2.1. Rule systems

Throughout the article, we fix a countably infinite set of propositional variables $\mathit {Prop}$ . For a signature $\nu $ (a finite set of propositional connectives), the set $\mathit {Frm}_\nu $ of $\nu $ -formulae is built from $\mathit {Prop}$ using the connectives in $\nu $ in the usual way. A substitution is a map $s:\mathit {Frm}_\nu (\mathit {Prop})\to \mathit {Frm}_\nu (\mathit {Prop})$ which commutes with the operators in $\nu $ .

A rule in signature $\nu $ is a pair $(\Gamma , \Delta )$ such that $\Gamma , \Delta $ are finite subsets of $\mathit {Frm}_\nu $ . In case $\Delta =\{\varphi \},$ we write $\Gamma /\Delta $ simply as $\Gamma /\varphi $ , and analogously if $\Gamma =\{\psi \}$ . Moreover, we write $/\varphi $ for the rule $\varnothing /\varphi $ . A rule is said to be single-conclusion when of the form $\Gamma /\varphi $ , and assumption free when of the form $/\Delta $ . We use $;$ to denote union between finite sets of formulae, so that $\Gamma; \Delta =\Gamma \cup \Delta $ and $\Gamma; \varphi =\Gamma \cup \{\varphi \}$ . We let $\mathit {Rul}_\nu $ be the set of all rules in $\nu $ .

If $\mathcal {S}$ is a set of rule systems and $\Sigma , \Xi $ are sets of rules, we write $\Xi \oplus _{\mathcal {S}}\Sigma $ for the least rule system in $\mathcal {S}$ , if it exists, extending both $\Xi $ and $\Sigma $ . A set of rules $\Sigma $ is said to axiomatize a rule system $\mathtt {S}\in \mathcal {S}$ over some rule system $\mathtt {S}'\in \mathcal {S}$ if $\mathtt {S}'\oplus _{\mathcal {S}}\Sigma =\mathtt {S}$ . Normally, we will take $\mathcal {S}$ to be the set of all extensions of some particular rule system. When $\mathcal {S}$ is clear from context, we write simply $\oplus $ instead of $\oplus _{\mathcal {S}}$ .

In this article, we will work with rule systems in five different signatures.

• • the modal signature $m:=\{\land , \neg , \bot , \square \}$ ;

• • the tense signature $t:=\{\land , \neg , \bot , \square , \blacklozenge \}$ ;

• • the superintuitionistic (si) signature $si:=\{\land , \lor , \to , \bot , \top \}$ ;

• • the bi-superintuitionistic (bsi) signature $bsi:=\{\land , \lor , \to , \leftarrow , \bot , \top \}$ ;

• • the modal superintuitionistic (msi) signature $msi:=\{\land , \lor , \to , \boxtimes , \top , \bot \}$ .

When working in the modal and tense signatures, we will treat the other Boolean and modal connectives as defined in the usual way. We will denote the duals of $\square $ and $\blacklozenge $ as $\lozenge $ and $\blacksquare ,$ respectively. In the bsi signature, we also use the abbreviations

For each unary propositional connective $\heartsuit $ , we define the rules

(K)

(Nec)

A normal modal rule system is a rule system in the signature m containing the rule $/\varphi $ whenever $\varphi $ is a theorem of the Classical Propositional Calculus, as well as the rules (K $_\square $ ), (Nec $_\square $ ) and

(MP) $$ \begin{align} & \varphi\to \psi, \varphi/\psi. \end{align} $$

A normal tense rule system is a rule system in the signature t, whose $\square $ -free and $\blacklozenge $ -free fragments are each a normal modal rule system (with respect to $\square $ and $\blacklozenge ,$ respectively) and which, in addition, contains the rule

(t) $$ \begin{align} & /\varphi\to \square \blacklozenge \varphi. \end{align} $$

We will henceforth omit the prefix “normal.”

An si rule system is a rule system in the signature $si$ containing the rule $/\varphi $ whenever $\varphi $ is a theorem of the intuitionistic propositional calculus $\mathtt {IPC}$ , as well as the rule (MP). A bsi rule system is a rule system in the signature $bsi$ containing the rule $/\varphi $ whenever $\varphi $ is a theorem of the bi-intutionistic propositional calculus $\mathtt {biIPC}$ , as well as the rules (MP) and (Nec ). We refer the reader to [Reference Chagrov and Zakharyaschev14, Chapter 12] and [Reference Rauszer36] for explicit axiomatizations of $\mathtt {IPC}$ and $\mathtt {biIPC,}$ respectively.

Finally, an msi rule system is a rule system in the signature $msi$ , whose $si$ fragment is an si rule system and which, in addition, contains the rules (K $_\boxtimes $ ) and (Nec $_\boxtimes $ ), as well as the following:

(1) $$ \begin{align} &/p\to \boxtimes p, \end{align} $$

(2) $$ \begin{align} &{/}{\boxtimes} p \to (q \lor (q\to p)). \end{align} $$

When $\mathtt {M}$ is a modal (resp., tense and msi) rule system, we write $\mathbf {NExt}(\mathtt {M})$ for the class of all modal (resp., tense and msi) rule systems extending $\mathtt {M}$ . Similarly, when $\mathtt {L}$ is an si or bsi rule system, we write $\mathbf {Ext}(\mathtt {L})$ for the class of all si or bsi rule systems extending $\mathtt {L}$ . We note that all these classes of rule systems form complete lattices, where the meet is intersection and the join is given by the $\oplus $ operation over the relevant class of rule systems.

A (modal, tense, si, bsi, or msi) logic is a (modal, tense, si, bsi, or msi) rule system which can be axiomatized, over the least rule system of the same kind, by a set of assumption-free, single conclusion rules. Logics in this sense correspond one-to-one with logics conceived of as sets of formulae closed under appropriate conditions, a conception that much of the literature in the field of modal and superintuitionitstic logic shares. For example, the (normal) modal logics in the standard sense [e.g., Reference Chagrov and Zakharyaschev14, p. 113] correspond one-to-one with the normal modal rule systems axiomatizable by assumption-free, single conclusion rules. When $\mathtt {M}$ is a modal logic in this sense, there is always a corresponding modal rule system $/\mathtt {M}$ axiomatized by $\{/\varphi :\varphi \in \mathtt {M}\}$ . Conversely, for any modal rule system $\mathtt {N,}$ the set $\{\varphi :/\varphi \in \mathtt {N}\}$ is always a modal logic in the standard sense.

The set $\mathcal {L}$ of rule systems of a given kind which admit an assumption-free, single conclusion axiomatization forms a complete lattice. Moving forward, when $\mathbf {NExt}(\mathtt {M})$ (resp., $\mathbf {Ext}(\mathtt {L}))$ is a lattice of rule systems, we denote the corresponding lattice of logics as $\mathbf {NExtL}(\mathtt {M})$ (resp., $\mathbf {ExtL}(\mathtt {L}))$ . The join $\oplus _{\mathbf {NExtL}(\mathtt {M})}$ coincides with $\oplus _{\mathbf {NExt}(\mathtt {M})}$ . However, the meets $\otimes _{\mathbf {NExtL}(\mathtt {M})}$ and $\otimes _{\mathbf {NExt}(\mathtt {M})}$ generally come apart: the meet in $\mathbf {NExt}(\mathtt {M})$ of two logics may itself fail to be a logic. Likewise in the si and bsi cases.

The meet $\otimes _{\mathbf {NExtL}(\mathtt {M})}$ can be characterized in terms of $\otimes _{\mathbf {NExt}(\mathtt {M})}$ as follows. If $\mathtt {N}\in \mathbf {NExt}(\mathtt {M})$ is a rule system, let $\mathsf {Taut}(\mathtt {N})$ be the logic axiomatized over $\mathtt {M}$ by all assumption-free, single conclusion belonging to $\mathtt {N}$ . Analogously, we define $\mathsf {Taut}(\mathtt {L})$ when $\mathtt {L}$ is an si or bsi rule system.

Proposition 2.3. Let $\mathcal {S}:=\mathbf {NExt}(\mathtt {M}) ($ resp., $\mathbf {Ext}(\mathtt {L}))$ and let $\mathcal {L}:=\mathbf {NExtL}(\mathtt {M}) ($ resp., $\mathbf {ExtL}(\mathtt {L}))$ . Then the identity

$$\begin{align*}\otimes_{\mathcal{L}}\{\mathtt{S}_i:i\in I\}=\mathsf{Taut}\left(\otimes_{\mathcal{S}}\{\mathtt{S}_i:i\in I\}\right) \end{align*}$$

holds for all logics $\{\mathtt {S}_i:i\in I\}\subseteq \mathcal {L}$ .

Throughout the article, we will refer to a number of standard rule systems. We collect all of them in Table 1.

Table 1

Standard rule systems.

2.2. Algebraic semantics

We interpret rule systems over algebras in the same signature. If $\mathfrak {A}$ is a $\nu $ -algebra, we denote its carrier as A. Let $\mathfrak {A}$ be some $\nu $ -algebra. A valuation on $\mathfrak {A}$ is a map $V:\mathit {Frm}_\nu \to A$ , satisfying the condition

$$\begin{align*}V(f(\varphi_1, \ldots, \varphi_n)):=f^{\mathfrak{A}}(V(\varphi_1), \ldots, V(\varphi_n))\end{align*}$$

for each $f\in \nu $ . A pair $(\mathfrak {A}, V),$ where $\mathfrak {A}$ is a $\nu $ -algebra and V a valuation on $\mathfrak {A}$ , is called a model. A model $(\mathfrak {A}, V)$ satisfies a rule $\Gamma /\Delta $ when the following holds: if $V(\gamma )=1$ for all $\gamma \in \Gamma $ , then $V(\delta )=1$ for some $\delta \in \Delta $ . In this case, we write $\mathfrak {A}, V\models \Gamma /\Delta $ . A rule $\Gamma /\Delta $ is valid on a $\nu $ -algebra $\mathfrak {A}$ when $\mathfrak {A}, V\models \Gamma /\Delta $ holds for all valuations V on $\mathfrak {A}$ . When this holds, we write $\mathfrak {A}\vDash \Gamma /\Delta $ , otherwise, we write $\mathfrak {A}\nvDash \Gamma /\Delta $ and say that $\mathfrak {A}$ refutes $\Gamma /\Delta $ . We can extend this notion of validity to classes of $\nu $ -algebras in the obvious way.

Write $\mathcal {A}_\nu $ for the class of all $\nu $ -algebras. For every rule system $\mathtt {S,}$ we define

$$\begin{align*}\mathsf{Alg}(\mathtt{S}):=\{\mathfrak{A}\in \mathcal{A}_\nu:\mathfrak{A}\vDash \mathtt{S}\}.\end{align*}$$

Conversely, if $\mathcal {K}$ is a class of $\nu $ -algebras, we set

$$ \begin{align*} \mathsf{ThR}(\mathcal{K})&:=\{\Gamma/\Delta\in \mathit{Rul}_\nu:\mathcal{K}\vDash \Gamma/\Delta\}. \end{align*} $$

A variety (resp., universal class) of $\nu $ -algebras is a class of $\nu $ -algebras closed under homomorphic images, subalgebras, and direct products (resp., under isomorphic copies, subalgebras, and ultraproducts). When $\mathcal {K}$ is a class of $\nu $ -algebras, we write $\mathbf {Var}(\mathcal {K})$ and $\mathbf {Uni}(\mathcal {K}),$ respectively, for the class of subvarieties and of universal subclasses of $\mathcal {K}$ . It is well known that both $\mathbf {Var}(\mathcal {K})$ and $\mathbf {Uni}(\mathcal {K})$ admit the structure of a complete lattice. The meet operations of $\mathbf {Var}(\mathcal {K})$ and $\mathbf {Uni}(\mathcal {K})$ , denoted $\otimes _{\mathbf {Var}(\mathcal {K})}$ and $\otimes _{\mathbf {Uni}(\mathcal {K}),}$ respectively, coincide. However, the joins $\oplus _{\mathbf {Var}(\mathcal {K})}$ and $\oplus _{\mathbf {Uni}(\mathcal {K})}$ generally come apart. We can characterize $\oplus _{\mathbf {Var}(\mathcal {K})}$ in terms of $\oplus _{\mathbf {Uni}(\mathcal {K})}$ . For $\mathcal {U}\in \mathbf {Uni}(\mathcal {K})$ , let $\mathsf {Var}(\mathcal {U})$ be the least variety in which $\mathcal {U}$ is contained.

Proposition 2.4. Let $\mathcal {K}$ be a class of $\nu $ -algebras. Then the identity

$$\begin{align*}\oplus_{\mathbf{Var}(\mathcal{K})}\{\mathcal{V}_i:i\in I\}=\mathsf{Var}(\oplus_{\mathbf{Uni}(\mathcal{K})}\{\mathcal{V}_i:i\in I\})\end{align*}$$

holds for any $\{\mathcal {V}_i:i\in I\}\subseteq \mathbf {Var}(\mathcal {K})$ .

Throughout the article, we study the structure of lattices of rule systems via semantic methods. This is made possible by the following fundamental result connecting the syntactic types of rule system to closure conditions on the classes of algebras that validate them. Item 1 is widely known as Birkhoff’s theorem, after [Reference Birkhoff9].

In this sense, $\nu $ -logics correspond to varieties of $\nu $ -algebras, whereas $\nu $ -rule systems correspond to universal classes of $\nu $ -algebras.

We now briefly describe the classes of algebras we shall use to interpret the rule systems under discussion in more detail, and review some of their basic properties. For further details on these structures, we point the reader to [Reference Chagrov and Zakharyaschev14, Reference Esakia20, Reference Esakia21, Reference Kowalski30, Reference Pedroso De Lima Martins35, Reference Rauszer37, Reference Venema, Blackburn, van Benthem and Wolter41].

A Heyting algebra is a tuple $\mathfrak {H}=(H, \land , \lor , \to , 0, 1)$ such that $(H, \land , \lor , 0, 1)$ is a bounded distributive lattice and for every $a, b, c\in A,$ we have

$$\begin{align*}c\leq a\to b\iff a\land c\leq b.\end{align*}$$

A bi-Heyting algebra is a tuple $\mathfrak {H}=(H, \land , \lor , \to ,\leftarrow , 0, 1)$ such that the $\leftarrow $ -free reduct of $\mathfrak {H}$ is a Heyting algebra, and such that for all $a, b, c\in H,$ we have

$$\begin{align*}a\leftarrow b\leq c\iff a\leq b\lor c.\end{align*}$$

Equivalently, a bi-Heyting algebra can be defined as a Heyting algebra $\mathfrak {H}$ whose order dual is also a Heyting algebra, whose implication is defined by the identity

$$\begin{align*}a\leftarrow b:=\bigwedge\{c\in H:a\leq b\lor c\}.\end{align*}$$

A modal algebra is a tuple $\mathfrak {M}=(M, \land , \lor , \neg , \square , 0, 1)$ such that $(M, \land , \lor , \neg , 0, 1)$ is a Boolean algebra and the following equations hold:

(3) $$ \begin{align} \square 1&=1, \end{align} $$

(4) $$ \begin{align} \square(a\land b)&=\square a\land \square b. \end{align} $$

In any modal algebra $\mathfrak {M,}$ we can define the compound modality

(5) $$ \begin{align} \square^+a:=\square a \land a. \end{align} $$

A tense algebra is a structure $\mathfrak {M}=(M, \land , \lor , \neg , \square , \blacklozenge , 0, 1)$ , such that both the $\square $ -free and the $\blacklozenge $ -free reducts of $\mathfrak {M}$ are modal algebras (the former with respect to the dual of $\blacklozenge $ ), and $\square , \blacklozenge $ form a residual pair. That is, for all $a, b\in M,$ we have the following identity:

(6) $$ \begin{align} \blacklozenge a \leq b\iff a\leq \square y. \end{align} $$

Finally, a frontal Heyting algebra is a structure $\mathfrak {H}=(H, \land , \lor , \to ,\boxtimes , 0, 1)$ whose $\boxtimes $ -free reduct is a Heyting algebra and such that $\boxtimes $ satisfies the identities (3) and (4), as well as the following inequalities:

(7) $$ \begin{align} a&\leq \boxtimes a, \end{align} $$

(8) $$ \begin{align} \boxtimes a&\leq b\lor (b\to a). \end{align} $$

We write $\mathsf {HA}, \mathsf {biHA}, \mathsf {MA}, \mathsf {Ten}$ , and $\mathsf {FHA}$ for the classes of Heyting algebras, bi-Heyting algebras, modal algebras, tense algebras, and frontal Heyting algebras, respectively. It is well known that all these classes are equationally definable, hence varieties by Theorem 2.5. What is more, their universal subclasses are algebraic counterparts of the rule systems introduced in the previous section, in the sense spelled out by the following theorem.

Lastly, we introduce some uniform notation to refer to the non truth-functional operations of a $\nu $ -algebra. For $\mathfrak {A}$ a $\nu $ -algebra, let

$$\begin{align*}\mathit{op}(\mathfrak{A}):=\begin{cases} \{\to \} &\text{if }\mathfrak{A}\in \mathsf{HA},\\ \{\to, \leftarrow \} &\text{if }\mathfrak{A}\in \mathsf{biHA},\\ \{\square \} &\text{if }\mathfrak{A}\in \mathsf{MA},\\ \{\square, \blacklozenge \} &\text{if }\mathfrak{A}\in \mathsf{Ten},\\ \{\to, \boxtimes \} &\text{if }\mathfrak{A}\in \mathsf{FHA.} \end{cases}\end{align*}$$

2.3. Geometric semantics and duality

All the rule systems mentioned so far also admit a more suggestive geometric-topological semantics, which we shall rely on in the proofs of several results. We sketch this semantics here and relate the basic topological structures it involves to their algebraic counterparts.

A Stone space is a compact Hausdorff space with a basis of clopens. The topological structures we shall work with are all expansions of Stone spaces with one or more binary relations satisfying various conditions. For each of the signatures $\nu $ presented earlier, there is a corresponding class of such spaces, which for the moment we call $\nu $ -spaces. When $\mathfrak {X}:=(X, \preceq _1, \ldots , \preceq _n, \mathcal {O})$ is a $\nu $ -space we let $\mathsf {Clop}(\mathfrak {X})$ denote the set of clopen subsets of $\mathfrak {X}$ , and let $\mathsf {ClopUp}_{\preceq _i}(\mathfrak {X})$ denote the set of clopen upsets of $\mathfrak {X}$ with respect to the relation $\preceq _i$ , i.e., those elements of $\mathsf {Clop}(\mathfrak {X})$ which are upward-closed with respect to the relation $\preceq _i$ . Moreover, for $U\subseteq X,$ we write

In case $U=\{x\},$ we write and instead of and . When the space in question is only equipped with one relation or when the relation in question is clear from context, we may omit the subscripts from any of these operations.

We now describe these spaces in more detail. An Esakia space is a triple $\mathfrak {X}=(X, \leq , \mathcal {O})$ such that $(X, \mathcal {O})$ is a Stone space and $\leq $ is a partial order satisfying the following conditions:

1. (1) is closed for every $x\in X$ ;

2. (2) for every $U\in \mathsf {Clop}(\mathfrak {X})$ .

If, in addition, the structure $\mathfrak {X}^{-1}=(X, \geq , \mathcal {O})$ is also an Esakia space, where $\geq $ is the converse of $\leq $ , then we call $\mathfrak {X}$ a bi-Esakia space.

A modal space is a triple $\mathfrak {X}=(X, R, \mathcal {O})$ such that $(X, \mathcal {O})$ is a Stone space and R is a binary relation—not necessarily reflexive and transitive—satisfying conditions (1) and (2) above. When the structure $\mathfrak {X}^{-1}=(X, \breve {R}, \mathcal {O})$ is also a modal space, where $\breve {R}$ is the converse of R, we call $\mathfrak {X}$ a tense space.

Finally, a modalized Esakia space is a quadruple $\mathfrak {X}=(X, \leq , \sqsubseteq , \mathcal {O})$ such that ${(X, \leq , \mathcal {O})}$ is an Esakia space and the following conditions hold:

1. (1) whenever ;

2. (2) the reflexive closure of $\sqsubseteq $ coincides with $\leq $ .

Let $\mathfrak {X}=(X, \preceq _1, \ldots , \preceq _n, \mathcal {O})$ and $\mathfrak {X}'=(X', \preceq ^{\prime }_1, \ldots , \preceq ^{\prime }_n, \mathcal {O}')$ be $\nu $ -spaces. A mapping $f:\mathfrak {X}\to \mathfrak {X}'$ is called a $\nu $ bounded morphism when it is continuous and satisfies the conditions below for all $x, y\in X$ and each $i\leq n$ :

1. (1) $x\preceq _i y$ only if $f(x)\preceq ^{\prime }_i f(y)$ ;

2. (2) $f(x)\preceq ^{\prime }_i f(y)$ only if there is $z\in f^{-1}(y)$ such that $x\preceq _i z$ .

In the special case where $\nu \in \{\mathit {bsi, ten}\}$ , we must, in addition, require that the above conditions hold for the converses of $\preceq , \preceq '$ .

We now describe how to interpret $\nu $ -rule systems over $\nu $ -spaces. Let $\mathfrak {X}$ be a $\nu $ -space. If $\nu \in \{\mathit {si, bsi, msi}\}$ , a $\nu $ -valuation on $\mathfrak {X}$ is a mapping that commutes with the connectives in $\nu $ in the usual way. On the other hand, if $\nu \in \{\mathit {md}, \mathit {ten}\}$ , a $\nu $ -valuation on $\mathfrak {X}$ is defined in a similar way, except that we require V to range over instead of . We list below how valuations commute with the most important connectives.

(9)

(10)

(11)

(12)

(13)

Here and throughout the article, we use $-$ and $\smallsetminus $ to denote, respectively, the set-theoretic relative complement and difference operations.

Let $\mathfrak {X}$ be a $\nu $ -space and V a valuation on it. A formula $\varphi $ is satisfied on a model $(\mathfrak {X}, V)$ at a point x if $x\in V(\varphi )$ . In this case, we write $\mathfrak {X}, V, x\vDash \varphi $ , otherwise, we write $\mathfrak {X}, V, x\nvDash \varphi $ and say that the model $(\mathfrak {X}, V)$ refutes $\varphi $ at a point x. A rule $\Gamma /\Delta $ is valid on a model $(\mathfrak {X}, V)$ when the following holds: if $\mathfrak {X}, V, x\vDash \gamma $ holds for each $x\in X$ and every $\gamma \in \Gamma $ , then there is some $\delta \in \Delta $ such that $\mathfrak {X}, V, x\vDash \delta $ holds for each $x\in X$ . In this case, we write $\mathfrak {X}, V\vDash \Gamma /\Delta $ , otherwise, we write $\mathfrak {X}, V\nvDash \Gamma /\Delta $ and say that the model $(\mathfrak {X}, V)$ refutes $\varphi $ . A rule $\Gamma /\Delta $ is valid on a $\nu $ -space $\mathfrak {X}$ if it is valid on the model $(\mathfrak {X}, V)$ for every valuation V on $\mathfrak {X}$ , otherwise $\mathfrak {X}$ refutes $\Gamma /\Delta $ . We write $\mathfrak {X}\vDash \Gamma /\Delta $ to mean that $\Gamma /\Delta $ is valid on $\mathfrak {X}$ , and $\mathfrak {X}\nvDash \Gamma /\Delta $ to mean that $\mathfrak {X}$ refutes $\Gamma /\Delta $ . The notion of validity generalizes to classes of $\nu $ -spaces, as well as to classes of rules, in the obvious way.

For each of the signatures $\nu $ we shall work with, there is a duality result connecting $\nu $ -algebras to $\nu $ -spaces. All these dualities are generalizations of Stone duality, which relates the category of Boolean algebras with homomorphisms to that of Stone spaces with continuous functions [Reference Johnstone28]. We list these dualities in the following theorem.

In each of these cases, we write $\mathfrak {A}_*$ for the space dual to an algebra $\mathfrak {A}$ and $\mathfrak {X}^*$ for the algebra dual to a space $\mathfrak {X}$ . The space $\mathfrak {\mathfrak {A}}_*$ is always an expansion of the space of prime filters of $\mathfrak {A}$ . We write $\beta $ for the map, called the Stone map, which takes element a and returns the set $\beta (a)$ of prime filters in that algebras that contain a. In the other direction, the algebra $\mathfrak {\mathfrak {X}}^*$ is constructed by taking clopen sets (if $\nu \in \{\mathit {md, ten}\}$ ) or clopen upsets (if $\nu \in \{\mathit {si, bsi, msi}\}$ ). We refer the reader to [Reference Castiglioni, Sagastume and San Martín12, Reference Esakia17, Reference Esakia18, Reference Sambin and Vaccaro38] for detailed descriptions and proofs of these dualities.

2.4. Kripke semantics

Besides spaces, in Sections 4 and 5, we shall also work with Kripke frames. We will only use Kripke frames to interpret si, bsi, modal, and tense rule systems. Thus we define a Kripke frame to be a set $\mathfrak {X}:=(X, \preceq )$ , where X is a non-empty set and $\preceq $ is a binary relation on X. An intuitionistic Kripke frame is a Kripke frame $\mathfrak {X}:=(X, \leq )$ , where $\leq $ is a partial order. The notions of $\nu $ bounded morphism for $\nu \in \{\mathit {md, ten, si, bsi}\}$ are defined the same way as for spaces, but omitting the requirement of continuity.

For $\nu \in \{\mathit {md}, \mathit {ten}\}$ , a $\nu $ -valuation on a Kripke frame $\mathfrak {X}$ is a mapping $V:\mathit {Frm}_\nu \to \wp (X)$ that commutes with the connectives in $\nu $ in the usual way. For $\nu \in \{\mathit {si, bsi}\}$ , $\nu $ -valuation on an intuitionistic Kripke frame $\mathfrak {X}$ is a mapping $V:\mathit {Frm}_\nu \to \wp (X)$ that commutes with the connectives in $\nu $ in the usual way, such that for every $\varphi \in \mathit {Frm}_\nu $ . We extend our notions of satisfaction and validity from spaces to Kripke frames in the obvious way.

We recall briefly the following duality results concerning Kripke frames, which were first proved, respectively, in [Reference de Jongh and Troelstra29, Reference Thomason40] (see also [Reference Litak33]).

We write $\mathfrak {A}_+$ for the Kripke frame dual to an algebra $\mathfrak {A}$ among those mentioned in the theorem above, and $\mathfrak {X}^+$ for the algebra dual to a Kripke frame $\mathfrak {X}$ . The signature of $\mathfrak {X}^+$ will be clear from context. The Kripke frame $\mathfrak {A}_+$ is constructed by expanding the set of completely join prime filters of $\mathfrak {A}$ with a binary relation, which is defined the same way as in the dualities from Theorem 2.8. The algebra $\mathfrak {X}^+$ is constructed the same way as $\mathfrak {Y}^*$ when $\mathfrak {Y}$ is a space, but taking subsets (resp., upsets) instead of clopen subsets (resp., clopen upsets).

2.5. Transitive structures

We close our preliminaries by reviewing some classes of transitive structures we shall encounter throughout the article. Let us first introduce some more notational conventions. We refer to an algebra in $\mathsf {Alg}(\mathtt {S})$ as an $\mathtt {S}$ -algebra. Similarly, we let an $\mathtt {S}$ -space be a space in $\mathsf {Spa}(\mathtt {S})$ , and an $\mathtt {S}$ -frame be a Kripke frame that validates every rule in $\mathtt {S}$ —with the additional requirement that an $\mathtt {S}$ -frame be intuitionistic when $\mathtt {S}$ is a (b)si logic.

We recall that the $\mathtt {K4(.t)}$ -spaces can be characterized as those modal (resp., tense) spaces with a transitive relation, and that the $\mathtt {S4(.t)}$ -spaces coincide with those $\mathtt {K4(.t)}$ -spaces where the relation is, in addition, reflexive. We recall some well-known properties of these spaces. Given a preordered set $(X, R)$ , we define:

We omit subscripts when they can be inferred from context.

Among $\mathtt {S4(.t)}$ -spaces, we shall pay particular attention to $\mathtt {Grz(.t)}$ -spaces. We recall some of their basic properties. Given a preordered set $(X, R)$ and $U\subseteq X$ , we call an element $x\in U$ passive in U when there is no $y\in X\smallsetminus U$ such that $Rxy$ and $Ryz$ for some $z\in U$ . In other words, x is passive in U when one cannot “leave” and “re-enter” U starting from x. A cluster in $(X, R)$ is a set $C\subseteq X$ which is maximal with the property that $Rxy$ and $Ryx$ whenever $x, y\in C$ . A set $U\subseteq {X}$ is said to cut a cluster $C\subseteq X$ when neither $C\subseteq U$ nor $C\cap U=\varnothing $ hold.

We mention another simple fact concerning clusters, which we shall appeal to several times.

Another class of $\mathtt {K4}$ -spaces we shall pay close attention to is the class of $\mathtt {GL}$ -spaces. These spaces display various similarities with $\mathtt {Grz}$ -spaces, as the reader can appreciate by comparing the following results with Proposition 2.11, Theorem 2.12, and Corollary 2.13.

3.1. Mappings on algebras

We begin by reviewing some well-known semantic transformations between algebras. Recall that the free Boolean extension of a Heyting algebra $\mathfrak {H}$ is the unique Boolean algebra $B(\mathfrak {H})$ in which $\mathfrak {H}$ embeds as a distributive lattice, such that the image of $\mathfrak {H}$ under this embedding generates $B(\mathfrak {H})$ as a Boolean algebra [Reference Esakia21, Definition 2.5.6, Constraint 2.5.7; Reference Balbes and Dwinger1, Section V]. For simplicity, we will generally identify $\mathfrak {H}$ with its image in $B(\mathfrak {H})$ . Note that this convention is used in the definitions to follow.

If $\mathfrak {H}$ is a Heyting algebra, the algebra $\sigma _{1}\mathfrak {H}$ is constructed by expanding $B(\mathfrak {H})$ with the operation

$$\begin{align*}\square a:=\bigvee \{b\in H: b\leq a\}.\end{align*}$$

If $\mathfrak {M}$ is a bi-Heyting algebra, we define $\sigma _{2}\mathfrak {H}$ the same way but also add the operation

$$\begin{align*}\blacklozenge a:=\bigwedge \{b\in H: a\leq b\}.\end{align*}$$

Finally, if $\mathfrak {M}$ is a frontal Heyting algebra, we define $\sigma _{3}\mathfrak {H}$ by expanding $B(\mathfrak {H})$ with the operation

$$\begin{align*}\square a:=\boxtimes Ia, \end{align*}$$

where

$$\begin{align*}Ia:=\bigvee\{b\in H:b\leq a\}. \end{align*}$$

It is known that $\sigma (\mathfrak {H})$ is a $\mathtt {Grz(.t)}$ -algebra whenever $\mathfrak {H}$ is a (bi-)Heyting algebra [Reference Esakia21, Corollary 3.5.9; Reference Wolter43, Lemma 16], and moreover that $\sigma _{3}(\mathfrak {H})$ is a $\mathtt {GL}$ -algebra whenever $\mathfrak {H}$ is a frontal Heyting algebra [Reference Esakia20, Corollary 20].

Conversely, if $\mathfrak {M}$ is an $\mathtt {S4}$ -algebra, the algebra $\rho _{1} \mathfrak {M}$ is constructed as follows. As the carrier, we take the bounded lattice $O(\mathfrak {M})$ of open elements of $\mathfrak {M}$ , that is, of those elements $a\in M$ with $\square a=a$ , or, equivalently, $\blacklozenge a=a$ . We expand this lattice with the operation

$$\begin{align*}a\to b:=\square (\neg a\lor b).\end{align*}$$

When $\mathfrak {M}$ is an $\mathtt {S4.t}$ -algebra, we define $\rho _{2} \mathfrak {M}$ the same way but also add the operation

$$\begin{align*}a\leftarrow b:=\blacklozenge ( a\land \neg b).\end{align*}$$

Likewise, if $\mathfrak {M}$ is a $\mathtt {K4}$ -algebra, the algebra $\rho _{3}\mathfrak {M}$ is constructed as follows. As the carrier, we take the bounded lattice $O^+(\mathfrak {M})$ of quasi-open elements of $\mathfrak {M}$ , i.e., those elements of $\mathfrak {M}$ with $\square ^+a=a$ , where $\square ^+a:=\square a\land a$ . We then expand this lattice with the following operations:

$$ \begin{align*} a\to b&:=\square^+ (\neg a\lor b),\\ \boxtimes a&:= \square a. \end{align*} $$

It is well known that $\rho \mathfrak {M}$ is a (bi-)Heyting algebra for every $\mathtt {S4(.t)}$ -algebra $\mathfrak {M}$ , and that $\rho _{3}\mathfrak {M}$ is a frontal Heyting algebra for every $\mathtt {K4}$ -algebra $\mathfrak {M}$ .

All these mappings can be lifted to universal classes. Given $s\in \{1, 2, 3\}$ , let $\mathcal {U, V}$ be universal classes of algebras on which, respectively, $\sigma _{s}$ and $\rho _{s}$ are defined. We then put

$$\begin{align*}\sigma_{s} \mathcal{U}:=\mathsf{Uni}\{\sigma_{s} \mathfrak{H}:\mathfrak{H}\in \mathcal{U}\},\qquad \rho_{s} \mathcal{V}:= \{\rho_{s}\mathfrak{A}:\mathfrak{A}\in \mathcal{V}\}.\end{align*}$$

We also introduce mappings $\tau _{s}:\mathsf {Uni}(\mathcal {I}_s)\to \mathsf {Uni}(\mathcal {C}_s)$ by setting

$$\begin{align*}\tau_{s} \mathcal{W}:=\{\mathfrak{M}\in\mathcal{C} :\rho_{s}\mathfrak{M}\in \mathcal{W}\}. \end{align*}$$

3.2. Mappings on spaces

We now describe the maps defined in the previous subsection dually. If $\mathfrak {X}$ is an Esakia space, we set

$$\begin{align*}\sigma_1 \mathfrak{X}:=(X, R, \mathcal{O}),\qquad\qquad R:= {\leq}.\end{align*}$$

When $\mathfrak {X}$ is a bi-Esakia space we let $\sigma _{2}\mathfrak {X}:=\sigma _{1}\mathfrak {X}$ . Thus $\sigma _{1}$ and $\sigma _{2}$ are just identity maps; we simply notate the relation differently to indicate that we are viewing the structure as a modal or tense space. Moreover, if $\mathfrak {X}$ is a modalized Esakia space, we let $\sigma _{3}\mathfrak {X}$ be the $\leq $ -free reduct of $\mathfrak {X}$ . By Corollary 2.14, if $\mathfrak {X}$ is a (bi-)Esakia space, then $\sigma \mathfrak {X}$ is always a $\mathtt {Grz(.t)}$ -space. Likewise, by Corollary 2.18, $\sigma _{3}\mathfrak {X}$ is always a $\mathtt {GL}$ -space whenever $\mathfrak {X}$ is a modalized Esakia space.

Conversely, let $\mathfrak {Y}:=(Y, R, \mathcal {O})$ be a $\mathtt {K4}$ -space. For $x, y\in Y,$ write $x\sim y$ iff $Rxy$ and $Ryx$ . Define a map $\varrho :Y\to \wp (Y)$ by setting $\varrho (x)=\{y\in Y:x\sim y\}$ . We call this map the skeleton map. When $\mathfrak {Y}$ is an $\mathtt {S4}$ -space we let $\rho _{1}\mathfrak {Y}:=(\varrho [Y], \leq , \mathcal {O}_{\varrho})$ where $\varrho (x)\leq \varrho (y)$ iff $Rxy$ , and $U\in \mathcal {O}_\varrho$ iff $\varrho^{-1}(U)\in \mathcal {O}$ . We let $\rho _{2}$ be the restriction of $\rho _{1}$ to $\mathtt {S4.t}$ spaces. When $\mathfrak {Y}$ is a $\mathtt {K4}$ -space we let $\rho _{3}\mathfrak {Y}:=(\varrho [Y], \leq , \sqsubseteq , \mathcal {O}_{\varrho})$ , where $\varrho (x)\leq \varrho (y)$ iff $R^+xy$ and $\varrho (x)\sqsubseteq \varrho (y)$ iff $Rxy$ , and $\mathcal{O}_\varrho$ is defined as before. Here $R^+$ denotes the reflexive closure of R.

In other words, when $\mathfrak {Y}$ is an $\mathtt {S4}$ -space, the space $\rho _{1}\mathfrak {Y}$ is obtained by collapsing the clusters of $\mathfrak {Y}$ , lifting the relation R clusterwise and endowing the result with the quotient topology under the mapping $\varrho $ . When $\mathfrak {Y}$ is a $\mathtt {K4}$ -space, $\rho _{3}\mathfrak {Y}$ is constructed in a similar way, except that the intuitionistic relation of $\rho _{3}\mathfrak {Y}$ is obtained by lifting the reflexive closure of R clusterwise, and the modal relation is obtained by lifting R itself clusterwise.

We note a simple property of the transformations $\rho _{s}$ , which we shall appeal to later on.

Routine arguments show that the transformations just defined are indeed dual to their algebraic counterparts defined in the previous subsection [see, e.g., Reference Esakia21, Proposition 3.4.15]. This is to say that given $s\in \{1, 2, 3\}$ , for any algebras $\mathfrak {H, M}$ on which the algebraic maps $\sigma _{s}, \rho _{s}$ are defined we have $(\sigma _{s} \mathfrak {H})_*\cong \sigma _{s} \mathfrak {H}_*$ and $(\rho _{s} \mathfrak {M})_*\cong \rho _{s} \mathfrak {M}_*$ . Consequently, for any spaces $\mathfrak {X, Y}$ on which the geometric maps $\sigma _{s}, \rho _{s}$ are defined we have $(\sigma _{s} \mathfrak {X})^*\cong \sigma _{s} \mathfrak {X}^*$ and $(\rho _{s} \mathfrak {Y})^*\cong \rho _{s} \mathfrak {Y}^*$ .

By appealing to these dualities, the following proposition easily follows.

3.3. Mappings on Kripke frames

For $s\in \{1, 2\}$ , we also define versions of the maps $\sigma _{s}, \rho _{s}$ on Kripke frames. When $\mathfrak {X}$ is an intuitionistic Kripke frame, we let $\sigma _{s} \mathfrak {X}=\mathfrak {X}$ . Conversely, if $\mathfrak {X}$ is a Kripke frame with a reflexive and transitive relation, we let $\rho _{s}\mathfrak {X}$ be defined as we did above for spaces, but omitting the conditions concerning topology. Since the maps are defined the same way for the two pairs of signatures $1$ and $2$ , we will always omit signature subscripts when working with Kripke frames. We do not define counterparts of the maps $\sigma _{3}, \rho _{3}$ on Kripke frames.

The $\rho $ transformation on Kripke frames corresponds quite closely with its topological version. In particular, for every Kripke frame $\mathfrak {F}$ on which the mapping $\rho $ is defined, we have $\mathfrak {(\rho \mathfrak {F})}^+\cong \rho \mathfrak {F}^+$ , and so for every perfect $\mathtt {S4(.t)}$ -algebra $\mathfrak {M,}$ we have $\mathfrak {(\rho \mathfrak {M})}_+\cong \rho \mathfrak {M}_+$ . On the other hand, the algebraic version of the map $\sigma $ fails to preserve atomicity [Reference Wolter44, p. 103], so in general, the identity $\sigma \mathfrak {(F)}^+\cong \sigma \mathfrak {F}^+$ may fail. Observe further that when $\mathfrak {F}$ is an intuitionistic Kripke frame, $\sigma \mathfrak {F}$ is not guaranteed to be a Kripke frame for $\mathtt {Grz(.t)}$ . As is well known, the Kripke frames for $\mathtt {Grz(.t)}$ are precisely those partially ordered Kripke frames that are conversely well founded (resp., well-founded and conversely well founded). However, intuitionistic Kripke frames need not be well founded nor conversely well founded.

3.4. Translations

All the mappings we have introduced are semantic counterparts to various translations between formulas in different signatures. These translations are all versions of the Gödel Translation [Reference Gödel23]. For present purposes, we shall define the Gödel translation as a mapping $T_{1}:\mathit {Frm}_{\mathit {si}}\to \mathit {Frm}_{\mathit {md}}$ defined recursively as follows:

$$ \begin{align*} T_{1}(\bot):=\bot, \qquad T_{1}(\varphi\lor \psi)&:=T_{1}(\varphi)\lor T_{1}(\psi),\\ T_{1}(\top):=\top, \qquad T_{1}(\varphi\to \psi)&:=\square (\neg T_{1}(\varphi)\lor T_{1}(\psi)),\\ T_{1}(p):=\square p, \qquad T_{1}(\varphi\land \psi)&:=T_{1}(\varphi)\land T_{1}(\psi). \end{align*} $$

We extend this translation to define two more mappings, $T_{2}:\mathit {Frm}_{\mathit {bsi}}\to \mathit {Frm}_{\mathit {ten}}$ and $T_{3}:\mathit {Frm}_{\mathit {msi}}\to \mathit {Frm}_{\mathit {md}}$ . These mappings are obtained by extending the definition above with one of the following additional conditions [cf. Reference Kuznetsov and Muravitsky31, Reference Wolter, Kracht, Rijke, Wansing and Zakharyaschev42]:

$$\begin{align*}T_{2}(\varphi\leftarrow \psi):=\blacklozenge (T_{2}(\varphi)\land \neg T_{2}(\psi)),\qquad T_{3}(\boxtimes \varphi):=\square T_{3}(\varphi).\end{align*}$$

$T_{2}$ was introduced in [Reference Wolter, Kracht, Rijke, Wansing and Zakharyaschev42], whereas $T_{3}$ is equivalent to the translation introduced in [Reference Kuznetsov and Muravitsky31] (see also [Reference Wolter and Zakharyaschev45, Reference Wolter and Zakharyaschev46]). We extend these mappings from formulae to rules by setting

$$\begin{align*}T_{s}(\Gamma/\Delta):=T_{s}[\Gamma]/T_{s}[\Delta].\end{align*}$$

We will refer to all of these mappings as “Gödel Translations.”

The interactions between the Gödel translations and the semantic mappings previously introduced are described in the next lemma.

Lemma 3.4 [Cf. Reference Jeřábek27, Lemma 3.13].

Let $s\in \{1, 2, 3\}$ and let $\mathfrak {M}$ be an algebra on which $\rho _{s}$ is defined. Then for every rule in the intuitionistic signature of s, we have

$$\begin{align*}\mathfrak{M}\models T_{s}(\Gamma/\Delta)\iff \rho_{s}\mathfrak{M}\models \Gamma/\Delta.\end{align*}$$

Let us now define mappings between logics in different signatures by means of the Gödel translations. For $s\in \{1, 2, 3\}$ , let $\mathtt {L}\in \mathbf {Ext}(\mathtt {I}_s)$ . We define:

$$\begin{align*}\tau \mathtt{L}:=\mathtt{C}_s\oplus \{T_{s}(\Gamma/\Delta):\Gamma/\Delta\in \mathtt{L}\}, \qquad \sigma \mathtt{L}:=\mathtt{C}^+_s\oplus \tau_{s} \mathtt{L}.\end{align*}$$

Conversely, if $\mathtt {M}\in \mathbf {NExt}(\mathtt {C}_s)$ , we put

$$\begin{align*}\rho_{s}\mathtt{M}:=\mathtt{I}_s\oplus \{\Gamma/\Delta: T(\Gamma/\Delta)\in \mathtt{M}\}.\end{align*}$$

Finally, let $\mathtt {L}\in \mathbf {Ext}(\mathtt {I}_s)$ and $\mathtt {M}\in \mathbf {NExt}(\mathtt {M}_s)$ . We say that $\mathtt {M}$ is a companion of $\mathtt {L}$ when $\rho _{s}\mathtt {M}=\mathtt {L}$ . We call the companions of si and msi rule systems modal companions, and the companions of bsi rule systems tense companions.

5.1. Novel proofs

The main problem one needs to deal with in order to prove the results just announced is showing that each syntactic mapping $\sigma _{s}$ is surjective on the codomain $\mathcal {C}^+_s$ (recall we are using the notation introduced in Convention 3.1.) The novelty of our approach lies in the use of stable canonical rules to establish that result.

Our strategy is centered on the following lemma.

In each of the three cases, the $(\Rightarrow )$ direction is immediate from Proposition 3.3. We give full proofs of the other direction in the cases $s=2$ and $s=3$ . A proof of the case $s=1$ can be derived from the proof of the case $s=2$ by making minimal adaptations, which we shall sketch.

Proof of Case s = 2.

We prove the dual statement that $\mathfrak {M}_*\nvDash \Gamma /\Delta $ implies $\sigma \rho \mathfrak {M}_*\nvDash \Gamma /\Delta $ . Let $\mathfrak {X}:=\mathfrak {M}_*$ . In view of Theorem 4.10, it is enough to consider the case $\Gamma /\Delta =\mu (\mathfrak {F}, {\mathfrak {D}})$ , for $\mathfrak {F}$ the dual of a finite $\mathtt {S4(.t)}$ -algebra. So suppose $\mathfrak {X}\nvDash \mu (\mathfrak {F}, {\mathfrak {D}})$ . By Proposition 4.5, there is a stable map $f:\mathfrak {X}\to \mathfrak {F}$ satisfying the BDC for $\mathfrak { D}:=(\mathfrak {D}^\Uparrow , \mathfrak {D}^\Downarrow )$ . We construct a stable map $g:\sigma \rho \mathfrak {X}\to \mathfrak {F}$ which satisfies the BDC for $\mathfrak { D}$ .

Let $C:=\{x_1, \ldots , x_n\}\subseteq F$ be some cluster and let $Z_C:=f^{-1}(C)$ . Since f is relation-preserving, $Z_C$ does not cut clusters. Therefore, by Proposition 3.2, $\varrho [Z_C]$ is clopen, and so is $f^{-1}(x_i)$ for each $x_i\in C$ . Now for each $x_i\in C$ let

$$ \begin{align*} M_i&:=\mathit{max}_R(f^{-1}(x_i)),\\ N_i&:=\mathit{min}_R(f^{-1}(x_i)). \end{align*} $$

By Proposition 2.11 and Corollary 2.13, both $M_i, N_i$ are closed, and moreover neither cuts any cluster. Consequently, by Proposition 3.2 again, both $\varrho [M_i], \varrho [N_i]$ are closed as well.

For each $x_i\in C$ let $O_i:=M_i\cup N_i$ . Clearly, $O_i\cap O_j=\varnothing $ for each distinct $i, j\leq n$ , and since no $O_i$ cuts any cluster this implies $\varrho [O_i]\cap \varrho [O_j]$ for each distinct $i, j\leq n$ . We shall now find disjoint clopens $U_1, \ldots , U_n\in \mathsf {Clop}(\sigma \rho \mathfrak {X})$ with $\varrho [O_i]\subseteq U_i$ and $\bigcup _i U_i=\varrho [Z_C]$ . Let $k\leq n$ and assume that $U_i$ has been defined for all $i<k$ . If $k=n$ , put $U_n=\varrho [Z_C]\smallsetminus \left (\bigcup _{i< k} U_i\right )$ and we are done. Otherwise set $V_k:=\varrho [Z_C]\smallsetminus \left (\bigcup _{i< k} U_i\right )$ and observe that it contains each $\varrho [O_i]$ for $k\leq i\leq n$ . By the separation properties of Stone spaces, for each i with $k<i\leq n,$ there is some $U_{k_i}\in \mathsf {Clop}(\sigma \rho \mathfrak {X})$ with $\varrho [O_k]\subseteq U_{k_i}$ and $\varrho [M_i]\cap U_{k_i}=\varnothing $ . Then set $U_k:=\bigcap _{k<i\leq n} U_{k_i}\cap V_k$ .

We can now define a map

$$ \begin{align*} g_C&: \varrho[Z_C]\to C\\ z&\mapsto x_i\iff z\in U_i. \end{align*} $$

Clearly, $g_C$ is relation preserving. Finally, define $g: \sigma \rho \mathfrak {X}\to \mathfrak{F}$ by setting

$$\begin{align*}g(\varrho(z)):=\begin{cases} f(z)&\text{ if } f(z)\text{ does not belong to any proper cluster},\\ g_C(\varrho(z))&\text{ if }f(z)\in C\text{ for some proper cluster }C\subseteq F. \end{cases} \end{align*}$$

Now, g is evidently relation preserving. Moreover, it is continuous because both f and each $g_C$ are. Thus, g is a stable map.

We must now show that g satisfies the BDC for $\mathfrak { D}$ . Suppose $Rg(\varrho (x)) y$ and $y\in \mathfrak {d}$ for some $\mathfrak {d}\in \mathfrak {D}^\Uparrow $ . By construction, $f(x)$ belongs to the same cluster as $g(\varrho (x))$ , so also $Rf(x)y$ . Since f satisfies the BDC $^\Uparrow $ for $\mathfrak {D}^\Uparrow $ , there must be some $z\in X$ such that $Rxz$ and $f(z)\in \mathfrak {d}$ . Since $f^{-1}(f(z))\in \mathsf {Clop}(\mathfrak {X})$ , by Proposition 2.11 and Corollary 2.13, there is $z'\in \mathit {max}(f^{-1}(f(z)))$ with $Rzz'$ . Then also $Rxz'$ and $f(z')\in \mathfrak {d}$ . But from $z'\in \mathit {max}(f^{-1}(f(z))),$ it follows that $f(z')=g(\varrho (z'))$ by construction, so we have $g(\varrho (z'))\in \mathfrak {d}$ . As clearly $R\varrho (x)\varrho (z')$ , we have shown that g satisfies the BDC $^\Uparrow $ for $\mathfrak {D}^\Uparrow $ . Analogous reasoning establishes that g satisfies the BDC $^\Downarrow $ for $\mathfrak {D}^\Downarrow $ . By Proposition 4.5, this implies $\sigma \rho \mathfrak {X}\not \models \mu (\mathfrak {F}, {\mathfrak {D}})$ .

Proof of Case s = 3.

As before, we prove the dual statement $\mathfrak {M}_*\not \models \Gamma /\Delta $ implies $\sigma \rho \mathfrak {M}_*\not \models \Gamma /\Delta $ . Let $\mathfrak {X}:=\mathfrak {M}_*$ . Using Theorem 4.10, wlog, we restrict attention to the case $\Gamma /\Delta =\mu (\mathfrak {F}, {\mathfrak {D}})$ , for $\mathfrak {F}$ the dual of a finite $\mathtt {K4}$ -algebra. By Proposition 4.5, we take a stable map $f:\mathfrak {X}\to \mathfrak {F}$ satisfying the BDC for $\mathfrak { D}$ .

Given a cluster $C:=\{x_1, \ldots , x_n\}\subseteq F$ we let $Z_C:=f^{-1}(C)$ . Then $Z_C$ does not cut clusters, so by Proposition 3.2, $\varrho [Z_C]$ is clopen, and so is $f^{-1}(x_i)$ for each $x_i\in C$ . For each $x_i\in C,$ we let

$$ \begin{align*} M_i&:=\mathit{max}_R(f^{-1}(x_i)). \end{align*} $$

By Proposition 2.11 and Corollary 2.17, each $M_i$ is clopen and does not cut any cluster. Consequently, by Proposition 3.2, each $\varrho [M_i]$ is clopen.

Note $M_i\cap M_j=\varnothing $ holds for each distinct $i, j\leq n$ , and since no $M_i$ cuts clusters we have $\varrho [M_i]\cap \varrho [M_j]=\varnothing $ for each distinct $i, j\leq n$ . Constructing the desired partition of $\varrho [Z_C]$ into clopen sets is now simpler. First, for $k<n$ let $U_k=\varrho [M_k]$ . Then let $U_n:=\varrho [Z_C]\smallsetminus (U_1\cup \cdots \cup U_n)$ . Then $U_1, \ldots , U_n$ are clopen sets which partition $\varrho [Z_C]$ , such that $\varrho [M_k]\subseteq U_k$ for each $1\leq k\leq n$ .

We define our map $g:\sigma _{3}\rho _{3}\mathfrak {X}\to \mathfrak {F}$ as before. First, we let

$$ \begin{align*} g_C&: \varrho[Z_C]\to C\\ z&\mapsto x_i\iff z\in U_i. \end{align*} $$

Clearly, $g_C$ is relation preserving. Finally, define $g: \sigma _{3}\rho _{3}\mathfrak {X}\to \mathfrak{F}$ by setting

$$\begin{align*}g(\varrho(z)):=\begin{cases} f(z)&\text{ if} f(z) \text{ belongs to no proper cluster}, \\ g_C(\varrho(z))&\text{ if }f(z)\in C\text{ for some proper cluster }C\subseteq F. \end{cases} \end{align*}$$

It is clear that g is a stable map. We show that it satisfies the BDC for $\mathfrak {D}$ .

Suppose $Rg(\varrho (x))y$ and $y\in \mathfrak {d}$ for some $\mathfrak {d}\in \mathfrak {D}$ . If belongs to no proper cluster, then $g(\varrho (x))=f(x)$ , so $Rf(x)z$ . If $f(x)$ belongs to a proper cluster, then $g(\varrho (x))$ belongs to the same proper cluster, so again $Rf(x)z$ . Either way, $Rf(x)z$ . Since f satisfies the BDC for $\mathfrak {d}$ , there must be some $x\in Z$ such that $Rxz$ and $f(z)\in \mathfrak {d}$ . Now, , so by Proposition 2.11 and Corollary 2.17, one of the following conditions hold:

1. (1) $z\in \mathit {max}(f^{-1}(f(z)))$ ;

2. (2) There is $z'\in \mathit {max}(f^{-1}(f(z)))$ with $Rzz'$ .

Either way, there is $z'\in \mathit {max}(f^{-1}(f(z)))$ such that $Rxz'$ . But by construction, since $z'\in \mathit {max}(f^{-1}(f(z)))$ , we have $f(z')=g(\varrho (z'))$ . Consequently, $g(\varrho (z'))\in \mathfrak {d}$ . As clearly $R\varrho (x)\varrho (z')$ , we have shown that g satisfies the BDC for $\mathfrak {D}$ .

Our main lemma has the following key consequence.

5.2. Main results

The main results of this section follow from Theorem 5.2 by routine arguments; we review them here for completeness. We begin by establishing that the syntactic mappings $\tau , \rho , \sigma $ commute with $\mathsf {Alg}(\cdot )$ .

Lemma 5.4. Given $s\in \{1, 2, 3\}$ , let $\mathtt {L}\in \mathbf {Ext}(\mathtt {I}_s)$ and let $\mathtt {M}\in \mathbf {NExt}(\mathtt {C}_s)$ . The following conditions hold:

(16) $$ \begin{align} \mathsf{Alg}(\tau_{s}\mathtt{L})\ &=\tau_{s} \mathsf{Alg}(\mathtt{L}), \end{align} $$

(17) $$ \begin{align} \mathsf{Alg}(\sigma_{s}\mathtt{L})\ &=\sigma_{s} \mathsf{Alg}(\mathtt{L}), \end{align} $$

(18) $$ \begin{align} \mathsf{Alg}(\rho_{s}\mathtt{M})\ &=\rho_{s} \mathsf{Alg}(\mathtt{M}). \end{align} $$

The result just proved leads straightforwardly to the following, purely semantic characterization of companions.

We can now prove the main results of this section. The first result asserts that the companions of a rule system form an interval.

The second result is an analogue of the Blok–Esakia theorem. We use the qualifier “general” to indicate that the theorem applies uniformly to three different pairs of signatures.

We note that Theorem 5.6 remains true when restricted to lattices of logics only. This result, in the case of pair of signatures $1$ , was established by Maksimova and Rybakov [Reference Maksimova and Rybakov34] (see also [Reference Chagrov and Zakharyaschev14, Section 9.6]). The same holds for Theorem 5.7. Thus we obtain, as corollaries, the original Blok–Esakia theorem (case $s=1$ ), Wolter’s [Reference Wolter43] generalization thereof to bsi and tense logics (case $s=2$ ), as well as the original Kuznetsov–Muravitsky isomorphism (case $s=3$ ).

Proof. By construction, $\sigma $ and $\rho $ preserve the property of being a logic, because the Gödel translation of an assumption free, single-conclusion rule is always an assumption free, single-conclusion rule. Since we already know that the restrictions of $\sigma _{s}$ and $\rho _{s}$ to logics are mutual inverses, it suffices to show that $\sigma _{s}$ commutes with meets and joins. This is obvious for joins, since the join of two logics is a logic. For meets, it is enough to show that $\sigma _{s}$ commutes with $\mathsf {Taut}$ : for each rule system $\mathtt {L}\in \mathbf {Ext}(\mathtt {I_s}),$ we have $\sigma _{s}\mathsf {Taut}(\mathtt {L})=\mathsf {Taut}(\sigma _{s}\mathtt {L})$ . For if this condition holds, then we have:

$$ \begin{align*} \sigma_{s}(\mathtt{L}\otimes_{\mathbf{ExtL}(\mathtt{I_s})}\mathtt{L'})&= \sigma_{s}(\mathsf{Taut}(\mathtt{L}\otimes_{\mathbf{Ext}(\mathtt{I_s})}\mathtt{L'})) &\text{By Proposition 2.3}\\ &=\mathsf{Taut}(\sigma_{s}(\mathtt{L}\otimes_{\mathbf{Ext}(\mathtt{I_s})}\mathtt{L'}))\\ &= \mathsf{Taut}(\sigma_{s}\mathtt{L}\otimes_{\mathbf{Ext}(\mathtt{I_s})}\sigma_{s}\mathtt{L'})&\text{By Theorem 5.7}\\ &=\sigma_{s}\mathtt{L}\otimes_{\mathbf{ExtL}(\mathtt{I_s})}\sigma_{s}\mathtt{L'}. &\text{By Proposition 2.3} \end{align*} $$

Note that the reasoning is analogous when we consider infinite meets.

The claim indeed holds. Since $\sigma _{s}$ is order-preserving and $\mathsf {Taut}(\mathtt {L})\subseteq \mathtt {L,}$ we have $\sigma _{s}\mathsf {Taut}(\mathtt {L})\subseteq \sigma _{s}\mathtt {L}$ . Since $\mathsf {Taut}$ is also order preserving and the left-hand side is already a logic, it follows that $\sigma _{s}\mathsf {Taut}(\mathtt {L})\subseteq \mathsf {Taut}(\sigma _{s}\mathtt {L})$ . Likewise, if $\mathtt {M}\in \mathbf {NExt}(\mathtt {I}_s^+)$ , then $\rho _{s}\mathsf {Taut}(\mathtt {M})\subseteq \mathsf {Taut}(\rho _{s}\mathtt {M})$ . Thus,

$$ \begin{align*} \mathsf{Taut}(\sigma_{s}\mathtt{L})&=\sigma_{s}\rho_{s}\mathsf{Taut}(\sigma_{s}\mathtt{L})\\ &\subseteq\sigma_{s}\mathsf{Taut}(\rho_{s}\sigma_{s}\mathtt{L})\\ &=\sigma_{s}\mathsf{Taut}(\mathtt{L}), \end{align*} $$

as desired.