

Let K_{\raisebox{-3.0pt}{\scriptstyle 1}}, K_{\raisebox{-3.0pt}{\scriptstyle 2}} be connected components of the ARS-fan  $(X,F)$. Let \mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\in Spec$(F)$ be the lowest levels of K_{\raisebox{-3.0pt}{\scriptstyle 1}}, K_{\raisebox{-3.0pt}{\scriptstyle 2}}, resp. $($cf. $\ref{componlowestlevel}\,)$. Then,

$(1)$* If  I_{\raisebox{-3.0pt}{\scriptstyle 2}} $\subseteq$ I_{\raisebox{-3.0pt}{\scriptstyle 1}}, then  K_{\raisebox{-3.0pt}{\scriptstyle 1}} endowed with the specialization order is $($order-$)$isomorphic to the root-system obtained by deleting all levels I\subset\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 1$}}} in K_{\raisebox{-3.0pt}{\scriptstyle 2}}.*

$(2)$* If  \mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}=\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}, then  K_{\raisebox{-3.0pt}{\scriptstyle 1}}, K_{\raisebox{-3.0pt}{\scriptstyle 2}} are order-isomorphic.*

Proof. (1) Use Theorem 2.13 (1) with $I$ = $M$ = the maximal ideal of $F$,  $J$ = I_{\raisebox{-3.0pt}{\scriptstyle 1}},  and h_{\raisebox{-3.0pt}{\scriptstyle 1}}, h_{\raisebox{-3.0pt}{\scriptstyle 2}} the  $\rightsquigarrow$  - top elements of K_{\raisebox{-3.0pt}{\scriptstyle 1}}, K_{\raisebox{-3.0pt}{\scriptstyle 2}}, resp. The ARS-embedding \raisebox{1.72218pt}{\mbox{\varphi}}:\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\longrightarrow}\,\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle 2$}}} constructed therein verifies \raisebox{1.72218pt}{\mbox{\varphi}}[\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}]=\{\,u\in\mbox{K_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,|\;\mbox{I_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\subseteq}\;Z(u)\>\}, which is exactly statement (1).

(2) follows from Theorem 2.13 (2). $\Box$

** 2.18**

Some impossible configurations.

The preceding results show that there are strong constraints on the order structure of ARS-fans, especially when there is more than one connected component. We include a few examples to help the reader visualize the extent of those restrictions.

(1) A configuration like

[TABLE]

contradicts Corollary 2.16 (1).

(2) The four-component configuration

[TABLE]

(where the components pairwise verify the conclusion of 2.16 (2)) is also impossible:  card (\mbox{S^{,3}_{\raisebox{-3.0pt}{$\scriptstyle 4$}}}) $=3$ is not a power of 2, and hence S^{\,3}_{\raisebox{-3.0pt}{\scriptstyle 4}} (shown with arrows) cannot be an AOS-fan (see Corollary 2.5). However, the same configuration with K_{\raisebox{-3.0pt}{\scriptstyle 3}} replaced by another copy of K_{\raisebox{-3.0pt}{\scriptstyle 4}} does not clash with either 2.16 or 2.17.

Note. Our notation here (and below) follows the convention introduced in 1.8 for finite fans. Thus, S^{\,3}_{\raisebox{-3.0pt}{\scriptstyle 4}} stands for the set S^{\,I_{3}}_{\raisebox{-3.0pt}{\scriptstyle,I_{4}}}, see 2.1 and 3.1.

(3) The two-component root-system

[TABLE]

1

2

3

4

5

[TABLE]

[TABLE]

contradicts both Corollary 2.16 (2) (card (\mbox{S^{,3}{\raisebox{-3.0pt}{$\scriptstyle\,4$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle 1$}}}))=4, but  card (\mbox{S^{,3}{\raisebox{-3.0pt}{$\scriptstyle\,4$}}}(\mbox{K{\raisebox{-3.0pt}{$\scriptstyle 2$}}}))=2) and Corollary 2.17 (K_{\raisebox{-3.0pt}{\scriptstyle 1}} and K_{\raisebox{-3.0pt}{\scriptstyle 2}} have the same “length” but are not order-isomorphic). $\Box$

3 The specialization root-system of finite ARS-fans

In this section we shall mostly deal with finite fans in the categories ARS and RS. Our main result is Theorem 3.11 —the isomorphism theorem for finite ARS-fans— which proves that, in this case, the order of specialization alone determines the isomorphism type. The proof depends on the notion of a “standard generating system” which we introduce in 3.4. $\Box$

** 3.1**

Notation and Reminder (a) Notation 1.8 for finite (ARS- and RS-)fans is used systematically in this section, adapted in a self-explanatory way; e.g., for 1\leq k\leq j\leq n=\ell(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle F$}}}), L_{\raisebox{-3.0pt}{\scriptstyle k}} (or L_{\raisebox{-3.0pt}{\scriptstyle k}}(X_{\raisebox{-3.0pt}{\scriptstyle F}}), if necessary), will stand for the level \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I_{k}$}}},\;\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}} for the set S^{I_{k}}_{\raisebox{-3.0pt}{\scriptstyle I{j}}}, etc.

(b) Recall that the AOSs have a combinatorial geometric (matroid) structure; it was introduced in [D1] and [D2] for spaces of orders of fields, and later generalized to abstract order spaces in [Li]. In general, ARSs do not possess such a structure. Thus, combinatorial geometric notions such as dependent set, independent set, basis, closed set, closure, dimension, etc., will always refer to the above-mentioned combinatorial geometric structure, and apply only to AOSs. For the definition and the mutual relationships, in the general context of matroid theory, of combinatorial notions such as those just mentioned, the reader is referred to [Wh].

Since the combinatorial geometric structure of any AOS is isomorphic to that of a set of vectors in a (possibly infinite-dimensional) vector space over the two-element field  \mathbb{F}_{\raisebox{-3.0pt}{\scriptstyle 2}}  with the structure induced by linear dependence (cf. [D1], Thm. 3.1, p. 618), the notions above coincide with the corresponding notions over vector spaces. For example, a subset A\,\mbox{\subseteq}\,X of an AOS $(X,G,-1)$ ($G$ a group of exponent 2) is dependent iff there are pairwise distinct elements g,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\in A\,(r\geq 2), such that  g=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}} (as characters of $G$). Since functions in $X$ send $-1$ to $-1$, this functional identity can only hold if $r$ is odd. Likewise, $A$ is closed iff the product of any odd number of members of $A$ belongs to $A$. $\Box$

Warning. In this section the words closed set and closure are used only in the combinatorial geometric sense just defined. $\Box$

Lemma 3.2

Let $(X,F)$ be an ARS-fan $($not necessarily finite$)$. Let J\,\mbox{\subseteq}\,I be in  Spec($F$$)$, and let A\,\mbox{\subseteq}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle J$}}}, B\,\mbox{\subseteq}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}, be sets such that:

$($i$)$  The unique  $\rightsquigarrow$ - successor in L_{\raisebox{-3.0pt}{\scriptstyle I}} of each $g\in A$ belongs to $B$.

$($ii$)$ Every  $h\in B$  has a unique  $\rightsquigarrow$ - predecessor in $A$.

Then, $A$ dependent  $\Rightarrow$  $B$ dependent.

Proof. By assumption there are pairwise distinct elements g,\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\in A such that  g=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}; as observed above, $r$ is odd $\geq 3$. Let h,\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}} be the unique successors of g,\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}, resp., in $B$ coming from (i); thus,  g\,\mbox{\rightsquigarrow\,}\,h and g_{\raisebox{-3.0pt}{\scriptstyle i}} $\rightsquigarrow$  h_{\raisebox{-3.0pt}{\scriptstyle i}}, for $i=1,\dots,r$. By 1.6(a) we have g=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}. Since \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}} ($r$ is odd) and $g$ has a unique  $\rightsquigarrow$  - successor in L_{\raisebox{-3.0pt}{\scriptstyle I}}, we get h=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}.

By assumption (ii), every element in $A$ is the unique predecessor of an element in $B$. Since \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\neq\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}}, we get \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\neq\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle j$}}} for $1\leq i\neq j\leq r$; likewise, h\neq\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}} for $i=1,\dots,r$. This proves that $h$ is the product of $r$ distinct elements in $B$, and hence that $B$ is dependent. $\Box$

Proposition 3.3

(Choice of basis).* Let $(X,F)$ be a finite ARS-fan; let $1\leq k<n=\ell(X)$. Let  $\cal G$  be an arbitrary AOS-subfan of \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}=\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(X). Let \mbox{\cal F}=\{\,h\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,|\, There is g\in\mbox{\cal G} such that g\,\mbox{\rightsquigarrow\,}\,h\,\} be the AOS-fan consisting of the depth-$k$ successors of elements of  $\cal G$  $(cf.\,\ref{fansS})$. Assume:*

(*)\;\;\;\;\mbox{\forall}\,h,h^{\prime}\in\mbox{\cal F},\;\;{\mathrm{card}}\,(\{g\in\,\mbox{\cal G}\,|\,g\,\mbox{\rightsquigarrow\,}\,h\})={\mathrm{card}}\,(\{g\in\,\mbox{\cal G}\,|\,g\,\mbox{\rightsquigarrow\,}\,h^{\prime}\}).

Let \mbox{\cal B}=\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\} be a basis of $\cal F$  $($as an AOS $)$, and let  $\cal C$ be a basis of the AOS-fan \mbox{P_{\raisebox{-3.0pt}{$\scriptstyle h_{1}$}}}=\{g\in\,\mbox{\cal G}\,|\,g\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\} $($see 2.11 (1). For  $i=2,\dots,r$, let \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\in\,\mbox{\cal G} be such that g_{\raisebox{-3.0pt}{\scriptstyle i}} $\rightsquigarrow$  h_{\raisebox{-3.0pt}{\scriptstyle i}}.

Then*,  \mbox{\cal C}\,\cup\{\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\dots,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\} is a basis of  $\cal G$.*

Proof. If $r$ = 1, then \mbox{\cal F}=\mbox{\cal B}=\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\}, whence \mbox{\cal G}=\{g\in\mbox{\cal G}\,|\,g\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\}, and the result holds by the choice of $\cal C$. Henceforth we assume $r\geq 2$. We observe:

—  r={\mathrm{card}}(\mbox{\cal B})={\mathrm{dim}}(\mbox{\cal F}). Since $\cal F$ is an AOS-fan,  {\mathrm{card}}(\mbox{\cal F})=\mbox{2^{r-1}}.

—  For every h\in\mbox{\cal F}, \mbox{A_{\raisebox{-3.0pt}{$\scriptstyle h$}}}=\{g\in\mbox{\cal G}\,|\,g\,\mbox{\rightsquigarrow\,}\,h\} is a AOS-fan; this follows from the assumption that $\cal G$  is an AOS-fan, since A_{\raisebox{-3.0pt}{\scriptstyle h}} is closed under the product of any three of its elements, cf. Lemma 1.6 (b).

—  \mbox{A_{\raisebox{-3.0pt}{$\scriptstyle h$}}}\cap\,\mbox{A_{\raisebox{-3.0pt}{$\scriptstyle h^{\prime}$}}}=\mbox{\emptyset}  for $h\neq h^{\prime}$  in  $\cal F$.

By assumption $(*)$, card (A_{\raisebox{-3.0pt}{\scriptstyle h}}) = card (A_{\raisebox{-3.0pt}{\scriptstyle h^{\prime}}}) (= $2^{p-1}$, say), for h,h^{\prime}\in\mbox{\cal F}. Since \mbox{\cal G}=\bigcup_{h\in\cal F}\mbox{A_{\raisebox{-3.0pt}{$\scriptstyle h$}}}, we get card ($\cal G$) = card ($\cal F$)$\cdot$ card (A_{\raisebox{-3.0pt}{\scriptstyle h}}) (any h\in\mbox{\cal F}), and then card ($\cal G$) = \mbox{2^{r-1}}\cdot\,\mbox{2^{p-1}}=\mbox{2^{p+r-2}}; hence dim ($\cal G$) = $p+r-1$. Since card (\,\mbox{\cal C}\,\cup\{\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\dots,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\}) = $p+r-1$, it suffices to prove:

(**)\;\;\;\mbox{\cal C}\,\cup\{\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\dots,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\}  is an independent set.

Proof of $(**)$. Assume false.

Case 1.  Some g_{\raisebox{-3.0pt}{\scriptstyle i_{0}}}, with 2\leq\mbox{i_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\leq r, is dependent on the rest, i.e., there are  \mbox{\cal C}^{\prime}\,\mbox{\subseteq}\;\mbox{\cal C}  and  J\,\mbox{\subseteq}\,\{2,\dots,r\}\setminus\{\mbox{i_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\} so that  \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i_{0}$}}}=\prod_{c\in\cal C^{\prime}}c\cdot\,\prod_{j\in J}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}},  i.e.,

(+)\;\;\;\prod_{c\in\cal C^{\prime}}c=\prod_{j\in J\cup\{i_{0}\}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}}.

— If card (\mbox{\cal C}^{\prime}) is odd, since A_{\raisebox{-3.0pt}{\scriptstyle h_{1}}} is an AOS-fan, and hence a closed set, the left-hand side of $(+)$ is an element g^{\prime}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}, and we have  g^{\prime}\cdot\,\prod_{j\in J\cup\{i_{0}\}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}}=1. Setting A=\{g^{\prime}\}\,\cup\,\{\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,|\, j\in J\cup\{i_{0}\}\,\}\,\mbox{\subseteq}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}} and B=\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\}\,\cup\,\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,|\,j\in J\cup\{i_{0}\}\,\}\,\mbox{\subseteq}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}, the assumptions of Lemma 3.2 are met. Since $A$ is dependent, so is $B$, contradicting that B\,\mbox{\subseteq}\,\mbox{\cal B} and $\cal B$  is a basis of $\cal F$, whence an independent set.

— If \mbox{\cal C}^{\prime} = $\emptyset$, the same argument works, yielding a contradiction.

— Assume card (\mbox{\cal C}^{\prime}) even $>\,0$. Fix \mbox{c_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\in\mbox{\cal C}^{\prime}. Then card (\mbox{\cal C}^{\prime}\setminus\{\mbox{c_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}) = odd, and g^{\prime}=\prod_{c\in{\cal C^{\prime}}\setminus\{c_{0}\}}c\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}; also g^{\prime}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}, and we have:

\mbox{c_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\cdot\,g^{\prime}\cdot\,\prod_{j\in J\cup\{i_{0}\}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}}=1.

Pick any index \mbox{j_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\in J\,\cup\,\{i_{0}\} (so, \mbox{j_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\geq 2). Since \mbox{c_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\,,\,g^{\prime}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}} and g_{\raisebox{-3.0pt}{\scriptstyle j_{0}}} $\rightsquigarrow$  h_{\raisebox{-3.0pt}{\scriptstyle j_{0}}}, Lemma 1.6(b) yields \mbox{g^{\prime}{\raisebox{-3.0pt}{$\scriptstyle j_{0}$}}}:=\mbox{c{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\,g^{\prime}\,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j_{0}$}}}\,\mbox{\rightsquigarrow\,}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle j_{0}$}}}, and \mbox{g^{\prime}{\raisebox{-3.0pt}{$\scriptstyle j_{0}$}}}\cdot\,\prod_{j\in(J\cup\{i_{0}\})\setminus\{j_{0}\}}\mbox{g{\raisebox{-3.0pt}{$\scriptstyle j$}}}=1. Hence, A=\{\mbox{g^{\prime}{\raisebox{-3.0pt}{$\scriptstyle j_{0}$}}}\}\,\cup\,\{\mbox{g{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,| j\in(J\,\cup\,\{\mbox{i_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\})\setminus\{\mbox{j_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}\} is a dependent subset of L_{\raisebox{-3.0pt}{\scriptstyle k+1}}. Setting B=\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,|\,j\in J\,\cup\,\{\mbox{i_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}\} the assumptions of Lemma 3.2 are met, and hence $B$ is also dependent, contradicting that B\,\mbox{\subseteq}\,\mbox{\cal B}.

Case 2.  Some \mbox{c_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\in\mbox{\cal C} is dependent on the rest.

Then, there are  \mbox{\cal C}^{\prime}\;\mbox{\subseteq}\;\mbox{\cal C}\setminus\{\mbox{c_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}  and  J\,\mbox{\subseteq}\,\{2,\dots,r\} so that

(++)\;\;\;\mbox{c_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}=\prod_{c\in\cal C^{\prime}}c\cdot\prod_{j\in J}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}}.

Note that J\neq\mbox{\emptyset} (otherwise  $\cal C$  would be dependent). Taking $J$ minimal so that $(++)$ holds, and picking \mbox{j_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\in J, it follows that  c_{\raisebox{-3.0pt}{\scriptstyle 0}}  is not in the closure of  \mbox{\cal C}^{\prime}\,\cup\,\{\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,| j\in J\setminus\{\mbox{j_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}\} (cf. Warning, end of 3.1 (b)). By the exchange property,  g_{\raisebox{-3.0pt}{\scriptstyle j_{0}}} is in the closure of  \mbox{\cal C}^{\prime}\,\cup\,\{\mbox{c_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}\,\cup\,\{\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,|\,j\in J\setminus\{\mbox{j_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}\}, contrary to the result of Case 1. $\Box$

** 3.4**

Standard generating systems.

For any finite ARS-fan, $(X,F)$, we will construct, by induction on $k$, $1\leq k\leq n=\ell(X)$, a class of bases \mbox{\cal B}_{\raisebox{-3.0pt}{\scriptstyle k}} of the AOS-fan \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(X). Each basis \mbox{\cal B}_{\raisebox{-3.0pt}{\scriptstyle k}} will be required to satisfy the additional condition:

$(*)\;\;$ For $k\leq j\leq n$,  \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}} is a basis of the AOS-fan S^{k}_{\raisebox{-3.0pt}{\scriptstyle j}}.

This additional requirement guarantees that the inductive construction of the \mbox{\cal B}_{\raisebox{-3.0pt}{\scriptstyle k}}’s is not interrupted before the $n$-th (and last) step. The construction uses Proposition 3.3 and the results from § 2 above. The set \mbox{\cal B}=\bigcup_{k=1}^{n}\mbox{\mbox{$\cal B$}_{\raisebox{-3.0pt}{$\scriptstyle k$}}} is called a standard generating system for $(X,F)$.

Construction of standard generating systems.

Level 1. It suffices to observe that a basis \mbox{\cal B}_{\raisebox{-3.0pt}{\scriptstyle 1}} of L_{\raisebox{-3.0pt}{\scriptstyle 1}} satisfying condition $(*)$ exists. Begin by choosing a basis \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(n) of the AOS-fan \mbox{S^{1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}=\mbox{C^{1}{\raisebox{-3.0pt}{$\scriptstyle n$}}} (cf. Corollary 2.5). S^{1}_{\raisebox{-3.0pt}{\scriptstyle n}} is a closed subset (cf. Warning, end of 3.1 (b)) of the (AOS-)fan \mbox{S^{1}{\raisebox{-3.0pt}{$\scriptstyle n-1$}}}=\mbox{S^{1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}\,\cup\,\mbox{C^{1}{\raisebox{-3.0pt}{$\scriptstyle n-1$}}}; hence,  \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(n) is an independent subset of S^{1}_{\raisebox{-3.0pt}{\scriptstyle n-1}}; choose \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(n-1) to be a basis of S^{1}_{\raisebox{-3.0pt}{\scriptstyle n-1}} extending \mbox{\mbox{$\cal B$}_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(n).

In general, assume that, for $1<j\leq n$ an increasing sequence \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(n)\,\mbox{\subseteq}\dots\mbox{\subseteq}\,\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(j) of independent subsets of L_{\raisebox{-3.0pt}{\scriptstyle 1}} has been chosen so that \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(\ell) is a basis of the AOS-fan  \mbox{S^{1}{\raisebox{-3.0pt}{$\scriptstyle\ell$}}}\;\,(j\leq\ell\leq n). As above, \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(j) is an independent subset of the fan \mbox{S^{1}{\raisebox{-3.0pt}{$\scriptstyle j-1$}}}=\mbox{S^{1}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,\cup\,\mbox{C^{1}{\raisebox{-3.0pt}{$\scriptstyle j-1$}}}. Let \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(j-1) be a basis of S^{1}_{\raisebox{-3.0pt}{\scriptstyle j-1}} extending \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(j). Set \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(1); by construction, \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\cap\,\mbox{S^{1}{\raisebox{-3.0pt}{$\scriptstyle j$}}}=\mbox{\mbox{$\cal B$}_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(j) is a basis of S^{1}_{\raisebox{-3.0pt}{\scriptstyle j}}.

Induction step. Given an integer $k$, $1\leq k<n$, assume there exists a basis  \mbox{\cal B}_{\raisebox{-3.0pt}{\scriptstyle k}} of L_{\raisebox{-3.0pt}{\scriptstyle k}} satisfying property $(*)$; thus, for $k\leq j\leq n$, \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(j)=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}} is a basis of S^{k}_{\raisebox{-3.0pt}{\scriptstyle j}}. Further, since \mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}\,\mbox{\subseteq}\dots\mbox{\subseteq}\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle k$}}}=\mbox{L{\raisebox{-3.0pt}{$\scriptstyle k$}}}, we have \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(n)\,\mbox{\subseteq}\dots\mbox{\subseteq}\,\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(k)=\mbox{\mbox{$\cal B$}_{\raisebox{-3.0pt}{$\scriptstyle k$}}}. Using Proposition 3.3 we define a subset \mbox{\cal B}_{\raisebox{-3.0pt}{\scriptstyle k+1}} of L_{\raisebox{-3.0pt}{\scriptstyle k+1}} as follows:

— Firstly, fix an element \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(n) (this set is non-empty because $n=\ell(X)$). Pick a basis \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}) of the AOS-fan \{g\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}\,|\;g\,\mbox{\rightsquigarrow\,}\,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}.

— Next, for each h\in(\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}})\setminus\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\} there is a maximal index $j$ $=j(h)$,  $k+1\leq j\leq n$,  so that h\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(j); clearly, h\ \mbox{\not\in}\ \mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j+1$}}}, whence h\in\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}=\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\setminus\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j+1$}}} (if $j$ = $n$, then h\in\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}=\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}). Since $j\geq k+1$, we have \{g\in\mbox{C^{k+1}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,|\;g\,\mbox{\rightsquigarrow\,}\,h\}\neq\mbox{\emptyset}. Choose an element \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle h$}}}\in\mbox{C^{k+1}{\raisebox{-3.0pt}{$\scriptstyle j$}}} so that \mbox{g{\raisebox{-3.0pt}{$\scriptstyle h$}}}\,\mbox{\rightsquigarrow\,}\,h.

— Finally, set

[*]   \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}})\,\cup\,\{\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle h$}}}\,|\,h\in(\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}})\setminus\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}\}.

Claim. For k+1\leq p\leq n,\;\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}\,\cap\,\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle p$}}} is a basis of S^{k+1}_{\raisebox{-3.0pt}{\scriptstyle p}}.

Proof of Claim. We apply Proposition 3.3 with the following choice of parameters:

— \mbox{\cal G}=\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle p$}}} (whence \mbox{\cal F}=\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle p$}}}, since $k+1\leq p$);

— \mbox{\cal B}=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle p$}}} (a basis of  $\cal F$);

— \mbox{\cal C}=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}}) (a basis of  \{g\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}\,|\,g\,\mbox{\rightsquigarrow\,}\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}})).

Proposition 2.10 (a) shows that the cardinality assumption

card (\{g\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle p$}}}\,|\;g\,\mbox{\rightsquigarrow\,}\,h\}) = card (\{g\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle p$}}}\,|\;g\,\mbox{\rightsquigarrow\,}\,h^{\prime}\}),    (h,h^{\prime}\in\mbox{S^{k}_{\raisebox{-3.0pt}{$\scriptstyle j$}}})

of 3.3 holds. We conclude that

\mbox{\cal D}:=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}})\,\cup\,\{\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle h$}}}\,|\,h\in(\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle p$}}})\setminus\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}\}

is a basis of S^{k+1}_{\raisebox{-3.0pt}{\scriptstyle j}}. The Claim follows from:

(†)   \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}\,\cap\,\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle p$}}}=\;\mbox{\cal D}.

Proof of $({\dagger})$. Since \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}})\,\mbox{\subseteq}\,\mbox{\cal D}\,\cap\mbox{\mbox{$\cal B$}_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}} (see [*]), we need only prove:

($\subseteq$)  If h\in(\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}})\setminus\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}\}  and  \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle h$}}}\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle p$}}}, then  h\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}_{\raisebox{-3.0pt}{$\scriptstyle p$}}}.

This clearly follows from \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle h$}}}\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle p$}}}\,,\,\mbox{g{\raisebox{-3.0pt}{$\scriptstyle h$}}}\,\mbox{\rightsquigarrow\,}h  and  h\in\mbox{S^{k}_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}.

($\supseteq$)  Since $k+1\leq p$, we have  \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle p$}}}=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(p)\,\mbox{\subseteq}\,\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(k+1)=\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}. On the other hand, if h\in(\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle p$}}})\setminus\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\} and, as above, $j(h)$ denotes the largest index $j$ so that $k+1\leq j\leq n$ and h\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(j), we have $p\leq j(h)$, whence S^{k+1}_{\raisebox{-3.0pt}{\scriptstyle j(h)}} $\subseteq$ S^{k+1}_{\raisebox{-3.0pt}{\scriptstyle p}}. By choice, \mbox{g{\raisebox{-3.0pt}{$\scriptstyle h$}}}\in\mbox{C^{k+1}{\raisebox{-3.0pt}{$\scriptstyle j(h)$}}}; it follows that \mbox{g{\raisebox{-3.0pt}{$\scriptstyle h$}}}\in\mbox{S^{k+1}_{\raisebox{-3.0pt}{$\scriptstyle p$}}}, as required. $\Box$

Remarks 3.5

(a) In general, there are many different standard generating systems for a finite ARS-fan $(X,F)$. The construction in 3.4 allows for several choices of the bases \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(j) $(1\leq j\leq n)$ and, at each successive step, $k$, for many choices of elements \mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(n), of bases \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}), and of elements \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle h$}}}\in\mbox{C^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}} under each h\in(\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}})\setminus\{\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}. In spite of this lack of uniqueness, we shall prove below that any standard generating system determines the isomorphism type of a finite ARS-fan.

(b) Some of the sets \mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}=\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\setminus\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j+1$}}} may be empty. However, if \mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\neq\mbox{\emptyset}, then, necessarily,  \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\neq\mbox{\emptyset}. Indeed, if $j$ = $n$, then \mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}\neq\mbox{\emptyset} (as $n=\ell(X)$) and \mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}=\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}} is an AOS-fan; since \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}} is a basis of C^{k}_{\raisebox{-3.0pt}{\scriptstyle n}}, it must contain at least one element. If $j<n$, since S^{k}_{\raisebox{-3.0pt}{\scriptstyle j+1}} is a fan, it is a closed set; as it is disjoint from C^{k}_{\raisebox{-3.0pt}{\scriptstyle j}}, then no element of C^{k}_{\raisebox{-3.0pt}{\scriptstyle j}} is dependent on S^{k}_{\raisebox{-3.0pt}{\scriptstyle j+1}}. Hence, any basis of \mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}=\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j+1$}}}\,\cup\,\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}} must contain an element of C^{k}_{\raisebox{-3.0pt}{\scriptstyle j}}. $\Box$

Any standard generating system for a finite ARS-fan has the following property:

Corollary 3.6

Let  $\cal B$  be a standard generating system for a finite ARS-fan $(X,F)$; let $n=\ell(X)$, and $1\leq k\leq m\leq n$. Then, for every  g\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle m$}}}=\mbox{\cal B}\,\cap\,\mbox{L{\raisebox{-3.0pt}{$\scriptstyle m$}}}(X), the unique depth-$k$ successor of  $g$ in $X$ belongs to  $\cal B$  $($hence to \mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}=\mbox{\cal B}\,\cap\,\mbox{L{\raisebox{-3.0pt}{$\scriptstyle k$}}}(X)).

Proof. By the construction in 3.4 this holds for $m=k+1$. In fact, let g\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}\,,\,h^{\prime}\in\mbox{L{\raisebox{-3.0pt}{$\scriptstyle k$}}} and g\,\mbox{\rightsquigarrow\,}h^{\prime}. By the definition of \mbox{\cal B}_{\raisebox{-3.0pt}{\scriptstyle k+1}} and uniqueness of the successor of $g$ in L_{\raisebox{-3.0pt}{\scriptstyle k}}, if g\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}}), then h^{\prime}=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\in\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}(n)\,\mbox{\subseteq}\,\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}; if g=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle h$}}}, with h\in(\mbox{\mbox{$\cal B$}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}})\setminus\{\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}, we get h^{\prime}=h\in\mbox{\mbox{$\cal B$}_{\raisebox{-3.0pt}{$\scriptstyle k$}}}. Then iterate. $\Box$

For the proof of the Isomorphism Theorem 3.11 below we shall need the characterizations of ARS-morphisms between fans proved in 3.9 and 3.10 below, which, in turn, follow from the Small Representation Theorem 3.8.

Definition 3.7

Let $G,H$ be RSs, let \mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}},\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle H$}}} be their character spaces, and let Z\,\mbox{\subseteq}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}} . A map F:Z\,\mbox{\longrightarrow}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle H$}}} preserves 3-products $($in Z$$) iff for all \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}\in Z,

\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}\in Z\;\;\mbox{\Rightarrow}\;\;F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}})=F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}). $\Box$

Proposition 3.8

(Small representation theorem). Let $G$ be a RS. The following conditions are equivalent for a map  f:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}\,\mbox{\longrightarrow}\,\bf 3 :

$(1)$* a$)$ $f$ is continuous in the constructible topology of X_{\raisebox{-3.0pt}{\scriptstyle G}}.*

b$)$ $f$ preserves $3$-products in X_{\raisebox{-3.0pt}{\scriptstyle G}}.

$(2)$* $f$ is represented by an element of $G$: there is $a\in G$ so that f=\mbox{\widehat{a}}.*

[ \mbox{\widehat{a}}:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}\mbox{\longrightarrow}\ \bf 3  denotes “evaluation at $a$”: for h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}},  \mbox{\widehat{a}}(h)=h(a).]

Proof. (2) $\Rightarrow$ (1) is clear since the evaluation maps have properties (1.a) and (1.b).

(1) $\Rightarrow$ (2). We use the representation theorem [M], Cor. 8.3.6, p. 162. It suffices to check that the assumptions of this theorem as well as one of the equivalent conditions in its conclusion hold under our hypotheses in (1). In our notation, the conditions to be checked are: for x,y\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}},

(${\dagger}$)     $f(x)=0$  and  Z(x)\,\mbox{\subseteq}\,Z(y)  implies  $f(y)=0$.

(${\dagger}{\dagger}$)   $f(x)\neq 0$  and  $x^{-1}[0,1]\supseteq y^{-1}[0,1]$  implies  $f(x)=f(y)$.

(${\dagger}{\dagger}{\dagger}$)  For any saturated prime ideal $I$ of $G$, either

(i)   f\lceil\{u\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}\,|\,Z(u)=I\}=0,  or

(ii)  \prod_{i=1}^{4}f(\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle i$}}})=1  for any 4-element AOS-fan \{\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 4$}}}\}  in  \{u\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}\,|\,Z(u)=I\}.

— Condition (${\dagger}$) follows at once from Lemma 0.2 (2) (as Z(x)\,\mbox{\subseteq}\,Z(y)\;\mbox{\Rightarrow}\;y=yx^{2}).

— Condition (${\dagger}{\dagger}$) follows from Lemma 0.1 (3),(5):  x^{-1}[0,1]\supseteq y^{-1}[0,1]\;\mbox{\Rightarrow}\;x=x^{2}y. Since f(x)\neq 0\;\mbox{\Rightarrow}\;f(x^{2})=1, assumption (1.b) implies  $f(x)=f(x^{2})f(y)=f(y)$.

— As for (${\dagger}{\dagger}{\dagger}$), if (i) does not hold, (${\dagger}$) implies  $f(u)\neq\,0$  for all  u\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}  such that  $Z(u)=I$. Let  \{\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 4$}}}\}  be an AOS-fan in \{u\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle G$}}}\,|\,Z(u)=I\}. Thus,  \mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 4$}}}=\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}  and  f(\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\neq 0 for $i=1,\dots,4$. Assumption (1.b) gives f(\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 4$}}})=f(\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})f(\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})f(\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle 3$}}})\neq\,0, i.e.,  \prod_{i=1}^{4}f(\mbox{x_{\raisebox{-3.0pt}{$\scriptstyle i$}}})=1. $\Box$

Corollary 3.9

A map F:(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\,\mbox{\longrightarrow}\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}) between ARS-fans is an ARS-morphism iff  $F$ is continuous for the constructible topology $($of both source and target $)$ and preserves $3$-products in X_{\raisebox{-3.0pt}{\scriptstyle 1}} $($cf. 3.7 $)$.

Proof. ($\Leftarrow$) If $F$ has the stated properties and a\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}, then \mbox{\widehat{a}}\;\mbox{\circ}\,F:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\longrightarrow}\,\bf 3 also has those properties, and, by Proposition 3.8, is represented by an element of F_{\raisebox{-3.0pt}{\scriptstyle 1}}; hence,  $F$ is an ARS-morphism (cf. 1.1 (c.i)).

($\Rightarrow$) Assume  $F$ is an ARS-morphism. For continuity it suffices to show that $F^{-1}[V]$ is open constructible in X_{\raisebox{-3.0pt}{\scriptstyle 1}} whenever $V$ is basic open for the constructible topology of X_{\raisebox{-3.0pt}{\scriptstyle 2}}, i.e., of the form V=U(\mbox{a_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{a_{\raisebox{-3.0pt}{$\scriptstyle n$}}})\,\cap\,Z(a) with a,\mbox{a_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{a_{\raisebox{-3.0pt}{$\scriptstyle n$}}}\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 2$}}} (see [M], p. 111). By the assumption on  $F$, there are  b,\mbox{b_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{b_{\raisebox{-3.0pt}{$\scriptstyle n$}}}\in\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}  such that  \mbox{\widehat{a}}\;\mbox{\circ}\,F=\mbox{\widehat{b}}  and  \mbox{\widehat{\mbox{a_{\raisebox{-3.0pt}{\scriptstyle i}}}}}\;\mbox{\circ}\,F=\mbox{\widehat{\mbox{b_{\raisebox{-3.0pt}{\scriptstyle i}}}}}  for  $i=1,\dots,n$. These functional identities imply F^{-1}[V]=U(\mbox{b_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{b_{\raisebox{-3.0pt}{$\scriptstyle n$}}})\,\cap\,Z(b), as required.

Preservation of 3-products by  $F$  follows easily from the same property for  $\widehat{a}$  and  $\widehat{b}$  using the functional identity  \mbox{\widehat{a}}\;\mbox{\circ}\,F=\mbox{\widehat{b}}. $\Box$

Lemma 3.10

Let (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}),\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}) be ARS-fans.

$(1)$* For a map F:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\longrightarrow}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}} the following are equivalent:*

$($i $)$  $F$ preserves $3$-products in X_{\raisebox{-3.0pt}{\scriptstyle 1}}.

$($ii $)$ a$)$ $F$ preserves $3$-products of elements of the same level: for all $I\in$ Spec(\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})  and all                  \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}\in\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}),  F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}})=F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}).

b$)$ $F$ is monotone for the specialization order: for  g,h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},  g\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,h\;\;\mbox{\Rightarrow}\;\;F(g)\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,F(h).

({\underaccent{i}{\,\mbox{\rightsquigarrow\,}}}* denotes specialization in X_{\raisebox{-3.0pt}{\scriptstyle i}}$$).*

$(2)$* If (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}) is finite, any map verifying one of the equivalent conditions $($i$)$ or $($ii$)$ in $(1)$ is a morphism of ARSs.*

$(3)$* If both (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}),\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}) are finite, any bijection F:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\longrightarrow}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}} verifiying one of the equivalent conditions in $(1)$ is an isomorphism of ARSs.*

Proof. (1). (i) $\Rightarrow$ (ii). (ii.a) is a special case of (i).

(ii.b)   g\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,h\;\;\mbox{\Leftrightarrow}\;\;h=h^{2}g (Lemma 0.1). By (i), $F(h)=F(h)^{2}F(g)$, and this equality (in X_{\raisebox{-3.0pt}{\scriptstyle 2}}) gives F(g)\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,F(h).

(ii) $\Rightarrow$ (i). Let \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}} be any three elements in X_{\raisebox{-3.0pt}{\scriptstyle 1}}; say Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}})\,\mbox{\subseteq}\,Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})\,\mbox{\subseteq}\,Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}). Let I=Z(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}) and for  $i=2,3$  let  h^{\prime}_{\raisebox{-3.0pt}{\scriptstyle i}}  be the unique successor of  h_{\raisebox{-3.0pt}{\scriptstyle i}}  in  \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle I$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}); Lemma 1.5 shows that  \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 3$}}}; then, assumption (ii.a) gives

F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}})=F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})F(\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}})F(\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 3$}}}).

By (ii.b) we have  F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,F(h^{\prime}_{i}),\;(i=2,3). Next, note that Z(F(h^{\prime}_{i}))\,\mbox{\subseteq}\,Z(F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})) for $i=2,3$. In fact, since \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}} belong to the same level L_{\raisebox{-3.0pt}{\scriptstyle I}}, Z(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}})=Z(\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), and Lemma 0.2 (2) yields \mbox{h^{2}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}={\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}}^{2}; scaling by h_{\raisebox{-3.0pt}{\scriptstyle 1}} gives \mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}}=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}{\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}}^{2}. Since $F$ preserves 3-products of the same level, F(\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 1$}}})=F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})F(\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}})^{2} which, by 0.2 (1), gives Z(F(\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}}))\,\mbox{\subseteq}\,Z(F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})). Same argument for $i=3$.

Using 1.5 again, these inclusions and F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,F(\mbox{h^{\prime}_{\raisebox{-3.0pt}{$\scriptstyle i$}}}),\;(i=2,3) prove:

F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})F(\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 2$}}})F(\mbox{h^{\prime}{\raisebox{-3.0pt}{$\scriptstyle 3$}}})=F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}),

as required.

(2) follows at once from Corollary 3.9, since the continuity requirement is automatically fulfilled in this case: the constructible topology in  X_{\raisebox{-3.0pt}{\scriptstyle 1}} is discrete.

(3) By (2) it only remains to prove that  F^{-1}:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\,\mbox{\longrightarrow}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}  preserves 3-products in  X_{\raisebox{-3.0pt}{\scriptstyle 2}}. Let  \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}  and let \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}=F^{-1}(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}),\;i=1,2,3. From (1.i) we have  F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}})=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}. Composing both sides of this equality with  $F^{-1}$  gives the desired conclusion:

F^{-1}(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 3$}}})=F^{-1}(F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}))=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}=F^{-1}(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})F^{-1}(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})F^{-1}(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 3$}}}).     $\Box$

Remark. Note that any isomorphism of ARS-fans preserves depth.

Theorem 3.11

(The isomorphism theorem for finite ARS-fans.)* Let (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}),\,(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},\mbox{F_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}) be finite ARS-fans and let {\underaccent{1}{\,\mbox{\rightsquigarrow\,}}},{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}} denote their respective specialization orders. If  (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}) and (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}) are order-isomorphic, then X_{\raisebox{-3.0pt}{\scriptstyle 1}} and X_{\raisebox{-3.0pt}{\scriptstyle 2}} are isomorphic ARSs.*

Proof. The order-isomorphism assumption implies:

(1) \ell(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})=\ell(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}) ( = $n$, say, fixed throughout the proof).

(2) For $1\leq k\leq j\leq n$, card(\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}})) = card(\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}})).

The proof of (2) is an easy exercise. Since \mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle\ell$}}}\cap\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle\ell^{\prime}$}}}=\mbox{\emptyset} for $k\leq\ell\neq\ell^{\prime}\leq n$ and \mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}=\bigcup_{\ell=j}^{\,n}\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle\ell$}}}, we get:

(3) For $1\leq k\leq j\leq n$,  {\mathrm{card}}(\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}}))=\,{\mathrm{card}}(\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}})).

(4) For $1\leq k<n$ and all h\in\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}}), h^{\prime}\in\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), we have:

{\mathrm{card}}(\{g\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\,|\;g\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,h\})=\,{\mathrm{card}}(\{g^{\prime}\,\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}})\,|\;g^{\prime}{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,h^{\prime}\}).

Proof of (4). Consider the two-variable formula in the language $\{\leq\}$ of order:

\raisebox{1.72218pt}{\mbox{\varphi}}(x,y)\;:=\;x\in\mbox{S^{k+1}_{\raisebox{-3.0pt}{$\scriptstyle n$}}}\,\mbox{\wedge}\;x\leq y.

(It is left as an exercise for the reader to write a first-order formula in $\{\leq\}$ expressing the notion x\in\mbox{S^{k+1}_{\raisebox{-3.0pt}{$\scriptstyle n$}}}; cf. 2.1.)

If $\sigma$ denotes the order isomorphism between (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}) and (\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}), for g,h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}} we have:

(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}})\models\,\raisebox{1.72218pt}{\mbox{\varphi}}[g,h]\;\;\mbox{\Leftrightarrow}\;\;(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}},{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}})\models\,\raisebox{1.72218pt}{\mbox{\varphi}}[\mbox{\sigma}(g),\mbox{\sigma}(h)].

It follows that  $\sigma$  maps  \{g\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\,|\;g\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,h\}  bijectively onto  \{g^{\prime}\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}})\,|\;g^{\prime}\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{\sigma}(h)\}. Now, if  h\in\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}}), then  \mbox{\sigma}(h)\in\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}}). If  h^{\prime}\in\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), apply Proposition 2.10 with \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}=h^{\prime} and \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}=\mbox{\sigma}(h) to conclude. $\Box$

Since the sets in item (4) are AOS-fans (Corollary 2.5), they have the same dimension, i.e., any bases of each of them have the same cardinality. If  ${\cal B}^{1},{\cal B}^{2}$ are standard generating systems for X_{\raisebox{-3.0pt}{\scriptstyle 1}}, X_{\raisebox{-3.0pt}{\scriptstyle 2}}, respectively, then {\cal B}^{i}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle i$}}}) is a basis of the fan \mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle i$}}}), for $1\leq k\leq j\leq n$ and $i=1,2$; from (3) we get:

(5) For $1\leq k\leq j\leq n$,  {\mathrm{card\,}}({\cal B}^{1}\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}}))=\,{\mathrm{card\,}}({\cal B}^{2}\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}})).

In particular, for \mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle k$}}}=\mbox{L{\raisebox{-3.0pt}{$\scriptstyle k$}}} we obtain:

(6) If $1\leq k\leq n$, then  {\mathrm{card\,}}(\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k$}}})=\,{\mathrm{card\,}}(\mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle k$}}})   (where \mbox{{\cal B}^{i}{\raisebox{-3.0pt}{$\scriptstyle k$}}}={\cal B}^{i}\cap\,\mbox{L{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle i$}}})).

Next, we choose an arbitrary standard generating system  ${\cal B}^{1}$ for X_{\raisebox{-3.0pt}{\scriptstyle 1}}. By induction on $k$, $1\leq k\leq n$, we construct a standard generating system  ${\cal B}^{2}$ of X_{\raisebox{-3.0pt}{\scriptstyle 2}} ({\cal B}^{2}=\bigcup_{k=1}^{\,n}\mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle k$}}}) and a map \mbox{f{\raisebox{-3.0pt}{$\scriptstyle k$}}}:\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\mbox{\longrightarrow}\,\mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle k$}}} so that:

(7) i) For $k\leq j\leq n$,   \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}[\,{\cal B}^{1}\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}})]=\,{\cal B}^{2}\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}}).

ii) If  k<n,\;g\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}},\,h\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k$}}}  and  g\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,h, then  \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(g)\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(h).

Construction of ${\cal B}^{2}$ and the maps f_{\raisebox{-3.0pt}{\scriptstyle k}}.

Level 1. {\cal B}^{2}_{\raisebox{-3.0pt}{\scriptstyle 1}} is built as in the level 1 step in 3.4; with notation therein, \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}:\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\longrightarrow}\,\mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle 1$}}} is any bijection mapping  \mbox{\cal B_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}(j)  onto  \mbox{\cal B_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}(j), for  $1\leq j\leq n$. Such a bijection exists by (5) above $(k=1)$.

Induction step. Assume \mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle k$}}} and \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}} already constructed, so that:

— For $1\leq j\leq k$ and $j\leq{\ell}\leq n$,  \mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\cap\,\mbox{S^{j}{\raisebox{-3.0pt}{$\scriptstyle\ell$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})  is a basis of the AOS-fan \mbox{S^{j}{\raisebox{-3.0pt}{$\scriptstyle\ell$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}})  and  \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle j$}}}[\,\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\cap\,\mbox{S^{j}{\raisebox{-3.0pt}{$\scriptstyle\ell$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})]=\,\mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle j$}}}\,\cap\,\mbox{S^{j}{\raisebox{-3.0pt}{$\scriptstyle\ell$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}).

— Condition (7.ii) holds for all $j$  such that $1\leq j<k$.

The basis {\cal B}^{2}_{\raisebox{-3.0pt}{\scriptstyle k+1}}, and along with it the map  f_{\raisebox{-3.0pt}{\scriptstyle k+1}}, are defined by performing the construction of the inductive step in 3.4, with the following choice of parameters:

— If  \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}), and  \mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}})  is a basis of the (AOS-)fan \{g\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\,|\;g\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}\}, then take  \mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{f{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}}))  to be a basis of the fan  \{g^{\prime}\in\mbox{S^{k+1}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}})\,|\;g^{\prime}\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}})\}. This is possible since \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}})\in\mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\cap\,\mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle n$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), by (7.i). Using item (4), we let \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}\lceil\,\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}}) be a bijection between \mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}}) and \mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{f{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}})).

— If  g\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}\cap\,\mbox{C^{k+1}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})  with $k+1\leq j\leq n$, but  g\not\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(n,\mbox{h{\raisebox{-3.0pt}{$\scriptstyle 0$}}}), then, by the construction performed in the inductive step of 3.4, if $h$ is the unique depth-$k$ successor of $g$, we have h\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\,\cap\,\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}),\,h\neq\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 0$}}} and g=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle h$}}}. In this case choose any element  g^{\prime}\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(h) such that g^{\prime}\in\mbox{C^{k+1}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), and set \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k+1$}}}(g)=g^{\prime}. This is possible since  \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(h)\in\mbox{{\cal B}^{2}{\raisebox{-3.0pt}{$\scriptstyle k$}}}\cap\,\mbox{C^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}) (which follows easily from (7.i)). Clearly, this construction guarantees that (7.i) and (7.ii) hold for $k+1$.

Note that (7.ii) implies, by iteration, its own generalization:

(7) iii) If  1\leq k<m\leq n,\,g\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle m$}}},\,h\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k$}}}  and  g\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,h, then  \mbox{f_{\raisebox{-3.0pt}{$\scriptstyle m$}}}(g)\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(h).

Since \mbox{{\cal B}^{i}{\raisebox{-3.0pt}{$\scriptstyle k$}}}={\cal B}^{i}\cap\,\mbox{L{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle i$}}}) is a basis of the AOS-fan \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle i$}}}), $i=1,2$, we get:

(8) The bijection  f_{\raisebox{-3.0pt}{\scriptstyle k}}  extends (uniquely) to an AOS-isomorphism  \mbox{\widetilde{\mbox{f_{\raisebox{-3.0pt}{\scriptstyle k}}}}}:\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\,\mbox{\longrightarrow}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})  mapping \mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 1$}}}) onto \mbox{S^{k}{\raisebox{-3.0pt}{$\scriptstyle j$}}}(\mbox{X{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), for all $j$ such that $k\leq j\leq n$.

Now set  F:\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\,\mbox{\longrightarrow}\,\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}  to be  F=\,\bigcup_{k=1}^{\,n}\mbox{\widetilde{\mbox{f_{\raisebox{-3.0pt}{\scriptstyle k}}}}}. We prove:

Claim. $F$ is an isomorphism of ARSs.

Proof of Claim.  Since  \mbox{X_{\raisebox{-3.0pt}{$\scriptstyle i$}}}=\,\bigcup_{k=1}^{\,n}\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle i$}}}) (disjoint union) for $i=1,2$, and \widetilde{\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}} maps \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}) bijectively onto \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}}), we have:

(a) $F$ is well-defined and bijective.

(b) For all $k$, $1\leq k\leq n$, $F$ preserves 3-products in L_{\raisebox{-3.0pt}{\scriptstyle k}}.

This is clear: by (8)  F\lceil\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})=\mbox{\widetilde{\mbox{f_{\raisebox{-3.0pt}{\scriptstyle k}}}}}:\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\,\mbox{\longrightarrow}\,\mbox{L_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 2$}}})  is an isomorphism of AOS-fans.

(c) $F$ is monotone for the specialization order.

Let g,h\in\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}} be such that g\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,h; say $d(g)=m\geq d(h)=k$. We must prove F(g)\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,F(h). Since  {\cal B}^{1}_{\raisebox{-3.0pt}{\scriptstyle m}}  generates  \mbox{L_{\raisebox{-3.0pt}{$\scriptstyle m$}}}(\mbox{X_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}), then  g=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}} with  \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}},\dots,\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle m$}}}  and $r$ necessarily odd (possibly = 1). By Corollary 3.6, if  h_{\raisebox{-3.0pt}{\scriptstyle i}} is the unique depth-$k$ successor of  g_{\raisebox{-3.0pt}{\scriptstyle i}}, then \mbox{h{\raisebox{-3.0pt}{$\scriptstyle i$}}}\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k$}}}. Also, \mbox{g{\raisebox{-3.0pt}{$\scriptstyle i$}}}\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}}  ($i=1,\dots,r$)  implies  g=\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}}\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}  (1.6 (a)). Since both  $h$  and  \mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}  are successors of $g$  of the same level $k$, we get  h=\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}}\cdot\ \dots\ \cdot\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}. As  $F$  preserves products of any odd number of elements of the same level, we have:

F(g)=F(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\cdot\ \dots\ \cdot F(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}})     and     F(h)=F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\cdot\ \dots\ \cdot F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}}).

Since  \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\,{\underaccent{1}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}},  \mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}}\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle m$}}}  and  \mbox{h{\raisebox{-3.0pt}{$\scriptstyle i$}}}\in\mbox{{\cal B}^{1}{\raisebox{-3.0pt}{$\scriptstyle k$}}}, item (7.iii) yields  F(\mbox{g{\raisebox{-3.0pt}{$\scriptstyle i$}}})=\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle m$}}}(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle i$}}})\,{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,\mbox{f_{\raisebox{-3.0pt}{$\scriptstyle k$}}}(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}})=F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle i$}}})  ($i=1,\dots,r$). Then, by 1.6 (a) again,

F(g)=F(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\cdot\ \dots\ \cdot F(\mbox{g_{\raisebox{-3.0pt}{$\scriptstyle r$}}})\;{\underaccent{2}{\,\mbox{\rightsquigarrow\,}}}\,F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle 1$}}})\cdot\ \dots\ \cdot F(\mbox{h_{\raisebox{-3.0pt}{$\scriptstyle r$}}})=F(h),

which proves (c). The Claim follows from (a)–(c) using Lemma 3.10 (3). This completes the proof of Theorem 3.11. $\Box$