Epimorphisms in varieties of subidempotent residuated structures

TL;DR

This paper proves that epimorphisms are surjective in certain varieties of subidempotent residuated structures, extending previous results to broader classes including De Morgan monoids.

Contribution

It establishes conditions under which epimorphisms are surjective in varieties of subidempotent residuated lattices, generalizing earlier findings to new algebraic structures.

Findings

Epimorphisms are surjective in the specified varieties.

Conditions on finitely subdirectly irreducible algebras are necessary.

The results extend to varieties with bounds and include De Morgan monoids.

Abstract

A commutative residuated lattice A is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra A*). It is proved here that epimorphisms are surjective in a variety K of such algebras A (with or without involution), provided that each finitely subdirectly irreducible algebra B in K has two properties: (1) B is generated by lower bounds of e, and (2) the poset of prime filters of B* has finite depth. Neither (1) nor (2) may be dropped. The proof adapts to the presence of bounds. The result generalizes some recent findings of G. Bezhanishvili and the first two authors concerning epimorphisms in varieties of Brouwerian algebras, Heyting algebras and Sugihara monoids, but its scope also encompasses a range of interesting varieties of De Morgan monoids.

Equations

$$\uparrow x=\{y\in X:x⩽y\}$$

$$g(q(a))=g(a/\mu )=g(f(a/(\mu {|}_A)))=h(f(a/(\mu {|}_A)))=h(q(a)),$$

$$x\cdot y⩽z⟺x⩽y\to z\text{(the law of residuation)}$$

$$x⩽e⟹x=x^2\text{ (mbox:=nolinebreakx⋅x)}\text{(subidempotence)},$$

$$(x\wedge e{)}^2=x\wedge e.$$

$$x\cdot (y\vee z)=(x\cdot y)\vee (x\cdot z)$$

$$x\to (y\wedge z)=(x\to y)\wedge (x\to z)$$

$$(x\vee y)\to z=(x\to z)\wedge (y\to z)$$

$$x⩽y⟺e⩽x\to y$$

$$x=y⟺e⩽x\leftrightarrow y$$

$$e⩽x\to x\text{ and }e\to x=x$$

$$(x⩽e\&y⩽e)⟹x\wedge y=x\cdot y.$$

$$x⩽e⟹x⩽x^2.$$

$$F\mapsto {\Omega }^{\mathbf{A}}F\text{mbox}:=\text{nolinebreak}\{⟨a,b⟩\in A^2:a\leftrightarrow b\in F\};$$

$$\theta \mapsto \{a\in A:a\wedge e{\equiv }_{\theta }e\}.$$

$$a\to b\in F$$

$${\Omega }^{\mathbf{B}}(B\cap F)=({\Omega }^{\mathbf{A}}F){|}_B.$$

$$a⩾x_1\wedge \dots \wedge x_n\text{ for some }{x}_1,\dots ,x_n\in X\cup \{e\},\text{ where }0<n\in \omega .$$

$$F\cap G\subseteq H$$

$${\uparrow }^{\mathbf{A}}F=\{H\in \mathrm{Pr}(\mathbf{A}):F\subseteq H\},$$

$$H\mapsto h^{\leftarrow }[H]\text{mbox}:=\text{nolinebreak}\{a\in A:h(a)\in H\}$$

$$h^{\leftarrow }[h[G]]=G\text{ for all deductive filters G of A such that K⊆G.}$$

$${\mathbf{A}}^{-}=⟨A^{-};\wedge {|}_{(A^{-}{)}^2},\vee {|}_{(A^{-}{)}^2},{\to }^{-},e⟩\text{,}$$

$$a/F\mapsto (a\wedge e)/(A^{-}\cap F)$$

$${\mathrm{Fg}}^{\mathbf{A}}F=\{a\in A:a⩾b\text{ for some }b\in F\}$$

$$x\in X$$

$$\{{\phi }^{\mathbf{A}}(a):a\in A\}\cup \{{\phi }^{\mathbf{A}}(a{)}^c:a\in A\}.$$

$${\mathbf{X}}^{\ast }=⟨\mathrm{Cpu}(\mathbf{X});\cap ,\cup ,\to ,X⟩\in \mathsf{BRA},$$

$$U\to V\text{mbox}:=\text{nolinebreak}X\setminus \downarrow (U\setminus V)$$

$$x\mapsto \{U\in \mathrm{Cpu}(\mathbf{X}):x\in U\}.$$

$${\phi }_F^{\mathbf{A}}:a/F\mapsto \{H\in \mathrm{Pr}(\mathbf{A}):F\cup \{a\}\subseteq H\}$$

$$k(H)=q[H]=H/F\text{mbox}:=\text{nolinebreak}\{a/F:a\in H\}(H\in {\uparrow }^{\mathbf{A}}F),$$

$$k^{\ast }\circ {\phi }^{\mathbf{A}/F}:\mathbf{A}/F\cong ({\uparrow }^{\mathbf{A}}F{)}^{\ast }.$$

$$(k^{\ast }\circ {\phi }^{\mathbf{A}/F})(a/F)=k^{\ast }(\{H\in \mathrm{Pr}(\mathbf{A}/F):a/F\in H\})$$

Full text

This work received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”). The first author was also supported by the project CZ.02.2.69/0.0/0.0/17_050/0008361, OPVVV MŠMT, MSCA-IF Lidské zdroje v teoretické informatice. The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS

1. Introduction

In a variety of algebras, if a homomorphism is surjective, then it is an epimorphism, but the converse need not hold. Indeed, rings and distributive lattices each form varieties in which non-surjective epimorphisms arise. As it happens, this reflects the absence of unary terms defining multiplicative inverses in rings, and complements in distributive lattices, despite the uniqueness of those entities when they exist.

Such constructs are said to be implicitly (and not explicitly) definable. In a variety of logic, they embody implicitly definable propositional functions that cannot be explicated in the corresponding logical syntax, and Beth-style ‘definability properties’ preclude phenomena of this kind. (The allusion is to E.W. Beth’s theorems for classical propositional and predicate logic in [3].)

In particular, when a logic $\mathbf{L}$ is algebraized, in the sense of [7], by a variety ${\mathsf{K}}$ of algebras, then the ES property for ${\mathsf{K}}$—i.e., the demand that all epimorphisms in ${\mathsf{K}}$ be surjective—amounts to the so-called infinite Beth definability property for $\mathbf{L}$. The most general version of this ‘bridge theorem’ was formulated and proved by Blok and Hoogland [5, Theorems 3.12, 3.17] (also see [39, Theorem 7.6] and the antecedents cited in both papers).

In this situation, the subvarieties of ${\mathsf{K}}$ algebraize the axiomatic extensions of $\mathbf{L}$, but the ES property need not persist in subvarieties. It is therefore a well-motivated (but often nontrivial) task to determine which subvarieties of ${\mathsf{K}}$ have surjective epimorphisms. The present paper addresses this question in the context of residuated structures, i.e., the algebraic models of substructural logics (see [17, 24]).

We consider algebras called SRLs, which are residuated lattice-ordered commutative monoids satisfying the subidempotent law $x\leqslant e\Longrightarrow x\mathbin{\bm{\cdot}}x=x$ (where $\mathbin{\bm{\cdot}}$ is the monoid operation and $e$ its neutral element) and possibly equipped with an involution (SIRLs) and/or lattice-bounds. They need not be integral, i.e., $e$ need not be the greatest element. They include De Morgan monoids, i.e., the algebraic models of the relevance logic $\mathbf{R^{t}}$ of [1]. The negative cone of an S[I]RL ${\bm{A}}$, which comprises the lower bounds of $e$, may be given the structure of a Brouwerian or Heyting algebra ${\bm{A}}^{-}$, to which the Esakia duality of [13] applies. In particular, the depth of ${\bm{A}}$ may be defined as that of ${\bm{A}}^{-}$.

Heyting and Brouwerian algebras model intuitionistic propositional logic and its positive fragment. A result of Kreisel [28] shows (in effect) that every variety consisting of such algebras has a weak form of the ES property, whereas Maksimova established that only finitely many enjoy a certain strong form; see [16, 32, 33]. Uncountably many varieties of Heyting [Brouwerian] algebras have finite depth [29], and all of these have surjective epimorphisms; the latter claim was proved recently by G. Bezhanishvili and the first two authors [4], using Esakia duality. On the other hand, the ES property fails in uncountably many further varieties of Heyting [Brouwerian] algebras [40].

Relational duals for non-integral S[I]RLs are sometimes available [46], but they are rather complicated. Also, the functor that constructs negative cones (and restricts morphisms accordingly) is not a category equivalence, except in quite special cases; see [15, 18, 20, 21]. Partly for these reasons, we cannot systematically reduce ES problems for arbitrary varieties of S[I]RLs to an examination of negative cones.

Nevertheless, ${\bm{A}}^{-}$ contains enough information about ${\bm{A}}$ to facilitate the main result of this paper (Theorem 14), which is a sufficient condition for the surjectivity of epimorphisms. It states that, in a variety ${\mathsf{K}}$ of S[I]RLs, epimorphisms will be surjective if each finitely subdirectly irreducible member of ${\mathsf{K}}$ has finite depth and is generated by elements of its negative cone. Neither hypothesis may be dropped.

The assumptions of Theorem 14 persist in subvarieties and under varietal joins, so the result is labour-saving. Apart from generalizing the aforementioned findings of [4], it settles the question of epimorphism-surjectivity for many interesting varieties of De Morgan monoids (see Sections 9 and 10), yielding definability results for a corresponding range of relevance logics.

Notation. In an indicated partially ordered set $\langle X;\leqslant\rangle$, we define

[TABLE]

for $U\cup\{x\}\subseteq X$, and if $U=\mathop{\,\uparrow}U$, we call $U$ an up-set of $\langle X;\leqslant\rangle$. We define $\mathop{\,\downarrow}x$ and $\mathop{\,\downarrow}U$ dually. For $x,y\in X$, the notation $x\prec y$ (‘$y$ covers $x$’) signifies that $x<y$ and there is no $z\in X$ such that $x<z<y$.

If $Y\subseteq X$ and $x\in X$, we sometimes need to refer to $Y\cap\mathop{\,\uparrow}{x}$, which we then denote as ${\uparrow}^{Y}x$, even if $x\notin Y$.

The universe of an algebra ${\bm{A}}$ is denoted by $A$. Thus, the congruence lattice $\bm{\mathit{Con}}\,{\bm{A}}$ of ${\bm{A}}$ has universe $\mathit{Con}\,{\bm{A}}$. For $\emptyset\neq X\subseteq A$, the subalgebra of ${\bm{A}}$ generated by $X$ is denoted by $\bm{\mathit{Sg}}^{\bm{A}}X$ (and its universe by $\operatorname{\mathit{Sg}}^{\bm{A}}X$).

The class operator symbols $\mathbb{H}$, $\mathbb{S}$, $\mathbb{P}$ and $\mathbb{P}_{\mathbb{U}}$ stand, respectively, for closure under homomorphic images, subalgebras, direct products and ultraproducts, while $\mathbb{V}$ denotes varietal generation, i.e., $\mathbb{V}=\mathbb{HSP}$. We abbreviate $\mathbb{V}(\{{\bm{A}}\})$ as $\mathbb{V}({\bm{A}})$.

Recall that an algebra ${\bm{A}}$ is finitely subdirectly irreducible (briefly, FSI) iff its identity relation $\textup{id}_{A}$ is meet-irreducible in ${\bm{\mathit{Con}}}\,{\bm{A}}$. For a variety ${\mathsf{K}}$, we use ${\mathsf{K}}_{FSI}$ to denote the class of all FSI members of ${\mathsf{K}}$. Of course, ${\mathsf{K}}=\mathbb{V}({\mathsf{K}}_{FSI})$, by the Subdirect Decomposition Theorem.

2. Epimorphisms

Given a class ${\mathsf{K}}$ of similar algebras, a ${\mathsf{K}}$-morphism is a homomorphism $f\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}$, where ${\bm{A}},{\bm{B}}\in{\mathsf{K}}$. It is called a ${\mathsf{K}}$-epimorphism provided that, whenever $g,h\colon{\bm{B}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{C}}$ are ${\mathsf{K}}$-morphisms with $g\circ f=h\circ f$, then $g=h$. Clearly, surjective ${\mathsf{K}}$-morphisms are ${\mathsf{K}}$-epimorphisms. We say that ${\mathsf{K}}$ has the epimorphism-surjectivity (ES) property if all ${\mathsf{K}}$-epimorphisms are surjective.

A subalgebra ${\bm{D}}$ of an algebra ${\bm{E}}\in{\mathsf{K}}$ is said to be ${\mathsf{K}}$-epic (in ${\bm{E}}$) if every ${\mathsf{K}}$-morphism with domain ${\bm{E}}$ is determined by its restriction to ${\bm{D}}$. (This means that the inclusion map ${\bm{D}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{E}}$ is a ${\mathsf{K}}$-epimorphism, assuming that ${\bm{D}}\in{\mathsf{K}}$.) Thus, a ${\mathsf{K}}$-morphism is a ${\mathsf{K}}$-epimorphism iff its image is a ${\mathsf{K}}$-epic subalgebra of its co-domain. And, when ${\mathsf{K}}$ is closed under subalgebras, it has the ES property iff none of its members has a ${\mathsf{K}}$-epic proper subalgebra.

Recall that if ${\bm{A}}$ is a subalgebra of an algebra ${\bm{B}}$, and $\mu\in\mathit{Con}\,{\bm{B}}$, then the relation $\mu|_{A}\mathrel{\mbox{\,:\!}=\nolinebreak}\,A^{2}\cap\mu$ is a congruence of ${\bm{A}}$.

Lemma 1**.**

Let ${\mathsf{K}}$ be a variety of algebras, and ${\bm{A}}$ a ${\mathsf{K}}$-epic subalgebra of ${\bm{B}}\in{\mathsf{K}}$.* Then, for any $\mu\in\mathit{Con}\,{\bm{B}}$, the map $f\colon{\bm{A}}/(\mu|_{A})\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}/\mu$ defined by $a/(\mu|_{A})\mapsto a/\mu$ is an injective ${\mathsf{K}}$-epimorphism.*

Proof.

As ${\mathsf{K}}$ is a variety, ${\bm{B}}/\mu,\,{\bm{A}}/(\mu|_{A})\in{\mathsf{K}}$. Let $i\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}$ be the inclusion homomorphism and $q\colon{\bm{B}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}/\mu$ the surjective homomorphism $b\mapsto b/\mu$. Clearly, $\mu|_{A}$ is the kernel of the homomorphism $q\circ i\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}/\mu$, so $f$ is an injective ${\mathsf{K}}$-morphism.

Suppose $g,h\colon{\bm{B}}/\mu\mathrel{-\!\!\!\!\!\rightarrow}{\bm{C}}\in{\mathsf{K}}$ are homomorphisms with $g\circ f=h\circ f$. Then $g\circ q$ and $h\circ q$ are homomorphisms from ${\bm{B}}$ to ${\bm{C}}$. For each $a\in A$,

[TABLE]

i.e., $g\circ q\circ i=h\circ q\circ i$. Therefore, $g\circ q=h\circ q$, as $i$ is a ${\mathsf{K}}$-epimorphism. Since $q$ is surjective, it follows that $g=h$, as required. ∎

A variety ${\mathsf{K}}$ is said to have EDPM if it is congruence distributive and ${\mathsf{K}}_{FSI}$ is a universal class (i.e., subalgebras and ultraproducts of FSI members of ${\mathsf{K}}$ are FSI). The acronym stands for ‘equationally definable principal meets’ and is motivated by other characterizations of the notion in [6, 9].

Theorem 2**.**

(Campercholi [8, Theorem 6.8])* If a congruence permutable variety ${\mathsf{K}}$ with EDPM lacks the ES property, then some FSI member of ${\mathsf{K}}$ has a ${\mathsf{K}}$-epic proper subalgebra.*

3. Residuated Structures

General information about residuated structures (and their connection with substructural logics) can be found in [17]. Here, we recall the basic definitions and facts that will be relied on below.

Definition 3**.**

An algebra ${\bm{A}}=\langle A;\wedge,\vee,\mathbin{\bm{\cdot}},\rightarrow,e\rangle$ will be called a (commutative) subidempotent residuated lattice, or briefly an SRL, if $\langle A;\wedge,\vee\rangle$ is a lattice, $\langle A;\mathbin{\bm{\cdot}},e\rangle$ is a commutative monoid and $\rightarrow$ is a binary operation (called the residual) such that ${\bm{A}}$ satisfies

[TABLE]

where $\leqslant$ is the lattice order. An equational paraphrase of (2) is

[TABLE]

Every SRL satisfies the postulates below. Here and subsequently, ${x\leftrightarrow y}$ abbreviates ${(x\rightarrow y)\wedge(y\rightarrow x)}$.

[TABLE]

Of these properties, only (9) relies on subidempotence. It follows from (3) that $\cdot$ is isotone in both arguments, and from (4) and (5) that $\rightarrow$ is isotone in its second argument and antitone in its first. In particular, the implication $x\leqslant e\;\Longrightarrow\;x^{2}\leqslant x$ does not rely on (2), so we could express (2) as

[TABLE]

The key step in the proof of (9) is $x\wedge y\wedge e=(x\wedge y\wedge e)^{2}\leqslant x\mathbin{\bm{\cdot}}y$.

An SRL ${\bm{A}}$ is said to be

• •

square-increasing if it satisfies $x\leqslant x^{2}$;

• •

idempotent if it satisfies $x=x^{2}$;

• •

distributive if its lattice reduct $\langle A;\wedge,\vee\rangle$ is distributive; and

• •

integral if $e$ is its greatest element.

By (9), an SRL is integral iff its operations $\cdot$ and $\wedge$ coincide.

Definition 4**.**

A Brouwerian algebra is an integral SRL ${\bm{A}}$; it is normally identified with its reduct $\langle A;\wedge,\vee,\rightarrow,e\rangle$ (in view of the previous remark).

It follows that a Brouwerian algebra is idempotent, distributive (by (3)) and determined by its lattice reduct, and that it satisfies $x\rightarrow e=e=x\rightarrow x$.

Definition 5**.**

An involutive SRL, briefly an SIRL, is the expansion ${\bm{A}}$ of an SRL by an involution, i.e., a unary operation $\neg$ such that ${\bm{A}}$ satisfies $\neg\neg x=x$ and $x\rightarrow\neg y=y\rightarrow\neg x$.  In these algebras, we define $f=\neg e$.

An SIRL satisfies $x\rightarrow y=\neg(x\mathbin{\bm{\cdot}}\neg y)$ and $\neg x=x\rightarrow f$. Every SRL can be embedded into an SIRL, in such a way that distributivity and the square-increasing law $x\leqslant x^{2}$ are preserved (see [34, 19] and Section 9 below).

The class of all S[I]RLs is a finitely axiomatizable variety. It is arithmetical, i.e., congruence distributive and congruence permutable. (See [17, p. 94], where a termwise-equivalent formulation is used.) The variety of Brouwerian algebras has the ES property [33] (also see [14, 16, 32]).

Distributive square-increasing SIRLs—a.k.a. De Morgan monoids—algebraize the principal relevance logic $\mathbf{R^{t}}$ of [1] (also see [10, 36, 37, 38]), while distributive square-increasing SRLs—a.k.a. Dunn monoids—algebraize the ‘positive’ (negationless) fragment of $\mathbf{R^{t}}$. Their non-distributive counterparts similarly match the subsystem $\mathbf{LR^{t}}$ of $\mathbf{R^{t}}$ and its positive fragment. ($\mathbf{LR^{t}}$ adds the double-negation axiom to the system $\mathbf{FL_{ec}}$ of [17]; also see [35, 45].) The square-increasing law embodies the logical contraction axiom $(p\rightarrow(p\rightarrow q))\rightarrow(p\rightarrow q)$. The idempotent De Morgan monoids are called Sugihara monoids. They model the logic $\mathbf{RM^{t}}$ (a.k.a. $\mathbf{R}$-mingle), and their structure is better understood than that of other De Morgan monoids; see [1, 11, 20, 21, 41]. In all of these cases, there is a transparent lattice anti-isomorphism between the axiomatic extensions of the logic and the subvarieties of the model class.

A bounded S[I]RL is the expansion of an S[I]RL by a distinguished element $\bot$, which is the least element of the order, whence $\bot\rightarrow\bot$ is the greatest element. A Heyting algebra is a bounded Brouwerian algebra. For simplicity, we defer further discussion of bounded S[I]RLs until Section 10.4.

4. Filters

Let ${\bm{A}}$ be an S[I]RL. By a filter of ${\bm{A}}$, we mean a filter of the lattice $\langle A;\wedge,\vee\rangle$, i.e., a non-empty subset $F$ of $A$ that is upward closed and closed under the binary operation $\wedge$. It is called a deductive filter of ${\bm{A}}$ if, moreover, $e\in F$. In that case, $F$ is a submonoid of $\langle A;\mathbin{\bm{\cdot}},e\rangle$ and, whenever $b\in A$ and $a,a\rightarrow b\in F$, then $b\in F$. The respective explanations are that ${\bm{A}}$ satisfies $x\mathbin{\bm{\cdot}}y\geqslant(x\wedge e)\mathbin{\bm{\cdot}}(y\wedge e)=x\wedge y\wedge e$ (by (9)) and $x\mathbin{\bm{\cdot}}(x\rightarrow y)\leqslant y$ (by (1)).

The lattice of deductive filters of ${\bm{A}}$ is isomorphic to ${\bm{\mathit{Con}}}\,{\bm{A}}$. The isomorphism and its inverse are given by

[TABLE]

We abbreviate ${\bm{A}}/\bm{\varOmega}^{\bm{A}}F$ as ${\bm{A}}/F$, and $a/\bm{\varOmega}^{\bm{A}}F$ as $a/F$, noting that

[TABLE]

Whenever ${\bm{B}}$ is a subalgebra of ${\bm{A}}$, and $F$ is a deductive filter of ${\bm{A}}$, then $B\cap F$ is a deductive filter of ${\bm{B}}$ and

[TABLE]

Because of the above isomorphism, ${\bm{A}}$ is FSI iff its smallest deductive filter $\{a\in A\colon e\leqslant a\}$ is meet-irreducible in its lattice of deductive filters, and that amounts to the join-irreducibility of $e$ in $\langle A;\wedge,\vee\rangle$. Since this last condition is expressible as a universal first order sentence, every variety of S[I]RLs has EDPM. Therefore, Theorem 2 applies to all such varieties.

For any subset $X$ of ${\bm{A}}$, the smallest deductive filter of ${\bm{A}}$ containing $X$ is denoted by $\operatorname{\mathit{Fg}}^{{\bm{A}}}X$. Thus, $\operatorname{\mathit{Fg}}^{\bm{A}}X$ consists of all $a\in A$ such that

[TABLE]

In particular, if $e\geqslant b\in A$, then $\operatorname{\mathit{Fg}}^{\bm{A}}\{b\}=\{a\in A:b\leqslant a\}$.

It follows that the deductive filters of a subalgebra ${\bm{B}}$ of ${\bm{A}}$ are just the sets $B\cap F$ such that $F$ is a deductive filter of ${\bm{A}}$. Note that the deductive filters of a Brouwerian algebra are exactly its (lattice-) filters.

A filter $F$ of a lattice $\langle L;\wedge,\vee\rangle$ is said to be prime if its complement $L{\backslash}F$ is closed under the binary operation $\vee$. For any filters $F,G,H$ of $\langle L;\wedge,\vee\rangle$,

[TABLE]

The Prime Filter Extension Theorem asserts that, when $\langle K;\wedge,\vee\rangle$ is a sublattice of a distributive lattice $\langle L;\wedge,\vee\rangle$, then the prime filters of $\langle K;\wedge,\vee\rangle$ are exactly the non-empty sets $K\cap F$ such that $F$ is a prime filter of $\langle L;\wedge,\vee\rangle$ [2, Theorem III.6.5].

We use $\Pr({\bm{A}})$ to denote the set of all prime deductive filters of an S[I]RL ${\bm{A}}$, including $A$ itself. We always consider $\Pr({\bm{A}})$ to be partially ordered by set inclusion. For a deductive filter $F$ of ${\bm{A}}$, we write

[TABLE]

i.e., ${\uparrow}^{{\bm{A}}}F$ abbreviates ${\uparrow}^{\,\Pr({\bm{A}})}F$, where $\Pr({\bm{A}})$ is considered as a subset of the lattice of deductive filters of ${\bm{A}}$.

Remark 6**.**

Suppose $h\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}$ is a surjective homomorphism of S[I]RLs. The kernel of $h$ is $\bm{\varOmega}^{\bm{A}}K$ for some deductive filter $K$ of ${\bm{A}}$. If $G$ is a deductive filter of ${\bm{A}}$, with $K\subseteq G$, then $h[G]\mathrel{\mbox{\,:\!}=\nolinebreak}\,\{h(g):g\in G\}$ is a deductive filter of ${\bm{B}}$, and by the Correspondence Theorem of Universal Algebra,

[TABLE]

is a lattice isomorphism from the deductive filter lattice of ${\bm{B}}$ onto the lattice of deductive filters of ${\bm{A}}$ that contain $K$; the inverse isomorphism is given by $G\mapsto h[G]$. In particular,

[TABLE]

Clearly, a deductive filter $H$ of ${\bm{B}}$ is prime iff the filter $h^{\leftarrow}[H]$ of ${\bm{A}}$ is prime, so $H\mapsto h^{\leftarrow}[H]$ also defines an isomorphism of partially ordered sets from $\Pr({\bm{B}})$ onto ${\uparrow}^{{\bm{A}}}K$ (both ordered by inclusion).

5. Negative Cones

Definition 7**.**

The negative cone of an S[I]RL ${\bm{A}}=\langle A;\wedge,\vee,\mathbin{\bm{\cdot}},\to,e\,[,\neg]\rangle$ is the Brouwerian algebra

[TABLE]

where $A^{-}=\{a\in A:a\leqslant e\}$ and $a\to^{-}b=(a\to b)\wedge e$ for all $a,b\in A^{-}$.

Lemma 8**.**

Let ${\bm{A}}$ and ${\bm{B}}$ be S[I]RLs, and $F$ a deductive filter of ${\bm{A}}$.**

1. (i)

If $h\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}$ is a homomorphism, then $h|_{A^{-}}$ is a homomorphism from ${\bm{A}}^{-}$ to ${\bm{B}}^{-}$. If, moreover, $h$ is surjective, then so is $h|_{A^{-}}$. 2. (ii)

$A^{-}\cap F$* is a filter of ${\bm{A}}^{-}$, and $({\bm{A}}/F)^{-}\cong{\bm{A}}^{-}/(A^{-}\cap F)$, the isomorphism and its inverse being*

[TABLE]

Proof.

(i) Since homomorphisms between S[I]RLs are isotone maps that preserve $e$, we have $h[A^{-}]\subseteq B^{-}$. Also, as $h$ is a homomorphism, it is clear from the definitions of the operations on the negative cone that $h|_{A^{-}}$ is a homomorphism from ${\bm{A}}^{-}$ to ${\bm{B}}^{-}$. Now suppose $h$ is onto. For each $b\in B^{-}$, there exists $a\in A$ with $h(a)=b$, and since $b\leqslant e$, we have $b=h(a)\wedge e=h(a\wedge e)$. As $a\wedge e\in A^{-}$, this shows that $B^{-}=h[A^{-}]$.

(ii) Clearly, $A^{-}\cap F$ is a filter of ${\bm{A}}^{-}$. Let $q\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}/F$ be the canonical surjection. By (i), $q|_{A^{-}}\colon{\bm{A}}^{-}\mathrel{-\!\!\!\!\!\rightarrow}({\bm{A}}/F)^{-}$ is a surjective homomorphism. For all $a,b\in A^{-}$, we have $a\leftrightarrow b\in F$ iff $(a\leftrightarrow b)\wedge e\in A^{-}\cap F$ (since $F$ is upward closed and contains $e$), i.e., the kernel of $q|_{A^{-}}$ is $\bm{\varOmega}^{{\bm{A}}^{-}}\!(A^{-}\cap F)$. Thus, by the Homomorphism Theorem, $a/(A^{-}\cap F)\mapsto a/F$ defines an isomorphism from ${\bm{A}}^{-}/(A^{-}\cap F)$ onto $({\bm{A}}/F)^{-}$. For any $a\in A$, if $a/F\in(A/F)^{-}$, then $a\wedge e\in A^{-}$ and $(a\wedge e)/F=(a/F)\wedge(e/F)=a/F$, so $a/F\mapsto(a\wedge e)/(A^{-}\cap F)$ defines the inverse of the above isomorphism. ∎

Given an S[I]RL ${\bm{A}}$, if $F$ is a filter of ${\bm{A}}^{-}$, then

[TABLE]

Definition 9**.**

An S[I]RL ${\bm{A}}$ is negatively generated if $A=\operatorname{\mathit{Sg}}^{\bm{A}}(A^{-})$.

As surjective homomorphisms always map generating sets onto generating sets, the following lemma applies.

Lemma 10**.**

If $h\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}$ is a surjective homomorphism of S[I]RLs and ${\bm{A}}$ is negatively generated, then so is ${\bm{B}}$.**

6. Duality for Brouwerian Algebras

A well-known duality between Heyting algebras and Esakia spaces was established in [12, 13]. It entails an analogous duality between the variety $\mathsf{BRA}$ of Brouwerian algebras (considered as a concrete category, equipped with all algebraic homomorphisms) and the category $\mathsf{PESP}$ of ‘pointed Esakia spaces’ defined below, i.e., there is a category equivalence between $\mathsf{BRA}$ and the opposite category of $\mathsf{PESP}$. This is explained, for instance, in [4, Section 3], but we recall the key definitions here.

A structure ${\bm{X}}=\langle X;\tau,\leqslant,m\rangle$ is a pointed Esakia space if $\langle X;\leqslant\rangle$ is a partially ordered set with a greatest element $m$, and $\langle X;\tau\rangle$ is a compact Hausdorff space in which

1. (i)

every open set is a union of clopen (i.e., closed and open) sets; 2. (ii)

${\uparrow}x$ is closed, for all $x\in X$; 3. (iii)

${\downarrow}V$ is clopen, for all clopen $V\subseteq X$.

In this case, the Priestley separation axiom of [42] holds: for all $x,y\in X$,

1. (iv)

if $x\nleqslant y$, then $x\in U$ and $y\notin U$, for some clopen up-set $U\subseteq X$.

The morphisms of ${\mathsf{PESP}}$ are the so-called Esakia morphisms between these objects. They are the isotone continuous functions $g\colon{\bm{X}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{Y}}$ such that,

[TABLE]

It follows that $g(m)=m$, and if $g$ is bijective then $g^{-1}\colon{\bm{Y}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{X}}$ is also an Esakia morphism, so $g$ is a (categorical) isomorphism.

Given ${\bm{A}}\in\mathsf{BRA}$ and $a\in A$, let $\varphi^{{\bm{A}}}(a)$ denote $\{F\in\Pr({\bm{A}}):a\in F\}$ and $\varphi^{{\bm{A}}}(a)^{c}$ its complement $\{F\in\Pr({\bm{A}}):a\notin F\}$. The dual (in ${\mathsf{PESP}}$) of ${\bm{A}}$ is ${\bm{A}}_{*}=\langle\Pr({\bm{A}});\tau,\subseteq,A\rangle$, where $\tau$ is the topology on $\Pr({\bm{A}})$ with sub-basis

[TABLE]

The dual of a morphism $h\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}$ in $\mathsf{BRA}$ is the Esakia morphism $h_{*}\colon{\bm{B}}_{*}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}_{*}$, defined by $F\mapsto h^{\leftarrow}[F]$. Thus, the contravariant functor $(-)_{*}\colon{\mathsf{BRA}}\mathrel{-\!\!\!\!\!\rightarrow}{\mathsf{PESP}}$ is given by ${\bm{A}}\mapsto{\bm{A}}_{*}$; $h\mapsto h_{*}$.

The contravariant functor $(-)^{*}\colon{\mathsf{PESP}}\mathrel{-\!\!\!\!\!\rightarrow}{\mathsf{BRA}}$ works as follows. Let $g\colon{\bm{X}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{Y}}$ be an Esakia morphism, where ${\bm{X}},{\bm{Y}}\in{\mathsf{PESP}}$. Then

[TABLE]

where $\operatorname{Cpu}({\bm{X}})$ is the set of all non-empty clopen up-sets of ${\bm{X}}$, and

[TABLE]

for all $U,V\in\operatorname{Cpu}({\bm{X}})$. The homomorphism $g^{*}\colon{\bm{Y}}^{*}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{X}}^{*}$ is given by $U\mapsto g^{\leftarrow}[U]$. We refer to ${\bm{X}}^{*}$ [resp. $g^{*}$] as the dual of ${\bm{X}}$ [resp. $g$] in $\mathsf{BRA}$.

The functors $(-)_{*}$ and $(-)^{*}$ establish the aforementioned duality. For ${\bm{A}}\in\mathsf{BRA}$ and ${\bm{X}}\in\mathsf{PESP}$, the respective canonical isomorphisms from ${\bm{A}}$ to ${{\bm{A}}_{*}}^{*}$ and from ${\bm{X}}$ to ${{\bm{X}}^{*}}_{*}$ are given by $a\mapsto\varphi^{{\bm{A}}}(a)$ and

[TABLE]

In ${\mathsf{PESP}}$, there is a notion of substructure: an E-subspace of ${\bm{X}}\in{\mathsf{PESP}}$ is a non-empty closed up-set $U$ of ${\bm{X}}$. It is the universe of a pointed Esakia space ${\bm{U}}$, with the restricted order and the subspace topology, so the inclusion ${\bm{U}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{X}}$ is an Esakia morphism.

Lemma 11**.**

([13])**

1. (i)

A homomorphism $h$ between Brouwerian algebras is surjective iff $h_{*}$ is injective. Also, $h$ is injective iff $h_{*}$ is surjective. 2. (ii)

The image of a morphism in ${\mathsf{PESP}}$ is an $E$-subspace of the co-domain.

The ES property for $\mathsf{BRA}$ is relied on in the right-to-left implication of the first assertion of Lemma 11(i). The forward implication in the second assertion of (i) employs the Prime Filter Extension Theorem. In the absence of a convenient reference, a proof of the next lemma is supplied below; the result is presumably well-known.

Lemma 12**.**

Let $F$ be a filter of a Brouwerian algebra ${\bm{A}}$,* and ${q\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}/F}$ the canonical surjection. Then $q_{*}$ is an isomorphism from $({\bm{A}}/F)_{*}$ onto the E-subspace $q_{*}[({\bm{A}}/F)_{*}]$ of ${\bm{A}}_{*}$ whose universe is ${\uparrow}^{\bm{A}}F$. Also, the map*

[TABLE]

is an isomorphism from ${\bm{A}}/F$ onto $(q_{*}[({\bm{A}}/F)_{*}])^{*}$ and the following diagram commutes, where $i_{1}\colon q_{*}[({\bm{A}}/F)_{*}]\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}_{*}$ is the inclusion map.

{{\bm{A}}_{*}}^{*}$$\xrightarrow{\hskip 30.0pt}$$\xrightarrow{\hskip 28.0pt}$$i_{1}^{*}$$(q_{*}[({\bm{A}}/F)_{*}])^{*}=({\uparrow}^{{\bm{A}}}F)^{*}$${\bm{A}}$$\xrightarrow{\hskip 45.0pt}$$\xrightarrow{\hskip 43.0pt}$$q$${\bm{A}}/F

$\xrightarrow{\hskip 15.0pt}$

$\sim$

$\varphi^{\bm{A}}$

$\xrightarrow{\hskip 15.0pt}$

$\sim$

$\varphi^{{\bm{A}}}_{F}$

Furthermore, if $G$ is a filter of ${\bm{A}}$,* with $F\subseteq G$, then the following diagram commutes, where $q^{\prime}\colon a/F\mapsto a/G$, and $i_{2}$ is the inclusion map.*

({\uparrow}^{{\bm{A}}}F)^{*}=(q_{*}[({\bm{A}}/F)_{*}])^{*}$$\xrightarrow{\hskip 20.0pt}$$\xrightarrow{\hskip 18.0pt}$$i_{2}^{*}$$(q_{*}[({\bm{A}}/G)_{*}])^{*}=({\uparrow}^{{\bm{A}}}G)^{*}$${\bm{A}}/F$$\xrightarrow{\hskip 45.0pt}$$\xrightarrow{\hskip 43.0pt}$$q^{\prime}$${\bm{A}}/G

$\xrightarrow{\hskip 15.0pt}$

$\sim$

$\varphi^{{\bm{A}}}_{F}$

$\xrightarrow{\hskip 15.0pt}$

$\sim$

$\varphi^{{\bm{A}}}_{G}$

Proof.

As $q\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}/F$ is a surjective $\mathsf{BRA}$-morphism, Lemma 11(i) shows that $q_{*}\colon({\bm{A}}/F)_{*}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}_{*}$ is an injective Esakia morphism; its image is the E-subspace ${\uparrow}^{\bm{A}}F$ of ${\bm{A}}_{*}$, by Lemma 11(ii) and Remark 6. Let k={q_{*}}^{-1}\big{|}_{{\uparrow}^{\bm{A}}F} , so $k\colon\,{\uparrow}^{\bm{A}}F\cong({\bm{A}}/F)_{*}$ is defined by

[TABLE]

and $k^{*}\colon{({\bm{A}}/F)_{*}}^{*}\cong({\uparrow}^{\bm{A}}F)^{*}$. Since $\varphi^{{\bm{A}}/F}\colon{\bm{A}}/F\cong{({\bm{A}}/F)_{*}}^{*}$, we have

[TABLE]

For each $a\in A$,

[TABLE]

so $k^{*}\circ\varphi^{{\bm{A}}/F}=\varphi^{\bm{A}}_{F}$, whence $\varphi^{\bm{A}}_{F}\colon{\bm{A}}/F\cong({\uparrow}^{\bm{A}}F)^{*}=(q_{*}[({\bm{A}}/F)_{*}])^{*}$.

Commutativity of the second diagram subsumes that of the first (after identification of ${\bm{A}}/\{e\}$ with ${\bm{A}}$, and $\varphi^{\bm{A}}_{\{e\}}$ with $\varphi^{\bm{A}}$).

Accordingly, let $G$ be a filter of ${\bm{A}}$, with $F\subseteq G$, so $q^{\prime}\colon a/F\mapsto a/G$ is a homomorphism from ${\bm{A}}/F$ onto ${\bm{A}}/G$. For each $a\in A$, the respective left and right hand sides of the equation $\varphi^{\bm{A}}_{G}(q^{\prime}(a/F))=i_{2}^{*}(\varphi^{\bm{A}}_{F}(a/F))$ are, by definition,

[TABLE]

which are clearly equal, so $i_{2}^{*}\circ\varphi^{{\bm{A}}}_{F}=\varphi^{{\bm{A}}}_{G}\circ q^{\prime}$. ∎

7. Depth

For ${\bm{X}}=\langle X;\tau,\leqslant,m\rangle\in{\mathsf{PESP}}$ and $x\in X$, we define $\operatorname{depth}{(x)}$ (the depth of $x$ in ${\bm{X}}$) to be the greatest $n\in\omega$ (if it exists) such that there is a chain

[TABLE]

in ${\bm{X}}$. (Thus, $m$ has depth [math] in ${\bm{X}}$.) If no greatest such $n$ exists, we set $\operatorname{depth}{(x)}=\infty$. We define $\operatorname{depth}({\bm{X}})=\sup\,\{\operatorname{depth}(x):x\in X\}$.

For ${\bm{A}}\in{\mathsf{BRA}}$ and any subvariety ${\mathsf{K}}$ of ${\mathsf{BRA}}$, we define

[TABLE]

If a subvariety of ${\mathsf{BRA}}$ is finitely generated (i.e., of the form $\mathbb{V}({\bm{A}})$ for some finite ${\bm{A}}\in{\mathsf{BRA}}$) then it has finite depth, and if it has finite depth then it is locally finite (i.e., its finitely generated members are finite). Both converses are false. For each $n\in\omega$, the class ${{\mathsf{BRA}}_{n}=\{{\bm{A}}\in{\mathsf{BRA}}:\operatorname{depth}({\bm{A}})\leq n\}}$ is a finitely axiomatizable variety. These claims are explained in [4, Section 4], where their antecedents are also discussed.

Definition 13**.**

For any S[I]RL ${\bm{A}}$ and any variety ${\mathsf{K}}$ of S[I]RLs, we define the depth of ${\bm{A}}$ to be the depth of its negative cone ${\bm{A}}^{-}$, and the depth of ${\mathsf{K}}$ to be $\sup\,\{\operatorname{depth}({\bm{B}}):{\bm{B}}\in{\mathsf{K}}\}$.

For each $n\in\omega$, an S[I]RL has depth at most $n$ iff it satisfies the equations that result from the axioms for ${\mathsf{BRA}}_{n}$ when we replace $\rightarrow$ by $\rightarrow^{-}$ and $x$ by $x\wedge e$, for every apparent variable $x$. Thus, the S[I]RLs of depth at most $n$ also form a finitely axiomatizable variety.

Consequently, every finitely generated variety of S[I]RLs has finite depth, and the class of S[I]RLs of depth greater than (any fixed) $n$ is closed under ultraproducts. Since the class of S[I]RLs that are FSI is also closed under ultraproducts, it follows that, when each FSI member of a variety ${\mathsf{K}}$ of S[I]RLs has finite depth, then so does ${\mathsf{K}}$.

8. The ES Property

We can now formulate the main result of this paper.

Theorem 14**.**

Let ${\mathsf{K}}$ be a variety of S[I]RLs, such that each FSI member of ${\mathsf{K}}$ has finite depth and is negatively generated. Then every ${\mathsf{K}}$-epimorphism is surjective.

The proof of Theorem 14 is by contradiction, and it proceeds via a sequence of claims. Let ${\mathsf{K}}$ be as postulated, and suppose that ${\mathsf{K}}$ lacks the ES property. By Theorem 2, some ${\bm{A}}\in{\mathsf{K}}_{FSI}$ has a proper ${\mathsf{K}}$-epic subalgebra. Now ${\bm{A}}/\theta\in{\mathsf{K}}$ for all $\theta\in\mathit{Con}\,{\bm{A}}$, as ${\mathsf{K}}$ is a variety. We shall define a congruence $\theta$ of ${\bm{A}}$ such that the following is true.

Claim 1**:**

There exist $a\in A$ and a ${\mathsf{K}}$-epic proper subalgebra ${\bm{C}}$ of ${\bm{A}}/\theta$ such that ${\bm{C}}$ is negatively generated and ${\bm{A}}/\theta$ is generated by $C^{-}\cup\{a/\theta\}$, and $a/\theta\prec e/\theta$ in ${\bm{A}}/\theta$.

Once $\theta$ has been identified and Claim 1 proved, we shall contradict the fact that ${\bm{C}}$ is ${\mathsf{K}}$-epic in ${\bm{A}}/\theta$, by constructing a non-identity homomorphism $\ell\colon{\bm{A}}/\theta\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}/\theta$, such that $\ell|_{C}={\textup{id}}_{C}$, as follows.

Let $b=a/\theta$. Then $\operatorname{\mathit{Fg}}^{{\bm{A}}/\theta}\{b\}=\{d\in A/\theta:b\leqslant d\}$, because $b<e/\theta$. Let $\alpha=\bm{\varOmega}^{{\bm{A}}/\theta}{\,\operatorname{\mathit{Fg}}^{{\bm{A}}/\theta}\{b\}}$. For any $u,v\in A/\theta$, we have

[TABLE]

In particular, $e/\theta\equiv_{\alpha}b$, by (6) and (8).

Let $\{a_{i}:i\in I\}$ be an indexing of $A/\theta$, and $\vec{c}=c_{0},c_{1},\dots$ a well-ordering of the elements of $C^{-}$. Since $C\neq A/\theta$ and ${\bm{A}}/\theta$ is generated by $C^{-}\cup\{b\}$, it follows that $b\notin C$ and, for each $i\in I$, we have $a_{i}=t_{i}^{{\bm{A}}/\theta}(b,\vec{c})$ for a suitable S[I]RL-term $t_{i}(x,\vec{y})$, where $\vec{y}=y_{0},y_{1},\dots\,\,$. When $a_{i}$ is some $c_{j}\in C^{-}$, we can (and do) choose $t_{i}$ to be the variable $y_{j}$.

We define $\ell\colon A/\theta\mathrel{-\!\!\!\!\!\rightarrow}A/\theta$ by

[TABLE]

By the above choice, $\ell(c_{j})=c_{j}$ for $j=0,1,\dots$, while $\ell[A/\theta]\subseteq C$, because ${\bm{C}}$ is a subalgebra of ${\bm{A}}/\theta$. We claim that $\ell$ is a homomorphism.

To see this, let $\sigma$ be a fundamental S[I]RL-operation symbol, and let $a_{i_{1}},\dots,a_{i_{n}}\in A/\theta$, where $n$ is the rank of $\sigma$. Then $\sigma^{{\bm{A}}/\theta}(a_{i_{1}},\dots,a_{i_{n}})=a_{j}$ for some $j\in I$. For this $j$, we perform the following calculation, where every term is evaluated in ${\bm{A}}/\theta$ :

[TABLE]

By (15), therefore,

[TABLE]

Note that

[TABLE]

(because ${\ell[A/\theta]\subseteq C}$), but $b\notin C$, so

[TABLE]

Since $b\prec e/\theta$ in ${\bm{A}}/\theta$, this forces

[TABLE]

i.e., $e/\theta\leqslant\sigma(\ell(a_{i_{1}}),\dots,\ell(a_{i_{n}}))\leftrightarrow\ell(\sigma(a_{i_{1}},\dots,a_{i_{n}}))$. Then, by (7),

[TABLE]

confirming that $\ell$ is a homomorphism.

For each $c\in C$, we have $c=t^{{\bm{A}}/\theta}(\vec{c})$ for some S[I]RL-term $t$ (as ${\bm{C}}$ is generated by $C^{-}$), so

[TABLE]

This shows that $\ell|_{C}={\textup{id}}_{C}$, but $\ell(b)\neq b$, since $\ell(b)\in C$. As intended, this contradicts the fact that ${\bm{C}}$ is ${\mathsf{K}}$-epic in ${\bm{A}}/\theta$.

It remains to construct $\theta$ and to prove Claim 1. The construction of $\theta$ exploits the assumption that members of ${\mathsf{K}}_{FSI}$ have finite depth.

Recall that ${\bm{A}}$ has a proper ${\mathsf{K}}$-epic subalgebra, ${\bm{B}}$ say. As ${\bm{A}}$ is FSI, so is ${\bm{B}}$ (because ${\mathsf{K}}$ has EDPM, as we noted after (10)). By assumption, therefore, ${\bm{A}}$ and ${\bm{B}}$ are both negatively generated, so ${\bm{B}}^{-}\neq{\bm{A}}^{-}$, because ${\bm{B}}\neq{\bm{A}}$.

For each $F\in\Pr({\bm{A}}^{-})$, we clearly have $B\cap F=B^{-}\cap F=i_{*}(F)$, where $i$ is the inclusion map $i\colon{\bm{B}}^{-}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}^{-}$, considered as a ${\mathsf{BRA}}$-morphism.

As $i$ is not surjective, its dual $i_{*}\colon({\bm{A}}^{-})_{*}\mathrel{-\!\!\!\!\!\rightarrow}({\bm{B}}^{-})_{*}$ is not injective, by Lemma 11(i), i.e., the following set is not empty:

[TABLE]

By assumption, ${\bm{A}}^{-}$ has finite depth, so

[TABLE]

is a non-empty subset of $\omega$, and therefore has a least element, $n$ say. Pick $F_{1}\in\Pr({\bm{A}}^{-})$ such that $\operatorname{depth}(F_{1})=n$ and $\langle F_{1},G\rangle\in W$ for some $G$. Now,

[TABLE]

Having fixed $F_{1}$ in this way, we similarly choose $F_{2}\in\Pr({\bm{A}}^{-})\setminus\{F_{1}\}$ such that $F_{1}\cap B=F_{2}\cap B$ and

[TABLE]

As $\langle F_{1},F_{2}\rangle\in W$, we have $\operatorname{depth}(F_{1})\leq\operatorname{depth}(F_{2})$ (by (16)), so $F_{1}$ is not a proper subset of $F_{2}$.

Lemma 15**.**

If $F_{1}\subsetneq G\in\Pr({\bm{A}}^{-})$,* then $F_{2}\subsetneq G$*.**

Proof.

Let $F_{1}\subsetneq G\in\Pr({\bm{A}}^{-})$, so $\operatorname{depth}(G)<\operatorname{depth}(F_{1})$. As $i_{*}$ is an Esakia morphism and $i_{*}(F_{2})=i_{*}(F_{1})\subseteq i_{*}(G)$, there exists $H\in\Pr({\bm{A}}^{-})$ such that $F_{2}\subseteq H$ and $i_{*}(G)=i_{*}(H)$, by (14), i.e., $G\cap B=H\cap B$. Therefore, if ${G\neq H}$, then $\operatorname{depth}(F_{1})\leq\operatorname{depth}(G)$, by (16). This is a contradiction, so $G=H$, whence $F_{2}\subseteq G$. If $F_{2}=G$, then

[TABLE]

contradicting the fact that $\operatorname{depth}(F_{1})\leq\operatorname{depth}(F_{2})$. Therefore, $F_{2}\subsetneq G$. ∎

Lemma 16**.**

If $F_{2}\subsetneq G\in\Pr({\bm{A}}^{-})$,* then $F_{1}\subseteq G$*.**

Proof.

Let $F_{2}\subsetneq G\in\Pr({\bm{A}}^{-})$. Again, $i_{*}(F_{1})=i_{*}(F_{2})\subseteq i_{*}(G)$, so there exists $H\in\Pr({\bm{A}}^{-})$ such that $F_{1}\subseteq H$ and $i_{*}(G)=i_{*}(H)$. Suppose $G\neq H$. If $F_{1}=H$, then $F_{1}\cap B=G\cap B$, so, by (17), $\operatorname{depth}(F_{2})\leq\operatorname{depth}(G)$, contradicting the fact that $F_{2}\subsetneq G$. Therefore, $F_{1}\subsetneq H$, so $\operatorname{depth}(H)<\operatorname{depth}(F_{1})$. Then, by (16), $\operatorname{depth}(F_{1})\leq\operatorname{depth}(H)$, since $G\cap B=H\cap B$. This is a contradiction, so $G=H$, whence $F_{1}\subseteq G$. ∎

Recalling that $F_{1},F_{2}$ are distinct and that $F_{1}$ is not properly contained in $F_{2}$, we make the following claim:

Claim 2**:**

There are just two possibilities:

1. (A)

$F_{2}\subsetneq F_{1}$, in which case $F_{2}\prec F_{1}$ (in fact, $F_{1}$ is the least strict upper bound of $F_{2}$ in $\Pr({\bm{A}}^{-})$); 2. (B)

$F_{1}$ and $F_{2}$ are incomparable, in which case they have the same depth, the same strict upper bounds and, therefore, the same covers in $\Pr({\bm{A}}^{-})$.

Proof.

If $F_{2}\subsetneq F_{1}$, then $F_{1}$ is the least strict upper bound of $F_{2}$, by Lemma 16, i.e., $F_{1}$ is the unique cover of $F_{2}$. We may therefore assume that $F_{2}$ is not a proper subset of $F_{1}$, i.e., that $F_{1}$ and $F_{2}$ are incomparable. Then, by Lemmas 15 and 16, $F_{1}$ and $F_{2}$ have the same strict upper bounds, and hence the same covers in $\Pr({\bm{A}}^{-})$, from which it clearly follows that they also have the same depth. ∎

By (13), $\operatorname{\mathit{Fg}}^{{\bm{A}}}(F_{1}\cap F_{2})=\{d\in A:d\geqslant c\textup{ for some }c\in F_{1}\cap F_{2}\}$. We define

[TABLE]

Claim 3**:**

The following diagram commutes, where the maps will be defined in the proof.

{\bm{Z}}^{*}$$\xrightarrow{\hskip 45.0pt}

$\sim$

(i_{*}|_{Y})^{*}$${\bm{Y}}^{*}

$\xrightarrow{\hskip 25.0pt}$

$\sim$

\varphi^{{\bm{A}}^{-}}_{F_{1}}$${\bm{A}}^{-}\!/F_{1}$${\bm{B}}^{-}\!/(F_{1}\cap B)$${\bm{X}}^{*}

$\xrightarrow{\hskip 15.0pt}$

$\sim$

\varphi^{{\bm{A}}^{-}}_{F_{1}\cap F_{2}}$${\bm{A}}^{-}\!/(F_{1}\cap F_{2})

$\xrightarrow{\hskip 15.0pt}$

$\sim$

$\varphi^{{\bm{B}}^{-}}_{F_{1}\cap B}$

$\xrightarrow{\hskip 15.0pt}$

$\xrightarrow{\hskip 14.0pt}$

$(i_{Y})^{*}$

$\xrightarrow{\hskip 15.0pt}$

$\xrightarrow{\hskip 14.0pt}$

q$$({\bm{B}}/(\theta|_{B}))^{-}$$\lhook$$\xrightarrow{\hskip 75.0pt}$$j$$({\bm{A}}/\theta)^{-}

$\xrightarrow{\hskip 15.0pt}$

$\sim$

$i_{2}$

$\xrightarrow{\hskip 15.0pt}$

$\sim$

$i_{1}$

Proof.

The map $j\colon{\bm{B}}/(\theta|_{B})\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}/\theta$, defined by $b/(\theta|_{B})\mapsto b/\theta$, is an injective ${\mathsf{K}}$-epimorphism, by Lemma 1. By Lemma 8(i), the restriction of $j$ to $(B/(\theta|_{B}))^{-}$ is a ${\mathsf{BRA}}$-morphism from $({\bm{B}}/(\theta|_{B}))^{-}$ into $({\bm{A}}/\theta)^{-}$. We shall not distinguish notationally between $j$ and this restriction. Whenever $b\in B$ and $b/(\theta|_{B})\leqslant e/(\theta|_{B})$, then $b/(\theta|_{B})=(b\wedge e)/(\theta|_{B})$ and $b/\theta=(b\wedge e)/\theta$, so

[TABLE]

and $j[(B/(\theta|_{B}))^{-}]=\{b/\theta:b\in B^{-}\}$.

Let $K=F_{1}\cap F_{2}$, so $\theta=\bm{\varOmega}^{\bm{A}}\operatorname{\mathit{Fg}}^{\bm{A}}K$. By (13), $A^{-}\cap\operatorname{\mathit{Fg}}^{{\bm{A}}}K=K$, so $B^{-}\cap\operatorname{\mathit{Fg}}^{\bm{A}}K=B^{-}\cap K=B\cap K=B\cap F_{1}$ (since $B\cap F_{1}=B\cap F_{2}$). By (10), $\theta|_{B}=\bm{\varOmega}^{{\bm{B}}}(B\cap\operatorname{\mathit{Fg}}^{{\bm{A}}}K)$, so Lemma 8(ii) supplies isomorphisms

[TABLE]

defined by $a/\theta\mapsto(a\wedge e)/(F_{1}\cap F_{2})$ and $b/(\theta|_{B})\mapsto(b\wedge e)/(F_{1}\cap B)$.

By Lemma 12, ${\uparrow}^{{\bm{B}}^{-}}\!(F_{1}\cap B)$ is the universe of an E-subspace, ${\bm{Z}}$ say, of ${{\bm{B}}^{-}}_{\!*}$, and $\varphi^{{\bm{B}}^{-}}_{F_{1}\cap B}\colon{\bm{B}}^{-}\!/(F_{1}\cap B)\cong{\bm{Z}}^{*}$. Also, $q\colon a/(F_{1}\cap F_{2})\mapsto a/F_{1}$ defines a homomorphism from ${\bm{A}}^{-}\!/(F_{1}\cap F_{2})$ onto ${\bm{A}}^{-}\!/F_{1}$. Let ${\bm{X}}$ [resp. ${\bm{Y}}$] be the E-subspace of ${{\bm{A}}^{-}}_{\!*}$ with universe ${\uparrow}^{{\bm{A}}^{-}}\!(F_{1}\cap F_{2})$ [resp. ${\uparrow}^{{\bm{A}}^{-}}\!F_{1}$]. Let $i_{Y}\colon{\bm{Y}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{X}}$ be the inclusion morphism in ${\mathsf{PESP}}$. By Lemma 12, the following diagram commutes.

{\bm{A}}^{-}\!/F_{1}$$\xrightarrow{\hskip 40.0pt}

$\sim$

\varphi^{{\bm{A}}^{-}}_{F_{1}}$${\bm{Y}}^{*}

$\xrightarrow{\hskip 15.0pt}$

$\xrightarrow{\hskip 14.0pt}$

$q$

$\xrightarrow{\hskip 15.0pt}$

$\xrightarrow{\hskip 14.0pt}$

(i_{Y})^{*}$${\bm{A}}^{-}\!/(F_{1}\cap F_{2})$$\xrightarrow{\hskip 25.0pt}

$\sim$

\varphi^{{\bm{A}}^{-}}_{F_{1}\cap F_{2}}$${\bm{X}}^{*}

Recall that the ${\mathsf{BRA}}$-morphism $i\colon{\bm{B}}^{-}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{A}}^{-}$ is the inclusion map. As $i$ is injective, its dual $i_{*}\colon{{\bm{A}}^{-}}_{\!*}\mathrel{-\!\!\!\!\!\rightarrow}{{\bm{B}}^{-}}_{\!*}$ is surjective, by Lemma 11(i).

The above definitions clearly imply that $i_{*}[Y]\subseteq Z$. To establish the reverse inclusion, let $G\in Z$. By the Prime Filter Extension Theorem, $G=H\cap B$ for some $H\in\Pr({\bm{A}}^{-})$. Now, $i_{*}[F_{1}]=B\cap F_{1}\subseteq G=i_{*}[H]$, so by (14), $i_{*}[H]=i_{*}[H^{\prime}]$ for some $H^{\prime}\in{\uparrow}^{{\bm{A}}^{-}}\!F_{1}=Y$, whence $G=i_{*}[H^{\prime}]\in i_{*}[Y]$. Therefore, $Z=i_{*}[Y]$.

We claim that $i_{*}|_{Y}$ is injective. Suppose, on the contrary, that ${H_{1},H_{2}\in Y}$, with $H_{1}\neq H_{2}$ and $i_{*}[H_{1}]=i_{*}[H_{2}]$. For each $k\in\{1,2\}$, (16) shows that $\operatorname{depth}(F_{1})\leq\operatorname{depth}(H_{k})$, but $F_{1}\subseteq H_{k}$, so $H_{k}=F_{1}$, whence $H_{1}=H_{2}$. This contradiction confirms that $i_{*}|_{Y}$ is injective, whence $i_{*}|_{Y}\colon{\bm{Y}}\cong{\bm{Z}}$ in ${\mathsf{PESP}}$. In ${\mathsf{BRA}}$, therefore, $({i_{*}|_{Y}})^{*}\colon{\bm{Z}}^{*}\cong{\bm{Y}}^{*}$.

A composition of isomorphisms in ${\mathsf{BRA}}$ is an isomorphism, so

[TABLE]

To show that the diagram in Claim 3 commutes, it remains to prove that $g\circ i_{2}=\varphi^{{\bm{A}}^{-}}_{F_{1}}\!\circ q\circ i_{1}\circ j$. And indeed, if $b\in B$ and $b/(\theta|_{B})\in(B/(\theta|_{B}))^{-}$, then

[TABLE]

Claim 4**:**

Suppose $k\in\{1,2\}$ and $a\in A^{-}$ and $b\in B^{-}$, where $a\equiv_{\theta}b$. Then $a\in F_{k}$ iff $b\in F_{k}$. Consequently, $a\notin(F_{1}{\backslash}F_{2})\cup(F_{2}{\backslash}F_{1})$.

Proof.

As $a\equiv_{\theta}b$, we have $a\leftrightarrow b\in\operatorname{\mathit{Fg}}^{\bm{A}}(F_{1}\cap F_{2})$, so

[TABLE]

by (13). As $a\rightarrow^{-}b,\,b\rightarrow^{-}a\geqslant a\leftrightarrow^{-}b$, it follows that $a\rightarrow^{-}b,\,b\rightarrow^{-}a\in F_{k}$. Thus, $a\in F_{k}$ iff $b\in F_{k}$. In particular, if $a\in(F_{1}{\backslash}F_{2})\cup(F_{2}{\backslash}F_{1})$ then $b\in B\cap((F_{1}{\backslash}F_{2})\cup(F_{2}{\backslash}F_{1}))$, contradicting the fact that $B\cap F_{1}=B\cap F_{2}$. ∎

By Claim 3, $h\mathrel{\mbox{\,:\!}=\nolinebreak}\,\varphi^{{\bm{A}}^{-}}_{F_{1}\cap F_{2}}\circ i_{1}\,\colon({\bm{A}}/\theta)^{-}\cong{\bm{X}}^{*}$ and, for each $a\in A$ such that $a/\theta\in(A/\theta)^{-}$, we have

[TABLE]

By Claim 2, $F_{1}{\backslash}F_{2}\neq\emptyset$. In fact,

[TABLE]

To confirm this, let $d\in F_{1}{\backslash}F_{2}$. Clearly, ${\uparrow}^{{\bm{A}}^{-}}\!F_{1}\subseteq h(d/\theta)$. Conversely, let $H\in h(d/\theta)$. If $F_{1}\not\subseteq H$, then since $F_{1}\cap F_{2}\subseteq H\in\Pr({\bm{A}}^{-})$, we have $F_{2}\subseteq H$, by (11). In that case, $F_{2}\subsetneq H$, because $d\in H{\backslash}F_{2}$, but this contradicts Lemma 16. Thus, $F_{1}\subseteq H$, and so $h(d/\theta)={\uparrow}^{{\bm{A}}^{-}}\!F_{1}$, as claimed.

In Case (A) of Claim 2, we have ${\uparrow}^{{\bm{A}}^{-}}\!F_{2}={\uparrow}^{{\bm{A}}^{-}}\!(F_{1}\cap F_{2})=X=h(e/\theta)$.

In Case (B), we have $F_{2}{\backslash}F_{1}\neq\emptyset$, and we claim that

[TABLE]

To see this, let $d\in F_{2}{\backslash}F_{1}$. It is clear that ${\uparrow}^{{\bm{A}}^{-}}\!F_{2}\subseteq h(d/\theta)$, so consider $H\in h(d/\theta)$. If $F_{2}\not\subseteq H$ then, since $F_{1}\cap F_{2}\subseteq H$, we have $F_{1}\subseteq H$, by (11). In that case, $F_{1}\subsetneq H$, as $d\in H{\backslash}F_{1}$, but this contradicts Lemma 15. Thus, $F_{2}\subseteq H$, and so $h(d/\theta)\subseteq{\uparrow}^{{\bm{A}}^{-}}\!F_{2}$.

We define $M=(A/\theta)^{-}\setminus j[(B/(\theta|_{B}))^{-}]$.

Claim 5**:**

Fix any $a_{1}\in F_{1}{\backslash}F_{2}$. Choose $a_{2}$ to be $e$ in Case (A) of Claim 2, and an arbitrary element of $F_{2}{\backslash}F_{1}$ in Case (B). Then

[TABLE]

Moreover, $a/\theta\prec e/\theta$ (in ${\bm{A}}/\theta$) for all $a\in A$ such that $a/\theta\in M$.

Proof.

Observe that $a_{1},a_{2}\leqslant e$ and, as we showed above,

[TABLE]

By Claim 4 and (18), we have $a_{1}/\theta\in M$ and, in Case (B), $a_{2}/\theta\in M$. In Case (A), $a_{2}/\theta\notin M$, since $e\in B$.

Because $h$ is an isomorphism, $h[M]=X^{*}\setminus h[j[(B/(\theta|_{B}))^{-}]]$ and, for the first assertion of Claim 5, it suffices to prove that $h[M]\subseteq\{{\uparrow}^{{\bm{A}}^{-}}\!F_{1},\,{\uparrow}^{{\bm{A}}^{-}}\!F_{2}\}$.

Suppose, with a view to contradiction, that there exists $\mathcal{U}\in h[M]$ with $\mathcal{U}\notin\{{\uparrow}^{{\bm{A}}^{-}}\!F_{1},\,{\uparrow}^{{\bm{A}}^{-}}\!F_{2}\}$. Then $\mathcal{U}\subseteq X$, but $\mathcal{U}\neq X$, because

[TABLE]

We show first that $\mathcal{U}\subsetneq{\uparrow}^{{\bm{A}}^{-}}\!F_{1}$.

Suppose $F_{1},F_{2}\in\mathcal{U}$. For each $H\in X$, we have $F_{1}\subseteq H$ or $F_{2}\subseteq H$, by (11), so $H\in\mathcal{U}$ (since $\mathcal{U}$ is upward closed). This shows that $X\subseteq\mathcal{U}$, a contradiction. Therefore, $F_{1}$ and $F_{2}$ don’t both belong to $\mathcal{U}$.

Suppose $F_{2}\in\mathcal{U}$. Then $F_{1}\notin\mathcal{U}$ and ${\uparrow}^{{\bm{A}}^{-}}\!F_{2}\subseteq\mathcal{U}$, as $\mathcal{U}$ is upward closed. If $H\in\mathcal{U}$, then $H\neq F_{1}$ and $F_{1}\cap F_{2}\subseteq H$ (as $\mathcal{U}\subseteq X$). In that case, $F_{2}\subseteq H$ (otherwise, $F_{1}\subseteq H$, by (11), whence $F_{1}\subsetneq H$, but then $F_{2}\subsetneq H$, by Lemma 15). This shows that $\mathcal{U}\subseteq{\uparrow}^{{\bm{A}}^{-}}\!F_{2}$, so $\mathcal{U}={\uparrow}^{{\bm{A}}^{-}}\!F_{2}$, contrary to our initial assumptions about $\mathcal{U}$. Therefore, $F_{2}\notin\mathcal{U}$.

We claim that $\mathcal{U}\subseteq{\uparrow}^{{\bm{A}}^{-}}\!F_{1}$. For otherwise, $F_{1}\not\subseteq H$ for some $H\in\mathcal{U}$, whence $H\neq F_{2}$ and, by (11), $F_{2}\subseteq H$, i.e., $F_{2}\subsetneq H$, whereupon Lemma 16 delivers the contradiction $F_{1}\subseteq H$. Thus, $\mathcal{U}\subsetneq{\uparrow}^{{\bm{A}}^{-}}\!F_{1}$ (since $\mathcal{U}\neq{\uparrow}^{{\bm{A}}^{-}}\!F_{1}$, by assumption).

Now we shall argue that $\mathcal{U}\in h[j[(B/\theta|_{B})^{-}]]$ (contradicting the fact that $\mathcal{U}\in h[M]$, and thereby confirming the relation $h[M]\subseteq\{{\uparrow}^{{\bm{A}}^{-}}\!F_{1},\,{\uparrow}^{{\bm{A}}^{-}}\!F_{2}\}$).

As $\mathcal{U}\in X^{*}$ and $\mathcal{U}\subsetneq{\uparrow}^{{\bm{A}}^{-}}\!F_{1}=Y$, we have

[TABLE]

so by Claim 3, there exists $b\in B$ with $b/(\theta|_{B})\in(B/(\theta|_{B}))^{-}$ such that

[TABLE]

where $g$ is as in (19), and $\mathcal{V}\mathrel{\mbox{\,:\!}=\nolinebreak}\,h(b/\theta)$. By (18), we may assume that $b\leqslant e$.

Now $\mathcal{V}\in X^{*}$, so $\mathcal{V}$ is an upward-closed subset of $X$. Note that $F_{1}\notin\mathcal{V}$ (otherwise $Y={\uparrow}^{{\bm{A}}^{-}}\!F_{1}\subseteq\mathcal{V}$, yielding the contradiction $\mathcal{U}\subsetneq Y=Y\cap\mathcal{V}=\mathcal{U}$). It follows that $Y\not\subseteq\mathcal{V}$ (as $F_{1}\in Y$).

For any $H\in\mathcal{V}$, if $F_{1}\subseteq H$, then $F_{1}\subsetneq H$ (as $F_{1}\notin\mathcal{V}$), whence $F_{2}\subsetneq H$ (by Lemma 15), whereas if $F_{1}\not\subseteq H$, then $F_{2}\subseteq H$ (by (11)). This shows that $\mathcal{V}\subseteq{\uparrow}^{{\bm{A}}^{-}}\!F_{2}$.

We now argue that $\mathcal{V}\subseteq Y$.

Suppose, on the contrary, that there exists $H\in\mathcal{V}{\backslash}Y$. Then $F_{1}\not\subseteq H$ (by definition of $Y$), so $F_{2}\subseteq H$, by (11). Now Lemma 16 prevents $F_{2}$ from being a proper subset of $H$, so $F_{2}=H$. In particular, $F_{2}\in\mathcal{V}$, so ${\uparrow}^{{\bm{A}}^{-}}\!F_{2}\subseteq\mathcal{V}$, whence $\mathcal{V}={\uparrow}^{{\bm{A}}^{-}}\!F_{2}$.

In Case (A) of Claim 2, it would follow that $Y={\uparrow}^{{\bm{A}}^{-}}\!F_{1}\subseteq{\uparrow}^{{\bm{A}}^{-}}\!F_{2}=\mathcal{V}$, a contradiction.

In Case (B), we have $e\geqslant a_{2}\in F_{2}{\backslash}F_{1}$ and $h(a_{2}/\theta)={\uparrow}^{{\bm{A}}^{-}}\!F_{2}=\mathcal{V}=h(b/\theta)$. Then, since $h$ is injective, $a_{2}/\theta=b/\theta$, contradicting Claim 4.

This confirms that $\mathcal{V}\subseteq Y$, and so $\mathcal{V}=Y\cap\mathcal{V}=\mathcal{U}$. Therefore,

[TABLE]

completing the proof that $M$ is $\{a_{1}/\theta\}$ in Case (A), and is $\{a_{1}/\theta,\,a_{2}/\theta\}$ in Case (B).

It remains to show that $a/\theta\prec e/\theta$ in ${\bm{A}}/\theta$, whenever $a/\theta\in M$.

To establish that $a_{1}/\theta\prec e/\theta$ in ${\bm{A}}/\theta$ (i.e., in $({\bm{A}}/\theta)^{-}$), it suffices to show that ${\uparrow}^{{\bm{A}}^{-}}\!F_{1}\prec X$ in ${\bm{X}}^{*}$, because $h$ is an isomorphism.

Suppose ${\uparrow}^{{\bm{A}}^{-}}\!F_{1}\subsetneq\mathcal{W}\subsetneq X$, where $\mathcal{W}\in X^{*}$. Then $F_{1}\not\subseteq H$ for some $H\in\mathcal{W}$, whence $F_{2}\subseteq H$, by (11). We cannot have $F_{2}=H$, otherwise ${\uparrow}^{{\bm{A}}^{-}}\!F_{2}\subseteq\mathcal{W}$, in which case every element $G$ of $X$ belongs to $\mathcal{W}$ (as $G$ contains $F_{1}$ or $F_{2}$, again by (11)). Therefore, $F_{2}\subsetneq H$, and so $F_{1}\subseteq H$, by Lemma 16. This contradiction confirms that $a_{1}/\theta\prec e/\theta$ in ${\bm{A}}/\theta$.

We may now assume that Case (B) applies. The desired conclusion $a_{2}/\theta\prec e/\theta$ amounts similarly to the claim that ${\uparrow}^{{\bm{A}}^{-}}\!F_{2}\prec X$ in ${\bm{X}}^{*}$. Suppose ${\uparrow}^{{\bm{A}}^{-}}\!F_{2}\subsetneq\mathcal{W}\subsetneq X$, where $\mathcal{W}\in X^{*}$. Then $F_{2}\not\subseteq H$ for some $H\in\mathcal{W}$. Now $H\in X$, so $F_{1}\subseteq H$, by (11). Then $F_{1}=H$, by Lemma 15, so $F_{1}\in\mathcal{W}$, whence ${\uparrow}^{{\bm{A}}^{-}}\!F_{1}\subseteq\mathcal{W}$. By (11) again, $X\subseteq({\uparrow}^{{\bm{A}}^{-}}\!F_{1})\cup({\uparrow}^{{\bm{A}}^{-}}\!F_{2})\subseteq\mathcal{W}$, contradicting the fact that $\mathcal{W}\subsetneq X$. Therefore, $a_{2}/\theta\prec e/\theta$ in ${\bm{A}}/\theta$. ∎

We are now in a position to prove Claim 1 (and hence Theorem 14).

Proof of Claim 1.

Since ${\bm{A}}$ and ${\bm{B}}$ are negatively generated, so are ${\bm{A}}/\theta$ and ${\bm{B}}/(\theta|_{B})$, by Lemma 10. The subalgebra

[TABLE]

of ${\bm{A}}/\theta$ is isomorphic to ${\bm{B}}/(\theta|_{B})$, so ${\bm{J}}$ is also negatively generated. By Lemma 8(i), $J^{-}=j[(B/(\theta|_{B}))^{-}]$, whence $M=(A/\theta)^{-}{\backslash}J^{-}$. As ${\bm{B}}$ is ${\mathsf{K}}$-epic in ${\bm{A}}$, Lemma 1 shows that ${\bm{J}}$ is ${\mathsf{K}}$-epic in ${\bm{A}}/\theta$. Moreover,

[TABLE]

is negatively generated (since $J^{-}\cup\{a_{1}/\theta\}\subseteq S^{-}$), and ${\bm{J}}$ is a subalgebra of ${\bm{S}}$ (as $J=\operatorname{\mathit{Sg}}^{{\bm{A}}/\theta}(J^{-})$), so ${\bm{S}}$ is ${\mathsf{K}}$-epic in ${\bm{A}}/\theta$ (because ${\bm{J}}$ is).

Observe that $J\neq A/\theta$, because $a_{1}/\theta\notin J$ (by Claim 4 and (18), since $a_{1}\in F_{1}{\backslash}F_{2}$), and that $A/\theta=\operatorname{\mathit{Sg}}^{{\bm{A}}/\theta}((A/\theta)^{-})=\operatorname{\mathit{Sg}}^{{\bm{A}}/\theta}(J^{-}\cup M)$.

We choose ${\bm{C}}={\bm{J}}$ and $a=a_{1}$ in Case (A). We make the same choices in Case (B) if $a_{2}/\theta\in S$. Under these conditions, $J^{-}\cup M=J^{-}\cup\{a_{1}/\theta\}$ (by Claim 5) and $A/\theta=\operatorname{\mathit{Sg}}^{{\bm{A}}/\theta}(J^{-}\cup M)\subseteq S=\operatorname{\mathit{Sg}}^{{\bm{A}}/\theta}(C^{-}\cup\{a/\theta\})$, so ${\bm{A}}/\theta$ is generated by $C^{-}\cup\{a/\theta\}$, as required.

In Case (B), if $a_{2}/\theta\notin S$ (whence $S\neq A/\theta$), we choose ${\bm{C}}={\bm{S}}$ and $a=a_{2}$, whereupon $J^{-}\cup M=J^{-}\cup\{a_{1}/\theta,a_{2}/\theta\}$ (by Claim 5) and

[TABLE]

so again, ${\bm{A}}/\theta$ is generated by $C^{-}\cup\{a/\theta\}$. ∎

9. Reflections and De Morgan Monoids

Given an SRL ${\bm{A}}$, let $A^{\prime}=\{a^{\prime}:a\in A\}$ be a disjoint copy of $A$, and let $\bot,\top$ be distinct non-elements of $A\cup A^{\prime}$. The reflection $\textup{R}({\bm{A}})$ of ${\bm{A}}$ is the SIRL with universe $\textup{R}(A)=A\cup A^{\prime}\cup\{\bot,\top\}$ such that ${\bm{A}}$ is a subalgebra of the SRL-reduct of $\textup{R}({\bm{A}})$ and, for all $a,b\in A$ and $x,y\in\textup{R}(A)$,

[TABLE]

Since $f=e^{\prime}$, we have $\top=f^{2}$ and $\bot=\neg(f^{2})$, so $\bot,\top$ belong to every subalgebra of $\textup{R}({\bm{A}})$.

The reflection construction originates with Meyer [34]; also see [19, Section 9]. It preserves (and reflects) distributivity and the square-increasing law, so ${\bm{A}}$ is a Dunn monoid iff $\textup{R}({\bm{A}})$ is a De Morgan monoid. Also, ${\bm{A}}$ is FSI iff $\textup{R}({\bm{A}})$ is.

The reflection of a variety $\mathsf{K}$ of SRLs is the variety

[TABLE]

of SIRLs. The structure of a member of $\mathbb{R}({\mathsf{K}})$ is illuminated by the following lemma, which is proved in [38, Section 6]. (The extra assumptions there were not relied on in the proof.)

Lemma 17**.**

(cf. [38, Lemma 6.5])*  Let ${\bm{A}}$ be an SRL.*

1. (i)

If ${\bm{B}}$ is a subalgebra of ${\bm{A}}$, then $B\cup\{b^{\prime}:b\in B\}\cup\{\bot,\top\}$ is the universe of a subalgebra of $\textup{R}({\bm{A}})$ that is isomorphic to $\textup{R}({\bm{B}})$, and every subalgebra of $\textup{R}({\bm{A}})$ arises in this way from a subalgebra of ${\bm{A}}$. 2. (ii)

If $\theta$ is a congruence of ${\bm{A}}$, then

[TABLE]

is a congruence of $\textup{R}({\bm{A}})$, and $\textup{R}({\bm{A}})/\textup{R}(\theta)\cong\,\textup{R}({\bm{A}}/\theta)$. Also, every proper congruence of $\textup{R}({\bm{A}})$ has the form $\textup{R}(\theta)$ for some $\theta\in{{\mathit{Con}}}\,{\bm{A}}$. 3. (iii)

If $\{{\bm{A}}_{i}:i\in I\}$ is a family of SRLs and $\mathcal{U}$ is an ultrafilter over $I$, then $\prod_{i\in I}\textup{R}({\bm{A}}_{i})/\mathcal{U}\,\cong\textup{R}\!\left(\prod_{i\in I}{\bm{A}}_{i}/\mathcal{U}\right)$.

Jónsson’s Theorem [25, 26] states that, for any subclass $\mathsf{L}$ of a congruence distributive variety, $\mathbb{V}(\mathsf{L})_{FSI}\subseteq\mathbb{HSP}_{\mathbb{U}}(\mathsf{L})$. Together with Lemma 17, this yields the next corollary (as every variety is generated by its FSI members).

Corollary 18**.**

Let ${\mathsf{K}}$ be a variety of SRLs, with ${\bm{E}}\in\mathbb{R}({\mathsf{K}})$.* Then ${\bm{E}}$ is FSI iff ${\bm{E}}\cong\textup{R}({\bm{D}})$ for some ${\bm{D}}\in{\mathsf{K}}_{FSI}$*.**

Theorem 19**.**

Let ${\mathsf{K}}$ be a variety of SRLs, let ${\bm{B}}$ be a subalgebra of ${\bm{A}}\in{\mathsf{K}}$,* and identify $\textup{R}({\bm{B}})$ with the subalgebra of $\textup{R}({\bm{A}})$ given in Lemma 17(i). Then*

1. (i)

${\bm{B}}$* is ${\mathsf{K}}$-epic in ${\bm{A}}$ iff $\textup{R}({\bm{B}})$ is $\mathbb{R}({\mathsf{K}})$-epic in $\textup{R}({\bm{A}})$;* 2. (ii)

${\mathsf{K}}$* has the ES property iff $\mathbb{R}({\mathsf{K}})$ has the ES property;* 3. (iii)

${\mathsf{K}}$* is locally finite iff $\mathbb{R}({\mathsf{K}})$ is locally finite.*

Proof.

(i) ($\Rightarrow$): Let $g,h\colon\textup{R}({\bm{A}})\mathrel{-\!\!\!\!\!\rightarrow}{\bm{E}}\in\mathbb{R}({\mathsf{K}})$ be homomorphisms that agree on $\textup{R}({\bm{B}})$. In showing that $g=h$, we may assume that ${\bm{E}}$ is subdirectly irreducible (by the Subdirect Decomposition Theorem), whence ${\bm{E}}=\textup{R}({\bm{D}})$ for some ${\bm{D}}\in{\mathsf{K}}_{FSI}$, by Corollary 18. Since $g,h$ preserve $e,\mathbin{\bm{\cdot}},\neg$, they preserve $\bot,\top$. If $a,b\in A$, then $g(a),h(a)\neq\bot$ (otherwise, the kernel of $g$ or $h$ would identify $\top=a\mathbin{\bm{\cdot}}\top$ with $\bot\mathbin{\bm{\cdot}}\top=\bot$), and $g(a),h(a)\neq\top$ (because the kernels don’t identify $\top=\top\rightarrow\top$ with $\top\rightarrow a=\bot$), while $g(a),h(a)\neq b^{\prime}$ (because the kernels don’t identify $a^{2}\in A$ with $\top=(b^{\prime})^{2}$). Thus, $g[A],h[A]\subseteq D$, and so $g|_{A},h|_{A}$ are homomorphisms from ${\bm{A}}$ to ${\bm{D}}$, which agree on ${\bm{B}}$. As ${\bm{D}}\in{\mathsf{K}}$ and ${\bm{B}}$ is ${\mathsf{K}}$-epic in ${\bm{A}}$, we conclude that $g|_{A}=h|_{A}$. Then $g|_{A^{\prime}}=h|_{A^{\prime}}$, since $g,h$ preserve $\neg$. Consequently, $g=h$.

($\Leftarrow$): Let $g,h\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{D}}\in{\mathsf{K}}$ be homomorphisms that agree on ${\bm{B}}$. Then $\textup{R}({\bm{D}})\in\mathbb{R}({\mathsf{K}})$. Let $\overline{g},\overline{h}\colon\textup{R}({\bm{A}})\mathrel{-\!\!\!\!\!\rightarrow}\textup{R}({\bm{D}})$ be the respective extensions of $g,h$, preserving $\bot,\top$, such that $\overline{g}(a^{\prime})=g(a)^{\prime}$ and $\overline{h}(a^{\prime})=h(a^{\prime})$ for all $a\in A$. Then $\overline{g},\overline{h}$ are homomorphisms that agree on $\textup{R}({\bm{B}})$, so by assumption, $\overline{g}=\overline{h}$, whence $g=h$.

(ii) Obviously, ${\bm{B}}={\bm{A}}$ iff $\textup{R}({\bm{B}})=\textup{R}({\bm{A}})$. Therefore, the implication from right to left follows from (i). For the converse, use Theorem 2, Corollary 18, Lemma 17(i) and item (i) of the present theorem.

(iii) ($\Rightarrow$): As ${\mathsf{K}}$ is locally finite, there is a function $p\colon\omega\mathrel{-\!\!\!\!\!\rightarrow}\omega$ such that, for each $n\in\omega$, every $n$-generated member of ${\mathsf{K}}_{FSI}$ has at most $p(n)$ elements. It suffices to show that, for each $n\in\omega$, every $n$-generated ${\bm{E}}\in\mathbb{R}({\mathsf{K}})_{FSI}$ has at most $2+2p(n)$ elements. By Corollary 18, any such ${\bm{E}}$ may be assumed to be $\textup{R}({\bm{D}})$ for some ${\bm{D}}\in{\mathsf{K}}_{FSI}$. Let $G$ be an irredundant generating set for ${\bm{E}}$, with $\left|G\right|\leq n$. Then $\bot,\top\notin G$. Let $H=(G\cap D)\cup\{\neg g:g\in G\cap D^{\prime}\}$, so $\left|H\right|\leq n$ and ${\bm{C}}\mathrel{\mbox{\,:\!}=\nolinebreak}\,\operatorname{\mathit{Sg}}^{\bm{D}}H$ has at most $p(n)$ elements. By Lemma 17(i), $\textup{R}({\bm{C}})$ may be identified with a subalgebra of ${\bm{E}}$, but then $G\subseteq\textup{R}(C)$, so $\textup{R}({\bm{C}})={\bm{E}}$, whence $\left|E\right|\leq 2+2p(n)$.

($\Leftarrow$): Use the fact that an SIRL of the form $\textup{R}({\bm{A}})$ is generated by $A$. ∎

As a function from the lattice of varieties of SRLs into that of SIRLs, the operator $\mathbb{R}$ is obviously isotone. Using Corollary 18, we can show that $\mathbb{R}$ is also $\subseteq$-reflecting (and therefore injective); the proof is the same as that of [38, Lemma 6.7] (where again, the extra assumptions play no role).

Consequently, as all varieties of Brouwerian algebras of finite depth have the ES property (by [4, Theorem 5.4] or by Theorem 14) and since there are $2^{\aleph_{0}}$ such distinct varieties (even of depth $3$) [29], the following can be inferred from Theorem 19(ii),(iii).

Theorem 20**.**

There are $2^{\aleph_{0}}$ distinct locally finite varieties of De Morgan monoids with the ES property.

10. Further Examples and Applications

10.1. The weak ES property

The weak ES property for a variety ${\mathsf{K}}$ rules out non-surjective ${\mathsf{K}}$-epimorphisms $h\colon{\bm{A}}\mathrel{-\!\!\!\!\!\rightarrow}{\bm{B}}$ in all cases where ${\bm{B}}$ is generated by the union of $h[A]$ and a finite set. By [39, Theorem 5.4], it is equivalent to the demand that no finitely generated member of ${\mathsf{K}}$ has a ${\mathsf{K}}$-epic proper subalgebra. In varieties of logic, it amounts to the so-called finite Beth property for the corresponding deductive system [5, Theorem 3.14, Corollary 3.15] (also see [39, Theorem 7.9]).

Therefore, by an argument of Kreisel [28], every variety of Brouwerian (or Heyting) algebras has the weak ES property. It follows from a result of Campercholi [8, Corollary 6.5] that, in any finitely generated variety with a majority term (e.g., one generated by a finite lattice-based algebra), the weak ES property entails the ES property. This provides a different explanation of the slightly earlier finding that all finitely generated varieties of Brouwerian (or Heyting) algebras have the ES property [4, Corollary 5.5, Theorem 7.2].

Every variety with the weak ES property and the amalgamation property has (a strong form of) the ES property; see [23, 27, 43] and [22, Section 2.5.3]. Consequently, in all varieties of Brouwerian (or Heyting) algebras, amalgamability entails epimorphism-surjectivity, and the amalgamable varieties of these kinds have been classified completely by Maksimova; see [16, 30, 31].

The situation is different for varieties of (possibly non-integral) S[I]RLs, as they may lack the weak ES property, even when they are finitely generated (see Section 10.2 below).

10.2. Hypotheses of the main theorem

If ${\mathsf{K}}$ and ${\mathsf{L}}$ are subvarieties of a congruence distributive variety, with ${{\mathsf{M}}=\mathbb{V}({\mathsf{K}}\cup{\mathsf{L}})}$, then ${\mathsf{M}}_{FSI}={\mathsf{K}}_{FSI}\cup{\mathsf{L}}_{FSI}$, by Jónsson’s Theorem. The hypotheses in Theorem 14 therefore persist in (binary) varietal joins, and of course in subvarieties. This is helpful in applications, because the ES property itself is not hereditary.

Neither of the two hypotheses in Theorem 14 can be dropped.

To see that the finite depth assumption cannot be dropped for varieties of SRLs, it suffices to exhibit a variety of Brouwerian algebras (of infinite depth) without the ES property. This was done in [4, Section 6]. It was subsequently shown in [40] that there are $2^{\aleph_{0}}$ distinct locally finite varieties of Brouwerian algebras that lack the ES property and that have width $2$ (i.e., $2$ is the maximum cardinality of an anti-chain in the dual of an FSI member of the variety). It follows, as in Section 9, that there are $2^{\aleph_{0}}$ locally finite varieties of De Morgan monoids without the ES property, but this could alternatively be deduced from older findings, discussed below.

The demand for negative generation is not redundant either, because some finitely generated varieties of De Morgan monoids (and of Dunn monoids) lack the ES property. Indeed, by an argument of Urquhart [47] (also see [5, Corollary 4.15]), a variety of De Morgan monoids lacks even the weak ES property if it contains a certain six-element algebra ${\bm{C}}$, called the crystal lattice (which is not negatively generated). That algebra is depicted below. (Deletion of $b$ leaves an epic subalgebra behind, owing to the uniqueness of existent relative complements in distributive lattices.) The argument adapts to Dunn monoids, using the SRL-reduct of ${\bm{C}}$.

f^{2}=a\mathbin{\bm{\cdot}}b$$f$$a^{2}=a=\neg a$$b=\neg b=b^{2}$$e$$\neg(f^{2})

In particular, ${\bm{C}}$ is absent from each of the $2^{\aleph_{0}}$ varieties ${\mathsf{K}}$ of De Morgan monoids in Theorem 20, while the corresponding locally finite varieties $\mathbb{V}({\mathsf{K}}\cup\{{\bm{C}}\})$ lack the weak ES property, and by Jónsson’s Theorem, they are distinct.

Although the finite depth assumption in Theorem 14 cannot be dropped, it is not a necessary condition for the ES property. Indeed, epimorphisms are surjective in ${\mathsf{BRA}}$ (hence in its reflection), and in the locally finite variety of relative Stone algebras (i.e., subdirect products of totally ordered Brouwerian algebras). Also, the smallest variety containing the De Morgan monoids (alternatively, the idempotent SRLs) that are totally ordered and negatively generated has the ES property and is locally finite [48]. All of the varieties mentioned in this paragraph have infinite depth.

Likewise, the demand that FSI members be negatively generated is not entailed by the ES property, even in varieties of S[I]RLs of finite depth. Witnessing examples can also be found in [48].

10.3. More varieties of De Morgan monoids

Sugihara monoids—i.e., idempotent De Morgan monoids—are always negatively generated, and the same applies to the SRLs that can be embedded into them (these are the positive Sugihara monoids of [41]). The ES property was established recently for all varieties of [positive] Sugihara monoids [4, Theorems 8.5, 8.6]. For the finitely generated varieties of this kind, the surjectivity of epimorphisms could alternatively be deduced (immediately) from Theorem 14.

Apart from reflections and idempotent cases, Theorem 14 yields further examples as follows.

By [37, Theorem 6.1], the lattice of varieties of De Morgan monoids has just four atoms, each of which is a finitely generated variety satisfying the hypotheses of Theorem 14. One of them is generated by the reflection of a trivial SRL, i.e., by the non-idempotent De Morgan monoid ${\bm{C}}_{4}$ on the chain

[TABLE]

The covers of $\mathbb{V}({\bm{C}}_{4})$ are distinctive, as ${\bm{C}}_{4}$ is the only [math]-generated nontrivial algebra onto which FSI De Morgan monoids may be mapped by non-injective homomorphisms [44, Theorem 1].

There is a largest variety $\mathsf{U}$ of De Morgan monoids consisting of homomorphic pre-images of ${\bm{C}}_{4}$ (along with trivial algebras), and in the subvariety lattice of $\mathsf{U}$, the variety $\mathbb{V}({\bm{C}}_{4})$ has just ten covers [38, Sections 4, 8], only two of which are generated by reflections of Dunn monoids.

Each of these ten varieties is generated by a finite De Morgan monoid that is itself generated by one of the lower bounds of its neutral element (and is thus negatively generated). Therefore, the conditions of Theorem 14 obtain in all ten covers, and hence in their varietal join, so all subvarieties of this join have the ES property.

10.4. Bounds

The original Esakia duality of [13] supplies an equivalence between the category ${\mathsf{HA}}$ of Heyting algebras (and their homomorphisms—which must preserve $\bot$) and the opposite of the category ${\mathsf{ESP}}$ of Esakia spaces.

The objects of ${\mathsf{ESP}}$ are like those of ${\mathsf{PESP}}$, except that they need not have maximum elements; the definition of morphisms is unaffected. For ${\bm{A}}\in{\mathsf{HA}}$ and ${\bm{X}}\in{\mathsf{ESP}}$, we re-define $\Pr({\bm{A}})$ as the set of prime proper filters of ${\bm{A}}$, and $\operatorname{Cpu}({\bm{X}})$ as the set of all clopen up-sets of ${\bm{X}}$, including $\emptyset$. After these changes, the definitions of ${\bm{A}}_{*}$, ${\bm{X}}^{*}$, the duals of morphisms, and the canonical isomorphisms remain the same (but note that ${\bm{A}}_{*}$ is empty when $\left|A\right|=1$). The definition of depth is adjusted so that a Heyting algebra and its Brouwerian reduct have the same depth. (In particular, the depth of the Esakia space reduct of a pointed Esakia space ${\bm{X}}$ exceeds that of ${\bm{X}}$ by $1$.)

Theorem 14 remains true for varieties of bounded S[I]RLs; its proof requires no further alteration. In this form, it generalizes the recent finding that every variety of Heyting algebras of finite depth has surjective epimorphisms [4, Theorem 5.3].