Unital Specker $\ell$-groups and boolean multispaces

TL;DR

This paper establishes a duality between boolean multispaces and unital Specker $ ell$-groups, extending Stone duality and revealing categorical properties of these algebraic structures.

Contribution

It introduces boolean multispaces as a topological generalization and proves their categorical duality with unital Specker $ ell$-groups, extending classical dualities.

Findings

Category of boolean multispaces is dually equivalent to unital Specker $ ell$-groups.

Unital Specker $ ell$-groups with singular units are equivalent to boolean algebras.

The category of unital Specker $ ell$-groups has finite colimits and products, but lacks some countable copowers and equalizers.

Abstract

As a topological generalization of the notion of a multiset, a boolean multispace is a boolean space $X$ with a continuous function $u\colon X\to \mathbb Z_{>0}$, where $\mathbb Z_{>0}=\{1,2,\dots\}$ has the discrete topology. In this paper the category of boolean multispaces and continuous multiplicity-decreasing morphisms with respect to the divisibility order is shown to be dually equivalent to the category of unital Specker $\ell$-groups and unital $\ell$-homomorphisms. This result extends Stone duality, because unital Specker $\ell$-groups whose distinguished unit is singular are equivalent to boolean algebras. Boolean multispaces, in turn, are categorically equivalent to the Priestley duals of the MV-algebras corresponding to unital Specker $\ell$-groups via the $\Gamma$ functor. Via duality, we show that the category of unital Specker $\ell$-groups has finite colimits and finite products, but lacks some countable copowers and equalizers.

Equations

$$x\oplus \neg (x\oplus \neg y)=y\oplus \neg (y\oplus \neg x).$$

$$\mathsf{uS}\ell \mathsf{g}$$

$$\{\mathfrak{m}\in \mu (G)\mid g\notin \mathfrak{m}\},$$

$$F_g=\{\mathfrak{m}\in \mu (G)\mid g\in \mathfrak{m}\},$$

$$(C_X,u)$$

$$g_S^{♮}:\mu (S)$$

$$\mathfrak{m}$$

$$g_S^{♮}(\mathfrak{m})=$$

$$\mathsf{supp}:s\in \mathsf{sing}(S)\mapsto \{\mathfrak{m}\in \mu (S)\mid s/\mathfrak{m}>0\}=\{\mathfrak{m}\in \mu (S)\mid s\notin \mathfrak{m}\}.$$

$$(g_S^{♮}{)}^{-1}(C)$$

$$=\{\mathfrak{m}\in \mu (S)\mid g-c{s}_S\in \mathfrak{m}\}.$$

$$nf\wedge g=(n+1)f\wedge g.$$

$${\zeta }_{\gamma }=\frac{u_X}{u_Y(\gamma )}$$

$$\mathcal{S}:(X,u)\mapsto (C_X,u),\text{ for any (X,u)∈Bms.}$$

$$\mathcal{S}_{\gamma}\colon f\in C_{V}\mapsto f(\gamma)\frac{u_{W}}{u_{V}(\gamma)}\in C_{W},\,\,\,\mbox{ for any $f\in C_{V}$.}$$

$$({\mathcal{S}}_{\gamma }(f))(y)=f(\gamma (y))\frac{u_W(y)}{u_V(\gamma (y))}=0,\text{mbox}but({\mathcal{S}}_{{\gamma }^{\prime }}(f))(y)=f({\gamma }^{\prime }(y))\frac{u_W(y)}{u_V({\gamma }^{\prime }(y))}\ne 0,$$

$$\mathfrak{m}\in \mu (C_X)\mapsto {\dot{Z}}_X\mathfrak{m}=x_{\mathfrak{m}}=\text{mbox}theuniqueelementof\bigcap \{Z_{X}f\mid f\in \mathfrak{m}\}$$

$$\bigcap \{Z_{X}f\mid f\in {\mathfrak{M}}_X(x)\}=\{x\}.$$

$$\mathfrak{m}={\mathfrak{M}}_X(x)$$

$$\bigcap \{Z_{X}f\mid f\in \mathfrak{m}\}=\{x\}.$$

$$\bigcap \{Z_{X}f\mid f\in S\}=Z_X\left(\sum _{f\in S}|f|\right),$$

$${\mathfrak{M}}_X^{-1}(C)$$

$$=\{x\in X\mid \{g\in C_X\mid g(x)=0\}\in F_f\}$$

$$=\{x\in X\mid f\in \{g\in C_X\mid g(x)=0\}\}$$

$$=\{x\in X\mid f(x)=0\}$$

$$=Z_{X}f,$$

$${\mathrm{ev}}_x:C_X$$

$$f$$

$$g^{♮}({\mathfrak{M}}_X(x))={\mathrm{ev}}_x(g)=g(x),$$

$${\psi }^{-1}:(\mu (T),v^{♮})$$

$$\mathfrak{m}$$

$$F_f=\{\mathfrak{n}\in \mu (S)\mid f\in \mathfrak{n}\},$$

$$({\psi }^{-1}{)}^{-1}(F_f)$$

$$=\{\mathfrak{m}\in \mu (T)\mid f\in {\psi }^{-1}(\mathfrak{m})\}$$

$$=\{\mathfrak{m}\in \mu (T)\mid \psi (f)\in \mathfrak{m}\},$$

Full text

The first author was funded by UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee (grant number EP/Y015029/1, Project “DCPOS”) during his affiliation at the University of Birmingham and by an FSR Incoming Postdoctoral Fellowship during his affiliation at the Université catholique de Louvain.

Honoring Hilary Priestley

1. Introduction

…and one may say that the invention of functors is one of the main goals of modern mathematicians, and one which usually yields the most startling results.

J. Dieudonné, [17, p. 236]

MV-algebras were introduced by C.C. Chang in his paper [8]. By definition, an MV-algebra $A$ is a structure $(A,0,\neg,\oplus)$ satisfying the equations $x\oplus(y\oplus z)=(x\oplus y)\oplus z$,    $x\oplus 0=x$,   $x\oplus\neg 0=\neg 0$,    $\neg\neg x=x$  and, characteristically,

[TABLE]

These equations are a terse equivalent reformulation of Chang’s original axiomatization in [8]. See [12, §1.7]. As shown in [23], the commutativity of $\oplus$ follows from the above five equations.

From Chang’s completeness theorem [9] it follows that an equation holds in the unit real interval $\operatorname{[0,1]}$ equipped with the constant 0 and the Łukasiewicz implication $x\to_{\mbox{\tiny\L }}y\,\,\,(=\neg x\oplus y)$ if and only if it holds in every MV-algebra. See [12, Theorem 2.5.3]. For short, the algebra $(\operatorname{[0,1]},0,\to_{\mbox{\tiny\L }})$ is a term-equivalent variant of the canonical generator $\operatorname{[0,1]}=(\operatorname{[0,1]},0,\neg,\oplus)$ of the equational class of MV-algebras.

As shown in [27], equation (1) is deeply related to the continuity of implication in a $\operatorname{[0,1]}$-valued logic. In the terminology of [20], this identity is not “canonical”. This makes it difficult to extend to MV-algebras Priestley duality, [31, 32, 33].

In the present paper we consider a special class of MV-algebras, called Specker MV-algebras. They are the counterpart of unital Specker $\ell$-groups [4, 15, 16, 30] via the $\Gamma$ functor, [25]. By [16, Theorem 3.16], Specker $\ell$-groups are uniquely determined by their underlying lattice structure together with the identity element. By Lemma 4.5, Specker MV-algebras are just finite products $\prod_{i}B_{i}\otimes\,$Łn(i) of tensor products of boolean algebras $B_{i}$ by finite Łukasiewicz chains Łn(i).

As a topological generalization of the notion of a multiset, a boolean multispace is a boolean space, (i.e., a totally disconnected compact Hausdorff space) $X$ equipped with a continuous function $u\colon X\to\mathbb{Z}_{>0}$, where $\mathbb{Z}_{>0}=\{1,2,\dots\}$ has the discrete topology. In Theorem 4.2 it is shown that boolean multispaces and continuous multiplicity-decreasing morphisms with respect to the divisibility order are dually equivalent to unital Specker $\ell$-groups and unital $\ell$-homomorphisms. Since, as proved in [28], boolean spaces are dual to unital Specker $\ell$-groups whose distinguished unit is singular, this result extends Stone duality. See Corollary 4.3 for details. Using the $\Gamma$ functor we obtain that boolean multispaces are categorically equivalent to the Priestley duals of Specker MV-algebras. See Proposition 4.6. This result should be compared with the duality obtained in the paper [19] for the class of all MV-algebras, where the dual of an MV-algebra $A$ is an enriched Priestley space on the set of prime lattice ideals of $A$.

Section 5 is devoted to the construction of a functor $\mathcal{B}$ that, together with $\mathcal{S}$, is part of the duality.

In Sections 6–8 it is shown, among other things, that boolean multispaces are finitely complete and have finite coproducts, but—differently from boolean spaces—lack some countable powers and some coequalizers.

In the final Section 9 we briefly discuss related work in the literature on multisets.

To the best of our knowledge, the present paper is the first one where a duality is constructed between a category of topological multispaces and a pre-existing, time-honored category of algebraic structures, namely unital Specker $\ell$-groups and their unital $\ell$-homomorphisms. Unital and non-unital Specker $\ell$-groups have well-known representation theorems in the literature. See, e.g., [5, §13.5], and [4] and references therein.

Last but not least, this paper gives an answer to Problem 5.5 in [29] concerning the construction of MV-algebraic categories of Ulam-Rényi games on infinite search spaces. As a matter of fact, the MV-algebras of compact Ulam-Rényi games ([29, Definition 5.2]) equipped with MV-algebraic homomorphisms are categorically equivalent to Specker MV-algebras, as well as to unital Specker $\ell$-groups. The deep connections between searching games with errors/lies and their underlying combinatorial-algebraic structures might be of help to enrich the theory of adaptive error correcting codes (see [10, and references therein] for a detailed account) with combinatorial game-theoretical tools from category theory.

2. Preliminary material

We refer to [5] for $\ell$-groups, to [2, 7] for category theory, and to [14, 12, 25] for MV-algebras and the functors $\Gamma$ and $\Xi$.

By an $\ell$-group we mean a lattice-ordered abelian group. A (distinguished, strong, order) unit of an $\ell$-group $G$ is an element $u\geq 0$ with the property that for every $g\in G$ there is $n\in\mathbb{Z}_{>0}=\{1,2,\dots\}$ such that $-nu\leq g\leq nu.$ The pair $(G,u)$ is said to be a unital $\ell$-group. Unless otherwise specified, every ideal of $(G,u)$ in this paper (also known as an $\ell$-ideal) is proper, i.e., it is different from $G$. A unital $\ell$-homomorphism between two unital $\ell$-groups $(G,u_{G})$ and $(H,u_{H})$ is an $\ell$-homomorphism $\psi\colon G\to H$ such that $\psi(u_{G})=u_{H}$.

Definition 2.1** ([4, 5, 15, 16, 35]).**

A singular element of $G$ is an element $0\leq s\in G$ such that, for all $0\leq a\leq s$,      $a\wedge(s-a)=0$.111In [5, Definition 11.2.7] one finds the additional assumption $s>0$. This will have no effect on the results of this paper. A Specker $\ell$-group $S$ is an $\ell$-group that is generated, as a group, by its singular elements. A unital Specker $\ell$-group is a pair $(S,u)$ where $S$ is a Specker $\ell$-group and $u$ is a unit of $S$.

By [4, Lemma 2.11], weak and strong units coincide in Specker $\ell$-groups. Any unital Specker $\ell$-group $(S,u)$ has a greatest singular element, denoted $s_{S}$. This immediately follows, e.g., from [4, Lemma 2.11 and Corollary 2.12]. We let

[TABLE]

denote the full subcategory of unital $\ell$-groups given by unital Specker $\ell$-groups and unital $\ell$-homomorphisms.

For any unital $\ell$-group $(G,u)$, the space $\operatorname{{\boldsymbol{\mu}}}(G)$ of maximal ideals of $(G,u)$ is equipped with the maximal spectral topology inherited by restriction of the spectral topology from prime to maximal ideals. In particular, by definition, a subbasis (in fact, a basis) of open sets in $\operatorname{{\boldsymbol{\mu}}}(G)$ is given by the family of sets of the form

[TABLE]

letting $g$ range over all elements of $G$. A basis of closed sets is then given by the family of sets of the form

[TABLE]

for all $g\in G$. See [5, Definition 10.1.2 and §10.2]. The closed sets in $\operatorname{{\boldsymbol{\mu}}}(G)$ are precisely the sets of the form $\{\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(G)\mid\mathfrak{j}\subseteq\mathfrak{m}\}$, for $\mathfrak{j}$ a (possibly improper) ideal of $G$. See [5, Proposition 10.1.7].222In Corollary 2.10, we will observe that the assignment $\mathfrak{j}\mapsto\{\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(G)\mid\mathfrak{j}\subseteq\mathfrak{m}\}$ from the set of possibly improper ideals of $(G,u)$ to the set of closed subsets of $\operatorname{{\boldsymbol{\mu}}}(G)$ is a bijection (in fact, an order anti-isomorphism), with inverse $C\mapsto\bigcap C$.

Lemma 2.2**.**

For any totally ordered nontrivial Specker $\ell$-group $S$ there is a unique $\ell$-isomorphism of $S$ onto $\mathbb{Z}$, once the latter is equipped with the natural order.

Proof.

This immediately follows from the fact that any two nonzero singular elements of $S$ coincide. As a matter of fact, let $a$ and $b$ be nonzero singular elements. Since $S$ is totally ordered, either $a\leq b$ or $b\leq a$. Without loss of generality, $a\leq b$. Since $b$ is singular, $a\land(b-a)=0$. Since $S$ is totally ordered, either $a=0$ or $b-a=0$. Since $a$ is nonzero, $b-a=0$, i.e., $a=b$. ∎

Lemma 2.3**.**

Any surjective $\ell$-homomorphism between $\ell$-groups maps singular elements to singular elements. Therefore, homomorphic images (i.e., quotients) of Specker $\ell$-groups are Specker $\ell$-groups.

Proof.

Let $f\colon S\twoheadrightarrow T$ be a surjective $\ell$-homomorphism between $\ell$-groups. Let $s\in S$ be a singular element. We show that $f(s)$ is a singular element. From $s\geq 0$ we get $f(s)\geq 0$. For any $b\in T$ such that $0\leq b\leq f(s)$ let us prove $b\land(f(s)-b)=0$. By surjectivity, there is $a\in S$ such that $f(a)=b$. Set $a^{\prime}=(a\lor 0)\land s$. Then, $0\leq a^{\prime}\leq s$, and hence, since $s$ is a singular element, we have $a^{\prime}\land(s-a^{\prime})=0$. Therefore, $f(a^{\prime})\land(f(s)-f(a^{\prime}))=0$. Moreover, $f(a^{\prime})=f((a\lor 0)\land s)=(f(a)\lor 0)\land f(s)=f(a)$, whence $f(a)\land(f(s)-f(a))=0$. This proves that $f$ maps singular elements to singular elements. The rest is immediate. ∎

Theorem 2.4**.**

Let $S$ be a Specker $\ell$-group and $\mathfrak{m}$ a maximal ideal of $S$. Then there is a unique $\ell$-homomorphism $\rho_{\mathfrak{m}}$ of $S$ onto $\mathbb{Z}$ whose kernel (i.e., the preimage of $\{0\}$) is $\mathfrak{m}$.

Proof.

Since any maximal ideal of $S$ is prime ([5, Corollary 9.2.4]), $S/\mathfrak{m}$ is a nontrivial totally ordered $\ell$-group. By Lemma 2.3, $S/\mathfrak{m}$ is a Specker $\ell$-group. By Lemma 2.2, there is a unique $\ell$-isomorphism $\iota_{\mathfrak{m}}\colon S/\mathfrak{m}\to\mathbb{Z}$. ∎

Lemma 2.5**.**

Let $(S,u)$ be a unital Specker $\ell$-group, $\mathfrak{m}$ a maximal ideal, $\rho_{\mathfrak{m}}\colon S\to\mathbb{Z}$ the unique surjective $\ell$-homomorphism whose kernel is $\mathfrak{m}$, and $s_{S}$ the greatest singular element of $S$. We then have:

1. (i)

$\rho_{\mathfrak{m}}(s_{S})=1$. 2. (ii)

For every $g\in S$, $\rho_{\mathfrak{m}}(g)$ is the unique $j\in\mathbb{Z}$ such that $g-js_{S}\in\mathfrak{m}$.

Proof.

(i) By Lemma 2.3, $\rho_{\mathfrak{m}}\colon S\to\mathbb{Z}$ preserves singular elements. Then any singular element of $S$ is mapped either to [math] or to $1$. Since the set of singular elements of $S$ generates $S$ and $\rho_{\mathfrak{m}}$ is surjective, the image under $\rho_{\mathfrak{m}}$ of the set of singular elements of $S$ generates $\mathbb{Z}$. Therefore, there is a singular element of $S$ whose image under $\rho_{\mathfrak{m}}$ is $1$. Since $\rho_{\mathfrak{m}}$ is order-preserving, the image under $\rho_{\mathfrak{m}}$ of the greatest singular element of $S$ is $1$.

(ii) This immediately follows from (i). ∎

Throughout, the set $\mathbb{Z}$ is equipped with the discrete topology. For any boolean space $X$ and continuous function $u\colon X\to\mathbb{Z}_{>0}$ we will write

[TABLE]

to denote the unital $\ell$-group $C_{X}$ of all continuous $\mathbb{Z}$-valued functions on $X$, equipped with the unit $u$. From the compactness of $X$ and the discreteness of the topology of $\mathbb{Z}$ it follows that the range of $u$ (and in general of any continuous function from $X$ to $\mathbb{Z}_{>0}$) is finite.

Notation 2.6**.**

Let $(S,u)$ be a unital Specker $\ell$-group. For any $g\in S$ we define the map

[TABLE]

where $\rho_{\mathfrak{m}}$ is the unique surjective $\ell$-homomorphism from $S$ to $\mathbb{Z}$ with kernel $\mathfrak{m}$, as given by Theorem 2.4. In other words (by Lemma 2.5),

[TABLE]

where $s_{S}$ is the greatest singular element of $S$.

Lemma 2.7**.**

Let $(S,u)$ be a unital Specker $\ell$-group. For any $g\in S$ and any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(S)$, we have $g^{\natural}(\mathfrak{m})=0$ if and only if $g\in\mathfrak{m}$.

Proof.

The value $g^{\natural}(\mathfrak{m})=\rho_{\mathfrak{m}}(g)$ is [math] if and only if $g\in\ker(\rho_{\mathfrak{m}})=\mathfrak{m}$. ∎

Theorem 2.8**.**

Let $(S,u)$ be a unital Specker $\ell$-group and $s_{S}$ its greatest singular element. We then have:

1. (i)

For any $\mathfrak{n}\in\operatorname{{\boldsymbol{\mu}}}(S)$ there is a unique unital $\ell$-isomorphism of the maximal quotient $(S,u)/\mathfrak{n}$ onto the unital $\ell$-group $(\mathbb{Z},u^{\natural}_{S}(\mathfrak{n}))$. 2. (ii)

For any $g\in S$, the map $g_{S}^{\natural}\colon\operatorname{{\boldsymbol{\mu}}}(S)\to\mathbb{Z}$ is continuous, i.e. $g^{\natural}_{S}\in C_{\operatorname{{\boldsymbol{\mu}}}(S)}$. 3. (iii)

$\operatorname{{\boldsymbol{\mu}}}(S)$* is a boolean space, and the map    ${}^{\natural}\colon g\in(S,u)\mapsto g_{S}^{\natural}\in\mathbb{Z}^{\operatorname{{\boldsymbol{\mu}}}(S)}$ is a unital $\ell$-isomorphism of $(S,u)$ onto $(C_{\operatorname{{\boldsymbol{\mu}}}(S)},u^{\natural})$.*333For a different proof see [29, Proposition 2.7(ii)], which makes use of [29, Lemma 2.6]. In this latter lemma the obviously necessary assumption that $C_{X}$ is separating (i.e., that the compact space $X$ is a boolean space) was inadvertently omitted. This has no effect on the results of that paper. 4. (iv)

For any boolean space $X$ the singular elements of $C_{X}$ are precisely the characteristic functions of clopen subsets of $X$. Consequently, the set   $\mathsf{sing}(S)$    of singular elements of $(S,u)$, with the lattice structure inherited from $S$ by restriction, is a boolean algebra with top element $s_{S}$ and bottom element the zero element of $S$. This boolean algebra is isomorphic to the algebra of clopen sets of $\operatorname{{\boldsymbol{\mu}}}(S)$. An isomorphism is given by the map

[TABLE] 5. (v)

Every boolean space is homeomorphic to the maximal spectral space of some unital Specker $\ell$-group.

Proof.

(i) Immediate from Lemma 2.5.

(ii) Pick a basic open set $C\subseteq\mathbb{Z}$, say $C=\{c\},\,\,c\in\mathbb{Z}$, with the intent of showing that $(g_{S}^{\natural})^{-1}(C)$ is an open subset of $\operatorname{{\boldsymbol{\mu}}}(S)$. As a matter of fact,

[TABLE]

Since $(S,u)$ is a unital Specker $\ell$-group, every element $l\in S$ can be written as a linear combination of finitely many singular elements $s_{1},\dots,s_{n}\in S$ with integer coefficients $k_{1},\dots,k_{n}$. For any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(S)$ and $i\in\{1,\dots,n\}$ the element ${s_{i}}^{\natural}_{S}(\mathfrak{m})$ is a singular element of $\mathbb{Z}$, and hence it is either [math] or $1$; this is so because ${s_{i}}^{\natural}_{S}(\mathfrak{m})$ is the image of $s_{i}$ under the surjective $\ell$-homomorphism $\rho_{\mathfrak{m}}$ of Theorem 2.4, which preserves singular elements by Lemma 2.3. Therefore, the range of the map $l_{S}^{\natural}\colon\operatorname{{\boldsymbol{\mu}}}(S)\to\mathbb{Z}$ is contained in $\{\sum_{i=1}^{n}k_{i}e_{i}\mid(e_{i})_{i=1}^{n}\in\{0,1\}^{n}\}$, which is a finite subset of $\mathbb{Z}$. This holds in the particular case when $l=g-cs_{S}$. By definition of $\operatorname{{\boldsymbol{\mu}}}(S)$, the set $\{\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(S)\mid{g-cs_{S}}\in{\mathfrak{m}}\}$ is a (subbasic) closed subset of $\operatorname{{\boldsymbol{\mu}}}(S)$. It is also an open subset of $\operatorname{{\boldsymbol{\mu}}}(S)$, because, as we have just seen, the map $(g-cs_{S})^{\natural}\colon\operatorname{{\boldsymbol{\mu}}}(S)\to\mathbb{Z}$ has only finitely many values, and each of them is attained over a closed subset of $\operatorname{{\boldsymbol{\mu}}}(S)$. Hence, each fiber of $(g-cs_{S})^{\natural}$ is clopen. We conclude that $(g^{\natural}_{S})^{-1}(C)=\{\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(S)\mid g^{\natural}_{S}(\mathfrak{m})=c\}$ is an open set of $\operatorname{{\boldsymbol{\mu}}}(S)$, as desired to prove that $g^{\natural}_{S}$ is continuous.

(iii) The boolean space $X$ in the representation [4, Corollary 2.12] of any unital Specker $\ell$-group $(G,u)$ is homeomorphic to the maximal spectral space of $(G,u)$. This follows, e.g., from [26, Theorem 4.16(iv)]. Surjectivity follows from [4, Corollary 2.12], or [5, Theorem 13.5.3].

(iv) That the singular elements of $C_{X}$ are precisely the characteristic functions of the clopen subsets of $X$ is straightforward. By (iii), $(S,u)$ is isomorphic to $(C_{\operatorname{{\boldsymbol{\mu}}}(S)},u^{\natural})$. The singular elements of $C_{\operatorname{{\boldsymbol{\mu}}}(S)}$ are the characteristic functions of the clopen sets of $\operatorname{{\boldsymbol{\mu}}}(S)$. By Lemma 2.7, for every $g\in S$ the support of the function $g^{\natural}\in C_{\operatorname{{\boldsymbol{\mu}}}(S)}$ is $\{\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(S)\mid g\notin\mathfrak{m}\}$. Therefore, for every singular element $s\in S$, $s^{\natural}$ is the characteristic function of $\{\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(S)\mid s\notin\mathfrak{m}\}$. (Compare with [5, Theorem 13.5.3].)

(v) Let $X$ be a boolean space. Let $(C_{X},1)$ be the unital Specker $\ell$-group of all continuous integer-valued functions on $X$, with the constant function 1 over $X$ as the distinguished unit of $C_{X}$. Since this is also the greatest singular element of $C_{X}$, from the main result of [28] it follows that the $\Gamma$ functor transforms $(C_{X},1)$ into the boolean algebra $B$ consisting of all continuous $\{0,1\}$-valued functions on $X$. Both $(C_{X},1)$ and $B$ are separating. As a particular case of [26, Theorem 4.16(iv)], $X$ is homeomorphic to $\operatorname{{\boldsymbol{\mu}}}(B)$. By [12, Theorem 7.2.2], the preservation properties of $\Gamma$ now yield a homeomorphism $\operatorname{{\boldsymbol{\mu}}}(C_{X})\cong\operatorname{{\boldsymbol{\mu}}}(B)$. In conclusion, we have the desired homeomorphisms $X\cong\operatorname{{\boldsymbol{\mu}}}(C_{X})\cong\operatorname{{\boldsymbol{\mu}}}(B).$ ∎

Corollary 2.9** ([15, Proposition 3.4]).**

Every unital Specker $\ell$-group $(S,u)$ is hyperarchimedean, in the sense that any prime ideal of $(S,u)$ is maximal.

Proof.

In view of Theorem 2.8(iii), it suffices to argue assuming that $(S,u)$ has the form $(C_{X},u_{X})$ for some boolean multispace $(X,u_{X})$. Since every map $h\in C_{X}$ is integer-valued and has a finite range, for any $0\leq f,g\in C_{X}$ there is $n\in\mathbb{Z}_{>0}$ such that

[TABLE]

By [5, Theorem 14.1.2 (i)$\Leftrightarrow$(vi)], this condition is equivalent to saying that $S$ is hyperarchimedean. ∎

Corollary 2.10**.**

Let $(S,u)$ be a unital Specker $\ell$-group. Every ideal of $S$ is the intersection of all maximal ideals of $S$ containing it.444This is true also for the improper ideal, as long as one uses the convention that the intersection of an empty family of subsets of $S$ is $S$.

Proof.

This follows from Corollary 2.9, since in any $\ell$-group every ideal is the intersection of all prime ideals containing it. ∎

3. Boolean multispaces and the functor $\SS$

Definition 3.1**.**

The category $\mathsf{Bms}$ of boolean multispaces is defined by:

$\mathsf{Bms}$-$\mathsf{OBJECT}$: A pair $(X,u_{X})$ with $X$ a boolean space and $u_{X}$ a continuous function from $X$ to $\mathbb{Z}_{>0}=\{1,2,\dots\}$.

For $x\in X$, $u(x)$ is called the multiplicity of $x$. The function $u$ is called the multiplicity (function) on $X$.

$\mathsf{Bms}$-$\mathsf{MORPHISM}$ from a $\mathsf{Bms}$-object $(X,u_{X})$ to a $\mathsf{Bms}$-object $(Y,u_{Y})$: a continuous function $\gamma\colon X\to Y$ such that the function $\zeta_{\gamma}$ defined by

[TABLE]

is $\mathbb{Z}_{>0}$-valued. In other words, for all $x\in X$, the positive integer $u_{Y}(\gamma(x))$ divides $u_{X}(x)$ (i.e., $\gamma$ decreases multiplicity with respect to the divisibility order of $\mathbb{Z}_{>0}$). We say that $\zeta_{\gamma}$ is the multiplicity of $\gamma$. Note that $\zeta_{\gamma}$ is continuous.

The composition of $\mathsf{Bms}$-morphisms $\eta_{1}\colon(X,u_{X})\to(Y,u_{Y})$ and $\eta_{2}\colon(Y,u_{Y})\to(Z,u_{Z})$ is the morphism $\eta_{2}(\eta_{1})\colon(X,u_{X})\to(Z,u_{Z})$.

We recall that every continuous function between compact Hausdorff spaces (in particular, every $\mathsf{Bms}$-morphism) is a closed map, meaning that it maps closed sets to closed sets [18, Theorem 3.1.12]. Therefore, a $\mathsf{Bms}$-isomorphism from $(X,u_{X})$ to $(Y,u_{Y})$ amounts to a continuous bijection $\eta\colon X\to Y$ such that $u_{Y}(\eta(x))=u_{X}(x)$ for all $x\in X$.

Definition 3.2**.**

1. (i)

The functor $\mathcal{S}\colon\mathsf{Bms}\to\mathsf{uS{\ell}g}^{\mathrm{op}}$ sends every $\mathsf{Bms}$-object $(X,u)$ to the $\mathsf{uS{\ell}g}$-object $(C_{X},u)$:

[TABLE] 2. (ii)

Furthermore, $\mathcal{S}$ transforms any $\mathsf{Bms}$-morphism $\gamma\colon(W,u_{W})\to(V,u_{V})$ into the unital $\ell$-homomorphism $\mathcal{S}_{\gamma}\colon\mathcal{S}(V,u_{V})\to\mathcal{S}(W,u_{W})$ given by

[TABLE]

This is indeed a well-defined assignment on morphisms:

Proposition 3.3**.**

Adopt the above notation.

1. (i)

$\mathcal{S}_{\gamma}(u_{V})=u_{V}(\gamma)\frac{u_{W}}{u_{V}(\gamma)}=u_{W}$, whence $\mathcal{S}_{\gamma}$ preserves units. 2. (ii)

$\mathcal{S}_{\gamma}(0)=0$. 3. (iii)

$\mathcal{S}_{\gamma}(f+g)=\mathcal{S}_{\gamma}(f)+\mathcal{S}_{\gamma}(g)$, and the like for $\vee,\wedge$ and $-$, which shows that $\mathcal{S}_{\gamma}$ is a unital $\ell$-homomorphism of $(C_{V},u_{V})$ into $(C_{W},u_{W})$. ∎

Proposition 3.4**.**

The functor $\mathcal{S}$ is essentially surjective, in the sense that for any $(S,u)\in\mathsf{uS{\ell}g}^{\mathrm{op}}$ there is $(X,v)\in\mathsf{Bms}$ such that $\mathcal{S}(X,v)\cong(S,u)$.

Proof.

By Theorem 2.8(iii) we have $(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural})\in\mathsf{Bms}$ and $(S,u)\cong\mathcal{S}(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural})$, whence $(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural})$ is the desired $\mathsf{Bms}$-object. ∎

Proposition 3.5**.**

The functor $\mathcal{S}$ is faithful: For any two distinct $\mathsf{Bms}$-morphisms $\gamma\not=\gamma^{\prime}\colon(W,u_{W})\to(V,u_{V})$, the unital $\ell$-homomorphisms $\mathcal{S}_{\gamma}$ and $\mathcal{S}_{\gamma^{\prime}}$ are distinct.

Proof.

$\gamma\not=\gamma^{\prime}$ means that there is $y\in W$ with $\gamma(y)\not=\gamma^{\prime}(y)$. Since $\gamma(y)$ and $\gamma^{\prime}(y)$ are distinct and $C_{V}$ is separating, there is a map $f\in C_{V}$ with $f(\gamma(y))=0$ and $f(\gamma^{\prime}(y))\not=0$. Now

[TABLE]

which shows that $\mathcal{S}_{\gamma}\not=\mathcal{S}_{\gamma^{\prime}}$. ∎

Lemma 3.6**.**

1. (i)

For $X$ a fixed but otherwise arbitrary boolean space, let $(C_{X},u)$ be the unital Specker $\ell$-group of all $\mathbb{Z}$-valued continuous functions on $X$, with the distinguished order unit $u$. Letting $Z_{X}f=\{x\in X\mid f(x)=0\}$, the map $\dot{Z}_{X}\colon\operatorname{{\boldsymbol{\mu}}}(C_{X})\to X$ given by

[TABLE]

is a homeomorphism of $\operatorname{{\boldsymbol{\mu}}}(C_{X})$ onto $X$. Its inverse $\mathfrak{M}_{X}$ maps any $x\in X$ to the maximal ideal $\mathfrak{M}_{X}(x)=\mathfrak{m}_{x}$ of $C_{X}$ given by all functions of $C_{X}$ vanishing at $x$. 2. (ii)

For every $x\in X$ and $g\in C_{X}$ we have the identity $g(x)=g^{\natural}(\mathfrak{M}_{X}(x))$. Thus the functions $\mathfrak{M}_{X}$ and $\dot{Z}_{X}$ yield $\mathsf{Bms}$-isomorphisms between $(X,u)$ and $(\operatorname{{\boldsymbol{\mu}}}(C_{X}),u^{\natural})$.

Proof.

(i) For every $x\in X$ one easily sees that $\mathfrak{M}_{X}(x)$ is a maximal ideal of $C_{X}$. Moreover, we have the identity

[TABLE]

For the $\subseteq$-inclusion, since $X$ is a boolean space, for every $y\in X$ different from $x$ there is $f\in C_{X}$ such that $f(x)=0$ and $f(y)\neq 0$. The $\supseteq$-inclusion is trivial.

We now proceed as follows:

Claim: For any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(C_{X})$ and $x\in\bigcap\{Z_{X}f\mid f\in\mathfrak{m}\}$ we have

[TABLE]

and

[TABLE]

Indeed, by hypothesis $\mathfrak{m}\subseteq\mathfrak{M}_{X}(x)$, whence (6) follows from $\mathfrak{m}$ being maximal. Then (7) follows from (6) and (5). Our claim is thus settled.

Next, we must prove that $\dot{Z}_{X}$ is well-defined. To this purpose, let $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(C_{X})$, with the intent of proving that the set $\bigcap\{Z_{X}f\mid f\in\mathfrak{m}\}$ is a singleton. By our claim, $\bigcap\{Z_{X}f\mid f\in\mathfrak{m}\}$ has at most one element. Let us prove that it has at least one element. For every finite subset $S\subseteq\mathfrak{m}$, we have

[TABLE]

which is nonempty because $\sum_{f\in S}\lvert f\rvert\in\mathfrak{m}$ and $\mathfrak{m}$ is proper. The compactness of $X$ now yields $\bigcap\{Z_{X}f\mid f\in\mathfrak{m}\}\neq\emptyset$. In conclusion, $\bigcap\{Z_{X}f\mid f\in\mathfrak{m}\}$ is a singleton, and $\dot{Z}_{X}$ is well-defined.

From (5) it follows that, for all $x\in X$,    $x=\dot{Z}_{X}\mathfrak{M}_{X}(x)$. On the other hand, by our claim, for all $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(C_{X})$,    $\mathfrak{m}=\mathfrak{M}_{X}\dot{Z}_{X}\mathfrak{m}$. We have thus shown that $\dot{Z}_{X}$ and $\mathfrak{M}_{X}$ are mutually inverse functions, yielding a bijection between $\operatorname{{\boldsymbol{\mu}}}(C_{X})$ and $X$.

There remains to prove that $\mathfrak{M}_{X}$ is a homeomorphism. Since $\mathfrak{M}_{X}$ is a bijection between compact Hausdorff spaces, it is enough to prove that $\mathfrak{M}_{X}$ is continuous. Let $C$ be a subbasic closed set in $\operatorname{{\boldsymbol{\mu}}}(C_{X})$, say $C=F_{f}=$ the set of all maximal ideals of $C_{X}$ to which $f$ belongs, where $f$ is a fixed but otherwise arbitrary element of $C_{X}$. Then

[TABLE]

a closed set by the continuity of $f$. This shows that $\mathfrak{M}_{X}$ is a homeomorphism.

(ii) For any $x\in X$ and $g\in C_{X}$ we will prove $g(x)=g^{\natural}(\mathfrak{M}_{X}(x))$. The evaluation function

[TABLE]

is surjective and its kernel coincides with $\mathfrak{M}_{X}(x)$. Thus the evaluation function coincides with the function $\rho_{\mathfrak{M}_{X}(x)}$ in Theorem 2.4 and in the definition of ♮ (Notation 2.6). Therefore, for every $g\in C_{X}$,

[TABLE]

as desired. In particular, $u(x)=u^{\natural}(\mathfrak{M}_{X}(x))$, whence both $\mathfrak{M}_{X}$ and its inverse $\dot{Z}_{X}$ are $\mathsf{Bms}$-isomorphisms. ∎

In what follows we will need the result stating that the preimage under a unital $\ell$-homomorphism of a maximal ideal of a unital $\ell$-group is maximal. While this is folklore, we will give a self-contained proof in Corollary 3.8. As a preliminary step, in the following lemma we will give an elementary characterization of maximal ideals. For readers familiar with commutative algebra, we mention that this characterization can be seen as an analogue in our setting of the following characterization of maximal ideals in a commutative ring $R$ with unit: an ideal $\mathfrak{j}$ is maximal if and only if $1\notin\mathfrak{j}$ and for all $a\in R\setminus\mathfrak{j}$ there is $b\in R$ such that $1-ba\in\mathfrak{j}$ (i.e., $R/\mathfrak{j}$ is a field) [3, p. 3].

Lemma 3.7**.**

Let $(G,u)$ be a unital $\ell$-group and $\mathfrak{j}$ a (possibly improper) ideal of $G$. The following conditions are equivalent:555It is well known that these conditions are also equivalent to the existence of an injective unital $\ell$-homomorphism from $(G/\mathfrak{m},u/\mathfrak{m})$ to $(\mathbb{R},1)$. Moreover, if $G$ is a Specker $\ell$-group, these conditions are also equivalent to $G/\mathfrak{j}$ being isomorphic to $\mathbb{Z}$ as an $\ell$-group (the nontrivial direction being Theorem 2.4).

1. (i)

$\mathfrak{j}$* is maximal.* 2. (ii)

$u\notin\mathfrak{j}$* (i.e., $\mathfrak{j}$ is proper) and for all $a\in G\setminus\mathfrak{j}$ there is $n\in\mathbb{Z}_{\geq 0}$ such that $(u-n\lvert a\rvert)\lor 0\in\mathfrak{j}$ (i.e., $n\lvert a\rvert/\mathfrak{j}\geq u/\mathfrak{j}$).*

Proof.

The ideal $\mathfrak{j}$ is maximal if and only if it is proper (a condition that is easily seen to be equivalent to $u\notin\mathfrak{j}$) and, for all $a\in G\setminus\mathfrak{j}$, the ideal generated by $\mathfrak{j}\cup\{a\}$ is $G$ (i.e., it contains $u$). The ideal generated by $\mathfrak{j}\cup\{a\}$ contains $u$ if and only if there are $n\in\mathbb{Z}_{\geq 0}$ and $j\in\mathfrak{j}$ such that $u\leq j+n\lvert a\rvert$, which is the case if and only if there is $n\in\mathbb{Z}_{\geq 0}$ such that $u-n\lvert a\rvert$ is below some element of $\mathfrak{j}$, which is equivalent to $(u-n\lvert a\rvert)\lor 0\in\mathfrak{j}$. ∎

While for commutative rings with unit it is not true that the preimage of a maximal ideal is a maximal ideal (as for the inclusion $\mathbb{Z}\hookrightarrow\mathbb{Q}$), for unital $\ell$-groups we have:

Corollary 3.8** (Preimage of maximal is maximal).**

Let $\psi\colon(G,u)\to(H,v)$ be a unital $\ell$-homomorphism between unital $\ell$-groups. For every maximal ideal $\mathfrak{m}$ of $H$ the set $\psi^{-1}(\mathfrak{m})=\{g\in G\mid\psi(g)\in\mathfrak{m}\}$ is a maximal ideal of $G$.

Proof.

This is an immediate consequence of Lemma 3.7. (For a proof in the context of MV-algebras see [12, Proposition 1.2.2]. For an alternative proof, combine [12, Proposition 1.2.16(i)] with [12, Corollary 7.2.3].) ∎

Lemma 3.9**.**

Let $\psi\colon(S,u)\to(T,v)$ be a unital $\ell$-homomorphism between unital Specker $\ell$-groups. The map

[TABLE]

is continuous. Moreover, for any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(T)$ there is $k\in\mathbb{Z}_{>0}$ such that for all $g\in S$ we have the identity $(\psi(g))^{\natural}(\mathfrak{m})=kg^{\natural}(\psi^{-1}(\mathfrak{m}))$. In particular, $v^{\natural}(\mathfrak{m})=ku^{\natural}(\psi^{-1}(\mathfrak{m}))$, whence $\psi^{-1}$ is a $\mathsf{Bms}$-morphism.

Proof.

The map is well-defined because, by Corollary 3.8, for any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(T)$ the set $\psi^{-1}(\mathfrak{m})=\{f\in S\mid\psi(f)\in\mathfrak{m}\}$ is a maximal ideal of $S$.

We next prove the continuity of $\psi^{-1}$. We recall that a subbasis for the closed sets of $\operatorname{{\boldsymbol{\mu}}}(S)$ is given by the sets of the form

[TABLE]

for $f$ ranging over arbitrary elements of $S$ (and analogously for $T$, mutatis mutandis). To prove that $\psi^{-1}\colon\operatorname{{\boldsymbol{\mu}}}(T)\to\operatorname{{\boldsymbol{\mu}}}(S)$ is continuous it is enough to prove that the preimage under the function $\psi^{-1}$ of any element of this subbasis for the closed sets of $\operatorname{{\boldsymbol{\mu}}}(S)$ is a closed subset of $\operatorname{{\boldsymbol{\mu}}}(T)$. Let $f\in S$. We have

[TABLE]

a closed subset of $\operatorname{{\boldsymbol{\mu}}}(T)$. This concludes the proof of the continuity of the map $\psi^{-1}\colon\operatorname{{\boldsymbol{\mu}}}(T)\to\operatorname{{\boldsymbol{\mu}}}(S)$.

There remains to prove

[TABLE]

Let $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(T)$. By Theorem 2.4, there is a unique surjective $\ell$-homomorphism $\rho_{\mathfrak{m}}\colon T\to\mathbb{Z}$ such that $\ker\rho_{\mathfrak{m}}=\mathfrak{m}$, and there is a unique surjective $\ell$-homomorphism $\rho_{\psi^{-1}(\mathfrak{m})}\colon S\to\mathbb{Z}$ such that $\ker\rho_{\psi^{-1}(\mathfrak{m})}=\psi^{-1}(\mathfrak{m})$. Since $\psi^{-1}(\mathfrak{m})$ is the kernel of the surjective $\ell$-homomorphism $\rho_{\psi^{-1}(\mathfrak{m})}\colon S\to\mathbb{Z}$ and is also the kernel of the composite $S\xrightarrow{\psi}T\xrightarrow{\rho_{\mathfrak{m}}}\mathbb{Z}$ (which need not be surjective), there is a unique $\ell$-homomorphism $\gamma\colon\mathbb{Z}\to\mathbb{Z}$ such that the following diagram commutes:

[TABLE]

Any $\ell$-homomorphism from $\mathbb{Z}$ to $\mathbb{Z}$ amounts to multiplication by a number in $\mathbb{Z}_{\geq 0}$. Therefore, there is $k\in\mathbb{Z}_{\geq 0}$ such that, for every $j\in\mathbb{Z}$, $\gamma(j)=kj$.

It remains to show that $k\not=0$. Since surjective $\ell$-homomorphisms preserve (strong order) units, $\rho_{\mathfrak{m}}(v)$ is a (strong order) unit of $\mathbb{Z}$, and hence it is nonzero. We have

[TABLE]

Therefore, $k\neq 0$, as desired. This settles (8) and completes the proof. ∎

Corollary 3.10** (of  Lemmas 3.6 and 3.9).**

For any boolean multispaces $(W,u_{W})$ and $(V,u_{V})$, and  $\mathsf{uS{\ell}g}$-morphism

[TABLE]

let   $\psi_{\rm dual}\colon W\to V$   be the composite function $\dot{Z}_{V}\psi^{-1}\mathfrak{M}_{W}$. In more detail,

[TABLE]

Then   $\psi_{\rm dual}$ is a $\mathsf{Bms}$-morphism of $(W,u_{W})$ into $(V,u_{V})$. $\Box$

4. $\SS$ is a categorical equivalence

Proposition 4.1** ($\mathcal{S}$ is full).**

For any boolean multispaces $(W,u_{W})$ and $(V,u_{V})$ and unital $\ell$-homomorphism

[TABLE]

there exists a (unique by the faithfulness of $\mathcal{S}$, Proposition 3.5) $\mathsf{Bms}$-morphism

[TABLE]

such that $\psi=\mathcal{S}_{\gamma}$. Specifically, $\gamma$ coincides with the map $\psi_{\rm dual}$.

Proof.

As per Corollary 3.10, the $\mathsf{Bms}$-morphism $\psi_{\rm dual}$ is the composition of the $\mathsf{Bms}$-morphisms

[TABLE]

To get the identity $\SS_{\psi_{\rm dual}}=\psi$ we must prove

[TABLE]

To this purpose, let us arbitrarily fix $f\in C_{V}$ and $x\in W$. By the definition of the functor $\SS$ on morphisms we can write

[TABLE]

There remains to prove

[TABLE]

By definition of $\psi_{\rm dual}$, the element $\psi_{\rm dual}(x)$ is the unique $y\in V$ such that for any $h\in C_{V}$,     $\psi(h)(x)=0$  implies   $h(y)=0$. Since $u_{W}(x)>0$,

[TABLE]

We then have the identities

[TABLE]

As a consequence,

[TABLE]

We have just proved

[TABLE]

which amounts to (9). ∎

Theorem 4.2**.**

The functor $\mathcal{S}$ is an equivalence between the category $\mathsf{Bms}$ of boolean multispaces and the opposite $\mathsf{uS{\ell}g}^{\rm op}$ of the category of unital Specker $\ell$-groups.

Proof.

As a joint effect of Propositions 3.4, 3.5, 4.1. ∎

Corollary 4.3**.**

Stone duality is the particular case of Theorem 4.2 when $\mathcal{S}$ is restricted to boolean multispaces with constant multiplicity $1$, and $\mathsf{uS{\ell}g}$ is restricted to the category of unital Specker $\ell$-groups whose distinguished unit is singular.

Proof.

In [28] it was proved that, by restriction, the functor $\Gamma$ of [25, §3] yields a categorical equivalence between unital Specker $\ell$-groups whose distinguished unit is singular, and boolean algebras. By Stone duality, the category $\mathsf{BA}$ of boolean algebras and their homomorphisms is dual to the full subcategory $\mathsf{Bms1}$ of boolean multispacs of the form $(X,\boldsymbol{1})$ with $\boldsymbol{1}$ the constant function $1$ over the boolean space $X$. Trivially, $\mathsf{Bms1}$ is categorically equivalent to $\mathsf{BA}$. ∎

Definition 4.4**.**

A Specker MV-algebra is an MV-algebra isomorphic to $\Gamma(S,u)$ for some unital Specker $\ell$-group $(S,u)$. We denote by

[TABLE]

the category of Specker MV-algebras with MV-algebraic homomorphisms.

Lemma 4.5** (Compare with [29, Theorem 4.5]).**

Up to isomorphism, any Specker MV-algebra $A$ is a finite product of MV-algebras of the form $C(X_{i},\mbox{\L }_{n(i)})$, where each  $X_{i}$  is a (nonempty) boolean space and  Łn(i)  is the Łukasiewicz chain with  $n(i)+1$ elements. Equivalently, $A$ is isomorphic to a finite product of tensor products $B_{i}\otimes$Łn(i), for boolean algebras $B_{i}$.

Proof.

By Theorem 2.8(iii) we may identify $(S,u)$ with $(C_{X},v)$, for some boolean space $X$, where $v\colon X\to\mathbb{Z}_{>0}$. The continuous map $v$ splits $(C_{X},v)$ into a finite product

[TABLE]

of unital Specker $\ell$-groups ($C_{X_{i}},v_{i})$, where $v_{i}$ takes a constant integer value (also denoted $v_{i}>0$) over each boolean space $X_{i}$. For short, $X_{1},\dots,X_{l}$ are the fibers of $v$. Direct inspection shows that $\Gamma$ transforms each ($C_{X_{i}},v_{i})$ into the MV-algebra of all continuous functions on $X_{i}$ taking values in Ł${}_{v_{i}}$. Now note that $\Gamma$ preserves finite products. For the second statement see [29, Corollary 4.6]. ∎

In [19, Cor. 4.5], the authors obtained an “extended Priestley” duality for MV-algebras; in this duality, the dual of an MV-algebra is the Priestley space dual to the underlying bounded distributive lattice of the MV-algebra, equipped with additional structure. The structures dual to MV-algebras according to this duality are called MV-spaces [19, Def. 4.4].

Proposition 4.6**.**

The category $\mathsf{Bms}$ of boolean multispaces is equivalent to the full subcategory of the category of MV-spaces consisting of those MV-spaces that correspond to Specker MV-algebras under the extended Priestley duality for MV-algebras in [19, Cor. 4.5].

Proof.

This follows from the duality between MV-algebras and MV-spaces in [19, Cor. 4.5] and the duality between boolean multispaces and unital Specker $\ell$-groups in Theorem 4.2. ∎

5. The functor $\SS$ as a part of a duality

Definition 5.1** ([34, Definitions 1.5.4, 1.4.1]).**

An equivalence of categories consists of functors $F\colon\mathsf{C}\leftrightarrows\mathsf{D}\penalty 10000\mskip 6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0muG$ together with natural isomorphisms

[TABLE]

Two functors $F\colon\mathsf{C}\leftrightarrows\mathsf{D}\penalty 10000\mskip 6.0mu plus 1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0muG$ are said to be part of an equivalence of $\mathsf{C}$ and $\mathsf{D}$ if there are natural isomorphisms $\eta$ and $\epsilon$ such that the quartet $(F,G,\eta,\epsilon)$ constitutes an equivalence of $\mathsf{C}$ and $\mathsf{D}$ in the above sense.

Thus in category theory the same term “equivalence” applies both to a functor $F$ and to a quartet $(F,G,\eta,\epsilon)$ with $F,G$ functors and $\epsilon,\eta$ natural isomorphisms.

We next introduce a functor which, in Theorem 5.6, will be shown to be a part with $\mathcal{S}$ of an equivalence between the categories $\mathsf{Bms}$ and $\mathsf{uS{\ell}g}^{\rm op}$.

Definition 5.2**.**

1. (i)

The functor $\mathcal{B}\colon\mathsf{uS{\ell}g}^{\rm op}\to\mathsf{Bms}$ transforms every unital Specker $\ell$-group $(S,u)$ into the boolean multispace

[TABLE]

As per Notation 2.6, the function $u^{\natural}\colon\operatorname{{\boldsymbol{\mu}}}(S)\to\mathbb{Z}$ maps any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(S)$ into the value at $u$ of the unique surjective $\ell$-homomorphism from $S$ to $\mathbb{Z}$ with kernel $\mathfrak{m}$; equivalently, into the unique integer $j=j_{u,\mathfrak{m}}$ such that $u-js_{S}\in\mathfrak{m}$. 2. (ii)

On any unital $\ell$-homomorphism $\psi\colon(S,u)\to(T,v)$ of unital Specker $\ell$-groups, the functor $\mathcal{B}$   sends $\psi$ to the morphism $\mathcal{B}_{\psi}\colon\operatorname{{\boldsymbol{\mu}}}(T)\to\operatorname{{\boldsymbol{\mu}}}(S)$ that maps any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(T)$ to the maximal ideal of $S$ given by $\psi^{-1}(\mathfrak{m})=\{f\in S\mid\psi(f)\in\mathfrak{m}\}$, as per Lemma 3.9.

A standard fact in category theory is that, for every full, faithful and essentially surjective functor $F\colon\mathsf{C}\to\mathsf{D}$ there is an equivalence $(F,G,\eta,\epsilon)$ with $F$ as the first component. For example, this is proved in the $(\Leftarrow)$-direction of [34, Theorem 1.5.9].

The proof runs as follows:

From the essential surjectivity of $F$ and the axiom of choice for classes one gets a family $(G_{Y})_{Y\in\mathsf{D}}$ of objects of $\mathsf{C}$ and a family $(\epsilon_{Y}\colon F(G_{Y})\to Y)_{Y\in\mathsf{D}}$ of isomorphisms in $D$. Using $F$ together with these two families, and the fact that $F$ is full and faithful, one constructs an equivalence $(F,G,\eta,\epsilon)$. In case we are already given a family $(G_{Y})_{Y\in\mathsf{D}}$ of objects of $\mathsf{C}$ and a family $(\epsilon_{Y}\colon F(G_{Y})\to Y)_{Y\in\mathsf{D}}$ of isomorphisms in $D$, one can dispense with both the essential surjectivity assumption for $F$ and the axiom of choice (still keeping the faithfulness and fullness assumptions), and jump directly to the point of the proof where one constructs the desired equivalence with these ingredients.

In more detail we have:

Lemma 5.3**.**

Let $F\colon\mathsf{C}\to\mathsf{D}$ be a full and faithful functor. For any $Y\in\mathsf{D}$ let $G_{Y}$ be an object of $\mathsf{C}$ and $\epsilon_{Y}\colon F(G_{Y})\to Y$ an isomorphism in $\mathsf{D}$. Then $(F,G,\eta,\epsilon)$ is a well-defined adjoint equivalence of categories666An adjoint equivalence of categories is an equivalence of categories $(F,G,\eta,\epsilon)$ that is also an adjunction; in other words, an adjunction where both the unit and the counit are natural isomorphisms. If two functors $F$ and $G$ are part of an equivalence $(F,G,\eta,\epsilon)$, they are also part of an adjoint equivalence. It suffices to appropriately modify either one of $\eta$ or $\epsilon$. See, e.g., [24, §IV.4] or [34, Proposition 4.4.5].

• •

$G\colon\mathsf{D}\to\mathsf{C}$* is the functor given by the following stipulations:*

—to any object $Y\in\mathsf{D}$ assign the object $G_{Y}$;

—to any morphism $g\colon Y\to Y^{\prime}$ in $\mathsf{D}$ assign the (existing and unique, by the fullness and the faithfulness of $F$) morphism $f\colon G_{Y}\to G_{Y^{\prime}}$ in $\mathsf{C}$ making the following diagram commute:

[TABLE]

• •

$(\epsilon_{Y}\colon F(G_{Y})\to Y)_{Y\in\mathsf{D}}$* is the given collection of isomorphisms, which is a natural transformation from $FG$ to $\mathrm{id}_{\mathsf{D}}$;*

• •

$(\eta_{X}\colon X\to G_{F(X)})_{X\in\mathsf{C}}$* is the natural transformation from $\mathrm{id}_{\mathsf{C}}$ to $GF$ whose component $\eta_{X}$ at $X$ is the (existing and unique, by the fullness and the faithfulness of $F$) morphism such that $F(\eta_{X})\colon F(X)\to F(G_{F(X)})$ is the inverse of $\epsilon_{F(X)}\colon F(G_{F(X)})\to F(X)$.*

Proof.

To see that $(F,G,\eta,\epsilon)$ is an equivalence, one can just follow the $(\Leftarrow)$-direction of the proof of [34, Theorem 1.5.9], starting from the line that reads:

“For each $\ell\colon d\to d^{\prime}$, Lemma 1.5.10 defines a unique morphism making the square […]”.

As observed in the proof of [24, Theorem IV.4 (iii)$\Rightarrow$(ii)], this equivalence is indeed an adjoint equivalence, because it satisfies the two triangle identities:

—The triangle identity $\epsilon_{F(X)}\circ F(\eta_{X})\,\,\,\,(=\epsilon_{F(X)}(F(\eta_{X})))\,\,=\,\,1_{F(X)}$, which holds by the definition of $\eta$;

—The triangle identity $G(\epsilon_{Y})\circ\eta_{G(Y)}=1_{G(Y)}$.

To prove this latter identity, since $F$ is a full and faithful functor it is enough to prove

[TABLE]

Applying to $\epsilon_{Y}$ the definition of $G$ on morphisms we obtain the following commutative diagram:

[TABLE]

Since $\epsilon_{Y}$ is an isomorphism, $FG(\epsilon_{Y})=\epsilon_{FG(Y)}$. The identity $\epsilon_{FG(Y)}\circ F(\eta_{G(Y)})=1_{FG(Y)}$ follows from the definition of $\eta_{G(Y)}$.

Having thus obtained the identity (11), the proof is complete. ∎

As we aim to establish a duality, we will need the following variant of Lemma 5.3:

Lemma 5.4**.**

Let $F\colon\mathsf{C}\to\mathsf{D}^{\mathrm{op}}$ be a full and faithful functor. For each $Y\in\mathsf{D}$ let $G_{Y}$ be an object of $\mathsf{C}$ and $\epsilon_{Y}\colon Y\to F(G_{Y})$ an isomorphism in $\mathsf{D}$. Then $(F,G,\eta,\epsilon)$ is an adjoint dual equivalence, where:

• •

$G\colon\mathsf{D}^{\mathrm{op}}\to\mathsf{C}$* is the functor given by the following stipulations:*

—to any object $Y\in\mathsf{D}$ assign the object $G_{Y}$;

—to any morphism $g\colon Y\to Y^{\prime}$ in $\mathsf{D}$ assign the uniquely determined morphism $f\colon G_{Y^{\prime}}\to G_{Y}$ in $\mathsf{C}$ making the following diagram commute:

[TABLE]

• •

$(\epsilon_{Y}\colon Y\to F(G_{Y}))_{Y\in\mathsf{D}}$* is the given collection of isomorphisms, which is a natural transformation from $\mathrm{id}_{\mathsf{D}}$ to $FG$;*

• •

$(\eta_{X}\colon X\to G_{F(X)})_{X\in\mathsf{C}}$* is the natural transformation from $\mathrm{id}_{\mathsf{C}}$ to $GF$ whose component $\eta_{X}$ at $X$ is the unique morphism such that $F(\eta_{X})\colon F(G_{F(X)})\to F(X)$ is the inverse of $\epsilon_{F(X)}\colon F(X)\to F(G_{F(X)})$. ∎*

Lemma 5.5**.**

For any morphism    $\psi\colon(S,u)\to(T,v)$    in  $\mathsf{uS{\ell}g}$, the (necessarily unique) morphism $\lambda\colon(\operatorname{{\boldsymbol{\mu}}}(T),v^{\natural})\to(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural})$ of Lemma 5.4 making the following diagram commute

[TABLE]

maps any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(T)$  to the set  $\psi^{-1}(\mathfrak{m})=\{f\in S\mid\psi(f)\in\mathfrak{m}\}$.

Proof.

By Theorem 2.8(iii), for every $(Q,w)$ in $\mathsf{uS{\ell}g}$ the map

[TABLE]

is an isomorphism in $\mathsf{uS{\ell}g}$. Therefore, the commutativity of (13) is equivalent to writing $\SS_{\lambda}=\natural\circ\psi\circ\natural^{-1}$. Since, by Propositions 3.5 and 4.1, $\SS$ is full and faithful, there is a unique morphism $\lambda\colon(\operatorname{{\boldsymbol{\mu}}}(T),v^{\natural})\to(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural})$ making (13) commute. Let $\lambda$ be such a morphism. Let $g\in S$, with the intent of proving $g\in\lambda(\mathfrak{m})\iff g\in\psi^{-1}(\mathfrak{m})$. We then have

[TABLE]

This latter equality will find use in the third line in the following chain of equivalences:

[TABLE]

This completes the proof. ∎

Applying this result to the full and faithful functor $\mathcal{S}\colon\mathsf{Bms}\to\mathsf{uS{\ell}g}^{\rm op}$ along with the collection of $\mathsf{uS{\ell}g}$-isomorphisms $(^{\natural}\colon(S,u)\to\SS(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural}))_{(S,u)\in\mathsf{uS{\ell}g}}$ (Theorem 2.8(iii)), we obtain:

Theorem 5.6**.**

The functors $\SS$ and $\mathcal{B}$ are part of an adjoint dual equivalence between the category $\mathsf{Bms}$ of boolean multispaces and the category $\mathsf{uS{\ell}g}$ of unital Specker $\ell$-groups, with the following unit and counit:

1. (i)

$(\mathfrak{M}_{X}\colon(X,u)\to(\operatorname{{\boldsymbol{\mu}}}(C_{X}),u^{\natural}))_{(X,u)\in\mathsf{Bms}}$* ;* 2. (ii)

$(^{\natural}\colon(S,u)\to(C_{\operatorname{{\boldsymbol{\mu}}}(S)},u^{\natural}))_{(S,u)\in\mathsf{uS{\ell}g}}$.

Proof.

We will use Lemma 5.4 replacing

• •

the category $\mathsf{C}$ by $\mathsf{Bms}$,

• •

the category $\mathsf{D}$ by $\mathsf{uS{\ell}g}$,

• •

the functor $F$ by $\SS$,

• •

the assignment $Y\in\mathsf{D}\mapsto G_{Y}\in\mathsf{C}$ by the assignment $(S,u)\in\mathsf{uS{\ell}g}\mapsto(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural})\in\mathsf{Bms}$, and

• •

the assignment $Y\mapsto(\epsilon_{Y}\colon Y\to F(G_{Y}))$ by the assignment $(S,u)\mapsto(\natural\colon(S,u)\to\SS(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural}))$.

By Propositions 3.5 and 4.1, $\SS$ is faithful and full. By Theorem 2.8(iii), for every object $(S,u)\in\mathsf{uS{\ell}g}$ the function ${}^{\natural}\colon(S,u)\to\SS(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural})$ is an isomorphism. In view of Lemma 5.4, a functor

[TABLE]

such that $\SS$ and $\mathcal{B}^{\prime}$ are part of a duality is obtainable by the following stipulations:

—let $\mathcal{B}^{\prime}$ send any object $(S,u)\in\mathsf{uS{\ell}g}$ to the object $(\operatorname{{\boldsymbol{\mu}}}(S),u^{\natural})$, and

—let $\mathcal{B}^{\prime}$ send any morphism $\psi\colon(S,u)\to(T,v)$ in $\mathsf{uS{\ell}g}$ to the uniquely determined (as specified below) morphism $\operatorname{{\boldsymbol{\mu}}}(T)\to\operatorname{{\boldsymbol{\mu}}}(S)$ making the following diagram commute:

[TABLE]

By Lemma 5.5, for any $\mathfrak{m}\in\operatorname{{\boldsymbol{\mu}}}(T)$ this morphism necessarily transforms the maximal ideal $\mathfrak{m}$ into the maximal ideal $\psi^{-1}(\mathfrak{m})=\{f\in S\mid\psi(f)\in\mathfrak{m}\}$. We conclude that $\mathcal{B}^{\prime}$ (is well defined and) coincides with the functor $\mathcal{B}$ of Definition 5.2. Therefore, $\mathcal{B}$ is part of a duality with $\SS$.

Finally, to prove that the unit is $(\mathfrak{M}_{X}\colon(X,u)\to(\operatorname{{\boldsymbol{\mu}}}(C_{X}),u^{\natural}))_{(X,u)\in\mathsf{Bms}}$, we must prove that, for every $(X,u)\in\mathsf{Bms}$, the function $\SS(\mathfrak{M}_{X})\colon\SS\mathcal{B}\SS(X,u)\to\SS(X,u)$ is the inverse of ${}^{\natural}\colon\SS(X,u)\to\SS\mathcal{B}\SS(X,u)$. Since these are isomorphisms, it is enough to prove that the following composite is the identity:

[TABLE]

The function $\SS(\mathfrak{M}_{X})\colon C_{\operatorname{{\boldsymbol{\mu}}}(C_{X})}\to C_{X}$, obtained by applying $\SS$ to the morphism $\mathfrak{M}_{X}\colon(X,u)\to(\operatorname{{\boldsymbol{\mu}}}(C_{X}),u^{\natural})$, maps any $g\in C_{\operatorname{{\boldsymbol{\mu}}}(C_{X})}$ to the function $\SS(\mathfrak{M}_{X})(g)\in C_{X}$ that maps $x\in X$ to $g(\mathfrak{M}_{X}(x))\frac{u(x)}{u^{\natural}(\mathfrak{M}_{X}(x))}$. The latter product equals $g(\mathfrak{M}_{X}(x))$ because, by Lemma 3.6(ii), $u^{\natural}(\mathfrak{M}_{X}(x))=u(x)$. In particular, for every $f\in C_{X}$,    $\SS(\mathfrak{M}_{X})$ maps $f^{\natural}$ to the function that maps $x$ to $f^{\natural}(\mathfrak{M}_{X}(x))$. The latter equals $f(x)$ by Lemma 3.6(ii). Therefore, for every $f\in C_{X}$, $\SS(\mathfrak{M}_{X})(f^{\natural})=f$. We have shown that the composite

[TABLE]

is the identity, and hence the unit is $(\mathfrak{M}_{X}\colon(X,u)\to(\operatorname{{\boldsymbol{\mu}}}(C_{X}),u^{\natural}))_{(X,u)\in\mathsf{Bms}}$. The proof is complete. ∎

6. Limits in $\mathsf{Bms}$ and colimits in the categories $\mathsf{uS{\ell}g}$ and $\mathsf{SMV}$

In this section we consider the existence problem for limits in $\mathsf{Bms}$, and for colimits in $\mathsf{uS{\ell}g}$ and $\mathsf{SMV}$.

We refer to [7, Definition 2.4.1] and [2, §12] for background on limits and colimits. For products in category theory we refer to [2, Definition 10.19] and [7, Definition 2.1.1]. For coproducts see [2, Table 10.63] and [7, Definition 2.2.1].

We show that $\mathsf{Bms}$ has all finite limits (i.e., limits of diagrams indexed by a category with a finite set of objects and a finite set of morphisms). All these limits have an easy description, as they are computed as in the category $\mathsf{Set}$ of sets and functions. Moreover, they are computed as in the category of boolean spaces. It will follow that the dual categories $\mathsf{uS{\ell}g}$, and $\mathsf{SMV}$ have finite colimits.

For a finite subset $I\subseteq\mathbb{Z}_{>0}$ we let $\mathsf{LCM}(I)$ denote the least common multiple of $I$, i.e., the supremum of $I$ in the poset $\mathbb{Z}_{>0}$ ordered by divisibility.

Theorem 6.1** (Description of finite limits).**

The category $\mathsf{Bms}$ of boolean multispaces is finitely complete. More generally, $\mathsf{Bms}$ has limits of all diagrams indexed by a category with a finite set of objects.

If $D\colon\mathsf{I}\to\mathsf{Bms}$ is one such diagram, then a limit $(p_{i}\colon(L,v)\to D(i))_{i\in\mathsf{I}}$ over $D$ is given by:

1. (i)

$L=\{(x_{i})_{i\in\mathsf{I}}\in\prod_{i\in I}D(i)\mid\text{for all }\mathsf{I}\text{-morphism }h\colon i\to j,\,D(h)(x_{i})=x_{j}\}$; 2. (ii)

for each $i\in\mathsf{I}$, $p_{i}\colon L\to D(i)$ maps $(x_{j})_{j\in\mathsf{I}}$ to $x_{i}$; 3. (iii)

the topology on $L$ is the coarsest topology making every $p_{i}$ continuous; equivalently, the subspace topology induced by the product topology of $\prod_{i\in I}D(i)$; 4. (iv)

for $x\in L$, $v(x)=\mathsf{LCM}\{u_{i}(p_{i}(x))\mid i\in\mathsf{I}\}$, where $u_{i}$ is the multiplicity of $D(i)$.

Proof.

The proof is straightforward. Items (i–ii) amount to the usual construction of limits in $\mathsf{Set}$. Likewise, (i–iii) amount to the usual construction of limits in the category of (topological spaces, as well as in its full subcategory of) boolean spaces. The finiteness of the set of objects of   $\mathsf{I}$   is used to show that the function

[TABLE]

is well-defined and continuous, because $(\mathbb{Z}_{>0})^{\mathrm{Ob}(\mathsf{I})}$ is discrete, being a finite power of discrete spaces. This guarantees that $v$ is well-defined and continuous. ∎

Remark 6.2*.*

Note that the limits in Theorem 6.1 (in particular, all finite limits) are computed in the same way as they are in $\mathsf{Set}$, in the category of topological spaces, and in the category of boolean spaces.

Remark 6.3*.*

From Theorem 6.1 we immediately have:

The category $\mathsf{Bms}$ of boolean multispaces has all finite products, equalizers and pullbacks (which are all special instances of finite limits).

Thus for instance:

1. (i)

The product $(X,u)\times(X^{\prime},u^{\prime})$ in $\mathsf{Bms}$ is given by $(X\times X^{\prime},\,\,\mathsf{LCM}(u,u^{\prime}))$, where the function $\mathsf{LCM}(u,u^{\prime})\colon X\times X^{\prime}\to\mathbb{Z}_{>0}$ maps any pair $(x,x^{\prime})\in X\times X^{\prime}$ to the least common multiple of $u(x)$ and $u^{\prime}(x^{\prime})$. The (canonical projection) morphism of $(X,u)\times(X^{\prime},u^{\prime})$ into $(X,u)$ (resp., into $(X^{\prime},u^{\prime})$) is determined by the projection of $X\times X^{\prime}$ onto $X$ (resp., onto $X^{\prime}$). 2. (ii)

The category $\mathsf{Bms}$ of boolean multispaces has a (unique up to isomorphism) terminal object $(X,u)$, namely the singleton $X=\{*\}$ equipped with the function $u$ defined by $u(*)=1$. 3. (iii)

Let $f,g\colon(X,u)\rightrightarrows(X^{\prime},u^{\prime})$ be two parallel $\mathsf{Bms}$-morphisms. The equalizer of $f$ and $g$ is $(E,e)$, where $E=\{x\in X\mid f(x)=g(x)\}$ and $e\colon E\to\mathbb{Z}_{>0}$ coincides with $u$ over $E$, together with the inclusion $(E,e)\hookrightarrow(X,u)$.

Furthermore, by [2, Theorem 12.4 (1)$\Rightarrow$(3)], from the finite completeness of $\mathsf{Bms}$ it follows that $\mathsf{Bms}$ has finite intersections777A category is said to have finite intersections if for every object $X$ the partially ordered class of subobjects of $X$ has finite infima. See, e.g., [2, Definition 12.1.(4)]..

Corollary 6.4**.**

The categories of unital Specker $\ell$-groups and Specker MV-algebras are finitely cocomplete. More generally, both categories have a colimit for every diagram indexed by a category with a finite set of objects.

We have just proved that the category $\mathsf{Bms}$ of boolean multispaces is finitely complete. We now prove that it is not complete, as it lacks some infinite products.

Proposition 6.5**.**

The category $\mathsf{Bms}$ is not complete. For example, there is no countable power of the boolean multispace $(\{a,b\},u)$ with $u(a)=1$ and   $u(b)=2$.

Proof.

We first sketch the main idea of the proof. Roughly speaking, the product of countably many copies of $(\{a,b\},u)$ does not exist, because the natural choice for the multiplicity function on the cartesian product $\{a,b\}^{\omega}$—namely the function $(x_{i})_{i}\mapsto\mathsf{LCM}\{u(x_{i})\mid i\in\omega\}$—is not continuous. This is essentially due to the fact that the function $\mathsf{LCM}=\max\colon\{1,2\}^{\omega}\to\{1,2\}$ is not continuous. This shows that the natural definition of product does not work.

We now prove that no choice works. By way of contradiction, let us suppose that one such countable power $(p_{i}\colon(P,v)\to(\{a,b\},u))_{i\in\omega}$ exists.

Claim: Up to a bijection, $P$ is the cartesian product $\{a,b\}^{\omega}$ and each $p_{i}$ is the projection on the $i$-th coordinate.

We define

[TABLE]

with the intent of showing that this function is a bijection. To prove that $h$ is injective let $x,y\in P$ be such that $h(x)=h(y)$. This means that, for every $i\in\omega$, $p_{i}(x)=p_{i}(y)$. Let $\{*\}$ be a singleton space, equipped with the multiplicity $\{*\}\to\mathbb{Z}_{>0}$ that maps $*$ to $\mathsf{LCM}(v(x),v(y))$. The functions

[TABLE]

are $\mathsf{Bms}$-morphisms, and for all $i\in\omega$ we have $p_{i}(f(*))=p_{i}(x)=p_{i}(y)=p_{i}(g(*))$, i.e., $p_{i}\circ f=p_{i}\circ g$. From the uniqueness in the universal property of $P$ it follows that $f=g$, whence $x=y$. This proves that $h$ is injective.

To prove that $h$ is surjective let $(x_{i})_{i\in\omega}$ be an element of $\{a,b\}^{\omega}$. Again let $\{*\}$ be a singleton space, equipped with the multiplicity $\{*\}\to\mathbb{Z}_{>0}$ that maps $*$ to the least common multiple of $\{u(x_{i})\mid i\in\omega\}$. As a consequence, for every $i\in\omega$, the function $t_{i}\colon\{*\}\to\{a,b\}$ mapping $*$ to $x_{i}$ is a morphism. The universal property of $P$ yields a unique morphism $s\colon\{*\}\to P$ such that for every $i\in\omega$   $p_{i}(s(*))=x_{i}$. Then $h(s(*))=(x_{i})_{i\in\omega}$ and $h$ is surjective.

We have just proved that $(p_{i}\colon P\to\{a,b\})_{i\in\omega}$ is (up to a bijection that will henceforth be ignored) the cartesian product with its canonical projections. Our claim is thus settled.

Since every $p_{i}$ is continuous, the topology on $P$ is finer than the product topology. Since any two comparable compact Hausdorff topologies on the same set are equal, the topology on $P$ is the product topology. Since every $p_{i}$ decreases multiplicities, for all $(x_{i})_{i\in\omega}\in P$ the number $v((x_{i})_{i\in\omega})$ is a multiple of $\mathsf{LCM}\{u(x_{i})\mid i\in\omega\}$. However, the proof that $h$ is surjective shows that $v((x_{i})_{i\in\omega})$ divides $\mathsf{LCM}\{u(x_{i})\mid i\in\omega\}$. As a consequence, $v((x_{i})_{i\in\omega})=\mathsf{LCM}\{u(x_{i})\mid i\in\omega\}$.

To reach a contradiction, we observe that $v$ is not continuous: Indeed, the preimage under $v$ of $\{1\}$ is the singleton $\{(1)_{i\in\omega}\}$, which is not an open subset of $P$. ∎

Corollary 6.6**.**

The categories $\mathsf{uS{\ell}g}$ and $\mathsf{SMV}$ are not cocomplete. For example, they lack the countable copower of $(\mathbb{Z}\times\mathbb{Z},(1,2))$ and of $\{0,1\}\times\{0,\frac{1}{2},1\}$, respectively. ∎

Remark 6.7*.*

We have seen that the category $\mathsf{Bms}$ is not complete. However, $\mathsf{Bms}$ can be embedded, as a full subcategory, into other complete categories already considered in the literature ([13, 11, 1]). To tweak $\mathsf{Bms}$ in order to get a complete category in the style of these papers, one may proceed as follows:

• •

Embed the lattice (with respect to the divisibility order) with bottom $\mathbb{Z}_{>0}$ into a complete lattice. For instance, the complete lattice of supernatural numbers as defined in [13], the complete lattice of additive subgroups of $\mathbb{R}$ containing $1$ following [11], or the complete lattice of natural numbers with respect to the divisibility order following [1].

• •

Equip this complete lattice with a topology $\mathcal{T}$ coarser than the discrete one, that makes the join of any arity a continuous function. (Thus, for example, in [1], $\mathcal{T}$ is the topology on $\mathbb{Z}_{\geq 0}$ whose closed sets are the finite unions of principal downsets. In [13] and [11], $\mathcal{T}$ is the Scott topology.) Consider continuity with respect to $\mathcal{T}$.

As a final remark in this section, while finite limits in $\mathsf{Bms}$ are computed as in $\mathsf{Set}$, this is not the case for all existing limits:

Theorem 6.8**.**

The forgetful functor from $\mathsf{Bms}$ to $\mathsf{Set}$ does not preserve limits.

Proof.

The product in $\mathsf{Bms}$ of countably many copies of a singleton, each copy with a different multiplicity, is the empty boolean multispace. ∎

7. Colimits in $\mathsf{Bms}$, and limits in $\mathsf{uS{\ell}g}$ and $\mathsf{SMV}$

In this section we show that the category $\mathsf{Bms}$ of boolean multispaces has finite coproducts but is not finitely cocomplete, as it lacks some pushouts, as well as some coequalizers. Therefore, dually, we get that $\mathsf{uS{\ell}g}$ and $\mathsf{SMV}$ have finite products but are not finitely complete, as they lack some pullbacks, as well as some equalizers.

Proposition 7.1** (Finite coproducts in $\mathsf{Bms}$).**

1. (i)

The coproduct $(X,u)\amalg(X^{\prime},u^{\prime})$ in the category $\mathsf{Bms}$ of boolean multispaces is given by the disjoint union $(X\sqcup X^{\prime},u\sqcup u^{\prime})$, together with the natural multiplicity-preserving injective morphisms of $(X,u)$ and $(X^{\prime},u^{\prime})$ into $(X\sqcup X^{\prime},u\sqcup u^{\prime})$. 2. (ii)

The category of boolean multispaces has a (unique up to isomorphism) initial object $(X,u)$, namely the empty space equipped with the unique map from $\emptyset$ to $\mathbb{Z}_{>0}$.

Therefore, $\mathsf{Bms}$ has finite coproducts.

Proof.

(i) We first show that $(X\sqcup X^{\prime},u\sqcup u^{\prime})$ has the required universal property. Consider a boolean multispace $(Y,v)$ and $\mathsf{Bms}$-morphisms $f\colon(X,u)\to(Y,v)$ and $f^{\prime}\colon(X^{\prime},u^{\prime})\to(Y,v)$. The continuous map $f\sqcup f^{\prime}\colon X\sqcup X^{\prime}\to Y$ whose restriction on $X$ is $f$ and whose restriction on $X^{\prime}$ is $f^{\prime}$ is the unique map making the following diagram commute:

[TABLE]

Moreover, this map decreases multiplicities, because so do $f$ and $f^{\prime}$ and the multiplicity on $X\sqcup X^{\prime}$ is the disjoint union of the multiplicities on $X$ and $X^{\prime}$.

(ii) This is immediate. ∎

From Proposition 7.1 it follows that the dual categories $\mathsf{uS{\ell}g}$ of unital Specker $\ell$-groups and $\mathsf{SMV}$ of Specker MV-algebras have finite products.

We next give an explicit description of such finite products, showing that they are computed as in $\mathsf{Set}$.

Proposition 7.2** (Finite products in $\mathsf{uS{\ell}g}$ and $\mathsf{SMV}$).**

The categories $\mathsf{uS{\ell}g}$ of unital Specker $\ell$-groups and $\mathsf{SMV}$ of Specker MV-algebras have finite products. They are computed as the usual products of algebras (i.e., they are computed as in the categories of unital $\ell$-groups and of MV-algebras):

1. (i)

The product of two unital Specker $\ell$-groups $(G,u)$ and $(G^{\prime},u^{\prime})$ is the cartesian product $(G\times G^{\prime},(u,u^{\prime}))$, and similarly for Specker MV-algebras. 2. (ii)

The terminal object in $\mathsf{uS{\ell}g}$ is the trivial algebra, and similarly for $\mathsf{SMV}$.

Proof.

(i) Given two boolean multispaces $(X,u)$ and $(X^{\prime},u^{\prime})$, the categorical product of $\SS(X,u)$ and $\SS(X^{\prime},u^{\prime})$ is isomorphic to $\SS((X,u)\amalg(X^{\prime},u^{\prime}))$, i.e., by Proposition 7.1(i), $\SS((X\sqcup X^{\prime},u\sqcup u^{\prime}))$, which is the cartesian product of $\SS(X,u)$ and $\SS(X^{\prime},u^{\prime})$.

(ii) It is easily seen that the trivial algebra is a unital Specker $\ell$-group. For an alternative proof, we can note that the terminal object in $\mathsf{uS{\ell}g}$ is obtained by applying $\SS$ to the initial object of $\mathsf{Bms}$, which, by Proposition 7.1, is the empty boolean multispace. It follows that the terminal object in $\mathsf{uS{\ell}g}$ is the trivial algebra. ∎

Theorem 7.3**.**

$\mathsf{Bms}$* lacks some pushouts, as well as some coequalizers.*

Proof.

Let $(\{*\},2)$ be a singleton space with multiplicity of $*$ equal to $2$. Let $(\alpha\mathbb{Z}_{\geq 0},2)$ be the one-point compactification $\alpha\mathbb{Z}_{\geq 0}=\mathbb{Z}_{\geq 0}\cup\{\infty\}$ of the set $\mathbb{Z}_{\geq 0}$, equipped with the multiplicity constantly equal to $2$. Let $(\{*\},1)$ be a singleton space with multiplicity of $*$ equal to $1$.

We will show that there is no pushout for the following diagram:

[TABLE]

where the function $(*\mapsto\infty)\colon\{*\}\to\alpha\mathbb{Z}_{\geq 0}$ maps $*$ to the accumulation point $\infty$ of $\alpha\mathbb{Z}_{\geq 0}$.

For a proof we will be guided by the idea that the natural candidate for a pushout would be $(\alpha\mathbb{Z}_{\geq 0},v)$ where $v(\infty)=1$ and for every isolated point $x$ we have $v(x)=2$. However, it turns out $v$ is not continuous. Let us transform this motivating idea into a proof.

By way of contradiction, suppose there is a pushout

[TABLE]

Claim: $g$ is a bijection.

We first prove that $g$ is injective. Let $(\alpha\mathbb{Z}_{\geq 0},1)$ be the boolean multispace whose underlying space is the one-point compactification of $\mathbb{Z}_{\geq 0}$ and whose multiplicity is constantly equal to $1$. The identity function $(\alpha\mathbb{Z}_{\geq 0},2)\to(\alpha\mathbb{Z}_{\geq 0},1)$ is a morphism, and so is the function $(*\mapsto\infty)\colon(\{*\},1)\to(\alpha\mathbb{Z}_{\geq 0},1)$ that maps $*$ to the accumulation point. As a consequence, in the following diagram the outer square commutes, whence there is a unique morphism $r\colon(P,v)\to(\alpha\mathbb{Z}_{\geq 0},1)$ making the following diagram commute:

[TABLE]

Since the identity $(\alpha\mathbb{Z}_{\geq 0},2)\to(\alpha\mathbb{Z}_{\geq 0},1)$ is injective, its factor $g$ is injective.

We next prove that $g$ is surjective. The image $\mathrm{im}(g)$ of $g$ with the restriction $v_{\mid}$ of $v$ is a boolean multispace. The universal property of $P$ yields a unique morphism $d\colon(P,v)\to(\mathrm{im}(g),v_{\mid})$ making the following diagram commute:

[TABLE]

With $\iota\colon\mathrm{im}(g)\to P$ shorthand for the inclusion map, also the following diagram commutes:

[TABLE]

By the uniqueness in the universal property of the pushout, the composite $\iota\circ d$ is the identity on $P$, and hence the inclusion $\iota$ is surjective. Therefore, $\mathrm{im}(g)=P$, whence $g$ is surjective. Our claim is thus settled.

It follows that $g$ is a bijective continuous function between boolean spaces, and hence it is a homeomorphism. Therefore, without loss of generality, we may assume the following:

—$P=\alpha\mathbb{Z}_{\geq 0}$,

—$g$ is the identity function (possibly not an identity morphism, as the multiplicity $v$ of $P$ may differ from the one of $(\alpha\mathbb{Z}_{\geq 0},2)$), and

—$h$ is the function that maps $*$ to the accumulation point $\infty$ of $\alpha\mathbb{Z}_{\geq 0}$.

We then have the diagram

[TABLE]

Since $(*\mapsto\infty)\colon(\{*\},1)\to(\alpha\mathbb{Z}_{\geq 0},v)$ decreases multiplicities, $v(\infty)=1$.

For every $n\in\mathbb{Z}_{\geq 0}$, let $v_{n}\colon\alpha\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{>0}$ be the function constantly equal to $1$ except on $n$, where it takes the value $2$. The universal property of the pushout yields a unique morphism $c_{n}\colon(\alpha\mathbb{Z}_{\geq 0},v)\to(\alpha\mathbb{Z}_{\geq 0},v_{n})$ making the following diagram commute, and showing that $c_{n}$ necessarily coincides with the identity function:

[TABLE]

For every $n\in\mathbb{Z}_{\geq 0}$, $v(n)$ is a multiple of $2$ since $c_{n}$ decreases multiplicities, and is a divisor of $2$ since $\mathrm{id}\colon(\alpha\mathbb{Z}_{\geq 0},2)\to(\alpha\mathbb{Z}_{\geq 0},v)$ decreases multiplicities. Therefore, $v(n)=2$.

But then $v$ is not continuous—a contradiction.

This proves that $\mathsf{Bms}$ lacks some pushouts.

Since $\mathsf{Bms}$ has finite coproducts and is not finitely cocomplete, it lacks some coequalizers. The example above shows that it lacks the coequalizer of the diagram $(\{*\},2)\rightrightarrows(\alpha\mathbb{Z}_{\geq 0},2)\sqcup(\{*\},1)$, where one function maps $*$ to the accumulation point of $\alpha\mathbb{Z}_{\geq 0}$, and the other function maps it to the unique element $*$ of $\{*\}$. ∎

Corollary 7.4**.**

The categories $\mathsf{uS{\ell}g}$ and $\mathsf{SMV}$ lack some pullbacks, as well as some equalizers. $\Box$

Remark 7.5*.*

Dualizing the proof of Theorem 7.3, we get that $\mathsf{uS{\ell}g}$ lacks the pullback of the following diagram:

[TABLE]

Intuitively, the obvious candidate for the pullback (obtained by computing it as in $\mathsf{Set}$) is the set of continuous functions from $\alpha\mathbb{Z}_{\geq 0}$ to $\mathbb{Z}$ whose evaluation at the accumulation point $\infty$ is in $2\mathbb{Z}$. However, this unital $\ell$-group is not Specker, because the only singular elements are the characteristic functions of clopen subsets of $\alpha\mathbb{Z}_{\geq 0}$ not containing $\infty$, and the constant function $2$ is not in the subgroup generated by these.

Expressing the same example in terms of equalizers, we see that $\mathsf{uS{\ell}g}$ lacks the equalizer of the diagram $(C_{\alpha\mathbb{Z}_{\geq 0}},2)\times(\mathbb{Z},1)\rightrightarrows(\mathbb{Z},2)$, where the two maps are

[TABLE]

8. Closure properties of $\mathsf{uS{\ell}g}$ inside $\mathsf{u{\ell}g}$

We let $\mathsf{{\ell}g}$ and $\mathsf{u{\ell}g}$ denote the categories respectively given by the equational class of $\ell$-groups with their $\ell$-homomorphisms, and the category of unital $\ell$-groups with their unital $\ell$-homomorphisms. In this section we are concerned with the mutual relationships between these categories and the category $\mathsf{uS{\ell}g}$.

While being categorically equivalent to the equational class of MV-algebras, unital $\ell$-groups are not definable by equations—in fact, a trivial compactness argument shows that the archimedean property of the strong unit is not even definable in first-order logic. The same undefinability property also holds for unital Specker $\ell$-groups and Specker MV-algebras. Since the equational class of MV-algebras has (finite and infinite) products, then so does its equivalent category $\mathsf{u{\ell}g}$ of unital $\ell$-groups. Naturally, the definition of “$\mathsf{u{\ell}g}$-product” does not coincide with the definition of a cartesian product in the equational class $\mathsf{{\ell}g}$ of $\ell$-groups, because cartesian products of unital $\ell$-groups, qua $\ell$-groups, need no longer be unital $\ell$-groups. The construction of $\mathsf{u{\ell}g}$-products is detailed in the proof of the following result:

Proposition 8.1**.**

The category $\mathsf{u{\ell}g}$ of unital $\ell$-groups has infinite products.

Proof.

Let $(G_{i},u_{i})_{i\in I}$ be a family of unital $\ell$-groups. Let $\prod_{i}G_{i}$ be the (cartesian) product of $\ell$-groups in the equational class of $\ell$-groups. Let us restrict $\prod_{i}G_{i}$ to the set $G$ of those elements of $\prod_{i}G_{i}$ whose absolute value is bounded by some positive integer multiple of the element $u=(u_{i})_{i\in I}$. Then the element $u$ becomes a unit of the restricted $\ell$-subgroup $G$. Since $\Gamma(G,u)\cong\prod_{i}(\Gamma(G_{i},u_{i})),$ the unital $\ell$-group $G$ equipped with the (distinguished strong order) unit $u$ is the categorical product of the family $(G_{i},u_{i})_{i}$ in the category of unital $\ell$-groups. ∎

Proposition 8.2**.**

There is a unital Specker $\ell$-group $(G,u)$ and a unital $\ell$-subgroup $(H,u)$ of $(G,u)$ that is not a unital Specker $\ell$-group.

Proof.

Let $(G,u)$ be the $\ell$-group of all continuous $\mathbb{Z}$-valued functions on the boolean space $\alpha\mathbb{Z}_{\geq 0}=$ the one-point compactification $\{\infty\}\cup\mathbb{Z}_{\geq 0}$ of $\mathbb{Z}_{\geq 0}=\{0,1,2,\dots\}$, with its distinguished unit $u$ given by the function constantly equal to 2 over $\alpha\mathbb{Z}_{\geq 0}$. Thus the singular elements are precisely the characteristic functions of the clopen subsets of $\alpha\mathbb{Z}_{\geq 0}$. See Theorem 2.8(iv). The constant 1 over $\alpha\mathbb{Z}_{\geq 0}$ is the greatest singular element of $G$. As a Specker $\ell$-group, $G$ is generated (both as a group and as an $\ell$-group) by the set $S_{G}$ of its singular elements.

Let $(H,u)\subseteq(G,u)$ be the unital $\ell$-group generated (as an $\ell$-group) by the unit $u$ along with the set $S^{\prime}_{G}\subseteq S_{G}$ of singular elements of $G$ except those having constant value 1 over an open neighborhood of $\infty$. In other words, $H=\{g\in G\mid g(\infty)\in 2\mathbb{Z}\}$. (Note that this is the unital $\ell$-group mentioned in Remark 7.5.) The singular elements of $H$ are precisely the characteristic functions of clopen sets not containing $\infty$. Then the constant function $2$ is not in the subgroup generated by the singular elements, whence $H$ is not a Specker $\ell$-group. ∎

A moment’s reflection shows that unital $\ell$-groups inherit from MV-algebras, via the equivalence $\Xi$ of [12, §7], infinite coproducts (also known as free products). By contrast we have:

Proposition 8.3**.**

The infinite coproduct $\amalg_{i\in I}(S_{i},u_{i})$ of unital Specker $\ell$-groups $(S_{i},u_{i})$ computed within the category $\mathsf{u{\ell}g}$ of unital $\ell$-groups need not be a unital Specker $\ell$-group—even assuming that all maps $u_{i}^{\natural}$ defined in 2.6 have values bounded by a fixed $n$.

Proof.

By way of contradiction let us assume that the infinite coproduct $\amalg_{i\in I}(S_{i},u_{i})$ of unital Specker $\ell$-groups $(S_{i},u_{i})$, computed within the category $\mathsf{u{\ell}g}$ of unital $\ell$-groups, is a unital Specker $\ell$-group. Then $\amalg_{i\in I}(S_{i},u_{i})$ is a coproduct in the category $\mathsf{uS{\ell}g}$ of unital Specker $\ell$-groups. However, by Corollary 6.6, $\mathsf{uS{\ell}g}$ lacks the countable copower of $(\mathbb{Z}\times\mathbb{Z},(1,2))$. ∎

For direct limits of algebras we refer to [22, §21].

Proposition 8.4**.**

The direct limit (with $\to$-arrows, also known as the “directed colimit”, [2, Examples 11.28(4)]) of a direct system of Specker MV-algebras $A_{j}$, as given in the category of MV-algebras and their homomorphisms, need not be a Specker MV-algebra—even assuming the boundedness condition for the homogeneity degree of all homogeneous components of all $A_{j}$.

Proof.

If the directed colimit of a direct system of Specker MV-algebras $A_{j}$, as given in the category of MV-algebras and their homomorphisms, were a Specker MV-algebra, then it would be a directed colimit in the category of Specker MV-algebras.

Recall that the category of Specker MV-algebras is finitely cocomplete (Corollary 6.4), but that $\{0,1\}\times\{0,\frac{1}{2},1\}$ lacks a countable copower. Let $\mathsf{I}$ be the category of finite subsets of $\mathbb{Z}_{\geq 0}$ and inclusions between them. Let $F\colon\mathsf{I}\to\mathsf{SMV}$ be the functor that, on objects, maps a finite subset $S$ to the $S$-fold copower of $\{0,1\}\times\{0,\frac{1}{2},1\}$ and, on morphisms, maps an inclusion $S\subseteq T$ to the canonical morphism of the $S$-fold copower of $\{0,1\}\times\{0,\frac{1}{2},1\}$ into the $T$-fold copower of $\{0,1\}\times\{0,\frac{1}{2},1\}$. It follows that this directed diagram has no colimit. For otherwise, by the dual of the exercise “Products as Projective Limits of Finite Products” [2, Exercise 11B], it would give a countable copower of $\{0,1\}\times\{0,\frac{1}{2},1\}$, a contradiction. ∎

9. Related work

Multisets and their topological variants (msets, bags, heaps, bunches, $M$-topological spaces, weighted sets, firesets, etc.) have an extensive literature. See, e.g., [6], [21], and references therein.

For an earlier paper on the relationships between multisets and MV-algebras see [13], which contains a duality for locally finite MV-algebras, of which Specker MV-algebras are special cases. For a more general result see [11], which constructs a duality for “locally weakly finite” MV-algebras. The latter class constitutes a generalization of locally finite MV-algebras that includes, among others, the standard MV-algebra $[0,1]$.

For recent work see [1], where a duality is constructed for the class of “metrically complete unital $\ell$-groups”. Every Specker MV-algebra is locally finite, and every unital Specker $\ell$-group is metrically complete. Dually, every boolean multispace $(X,u\colon X\to\mathbb{Z}_{>0})\,\,\,$ is a multiset in the sense of [13] (up to identifying $\mathbb{Z}_{>0}$ with a subset of the set of supernatural numbers), a “normal a-space” in the sense of [1], and a real-valued multiset $(X,u^{\prime}\colon X\to\mathrm{Sub}([0,1]))$ in the sense of [11] (up to identifying $n\in\mathbb{Z}_{>0}$ with $\{0,\frac{1}{n},\dots,\frac{n-1}{n},1\}$).