2.1 General machinery used in the paper

We work in ZFC set theory and use standard set-theoretic material, including maps or functions, ordinals, and cardinals. A cardinal is an (initial) ordinal. We write the first infinite cardinal as both $\omega $ and $\aleph _0$ ; countable will mean of cardinality at most this cardinal. The cardinality of a set X is denoted by $|X|$ , and the power set of X by $\wp (X)$ . We write ${ \mathrm{dom}}(f)$ for the domain of a map f.

2.2 Multimodal logic

We fix a nonempty set ${\mathsf {M}}$ of unary diamond-modalities, and a set ${\mathsf {Q}}=\{q_i:i<\omega \}$ of propositional atoms. Modal ${\mathsf {M}}$ -formulas, or simply modal formulas, are then defined as usual. Each atom in ${\mathsf {Q}}$ is a formula, as is $\top $ , and if $\varphi ,\psi $ are formulas, so are $\neg \varphi $ , $\varphi \vee \psi $ , and $\lozenge \varphi $ , for each $\lozenge \in {\mathsf {M}}$ .

We adopt standard conventions. We let $p,q,r,\ldots $ stand for arbitrary distinct atoms in ${\mathsf {Q}}$ . We let $\bot $ abbreviate $\neg \top $ ; $\varphi \wedge \psi $ abbreviate $\neg (\neg \varphi \vee \neg \psi )$ ; $\varphi \to \psi $ abbreviate $\neg \varphi \vee \psi $ ; $\varphi \leftrightarrow \psi $ abbreviate $(\varphi \to \psi )\wedge (\psi \to \varphi )$ ; and $\Box \varphi $ abbreviate $\neg \lozenge \neg \varphi $ , for each $\lozenge \in {\mathsf {M}}$ . We adopt the usual binding conventions, whereby $\neg ,\lozenge ,\wedge ,\vee ,\to ,\leftrightarrow $ (for each $\lozenge \in {\mathsf {M}}$ ) are in decreasing order of tightness.

A normal modal ${\mathsf {M}}$ -logic is a set of ${\mathsf {M}}$ -formulas containing all propositional tautologies, the axioms $\Box (p\to q)\to (\Box p\to \Box q)$ and $\lozenge p\leftrightarrow \neg \Box \neg p$ for each $\lozenge \in {\mathsf {M}}$ ,Footnote ¹ and closed under the inference rules of modus ponens (from $\varphi $ and $\varphi \to \psi $ , infer $\psi $ ), generalisation (from $\varphi $ , infer $\Box \varphi $ ), and substitution (from $\varphi $ , infer any substitution instance of it—any formula obtained from $\varphi $ by replacing its atoms by arbitrary formulas).

2.3 Structures

We assume some familiarity with basic first-order model theory. For unfamiliar terms, see, e.g., [Reference Chang and Keisler5, Reference Hodges13]. A signature is a set of non-logical symbols (function symbols, relation symbols, and constants) together with their types and arities. Others call it a similarity type, alphabet, or vocabulary. Equality is regarded as a logical symbol and is always available. We will be using various model-theoretic structures, some of which will be two-sorted. We take it as read that standard model-theoretic results apply to two-sorted structures. We identify (notationally) a structure ${\mathfrak M}$ with its domain ${ \mathrm{dom}}({\mathfrak M})$ . We write $S^{\mathfrak M}$ for the interpretation of a symbol or term S in ${\mathfrak M}$ . We write ${\bar a}\in {\mathfrak M}$ to indicate that ${\bar a}$ is a tuple of elements of the domain of ${\mathfrak M}$ . For a map f defined on ${\mathfrak M}$ , we write $f({\bar a})$ for the tuple obtained from ${\bar a}$ by applying f to its elements in order.

2.4 Kripke frames, models, semantics

Kripke frames are formulated as structures, as follows. We extend this slightly to cover Kripke models too.

We now recall the standard modal definitions of truth, validity, etc.

2.5 Point-generated inner subframes

Definition 2.4. For an ordinal $\alpha $ , we put $^{\alpha } {\mathsf {M}}=\{s\mid s:\alpha \to {\mathsf {M}}\}$ , the set of functions from $\alpha $ into ${\mathsf {M}}$ , and $^{<\alpha }{\mathsf {M}}=\bigcup _{\beta <\alpha }{}^{\beta } {\mathsf {M}}$ . One can think of the elements of $^{<\omega }{\mathsf {M}}$ as the finite sequences of diamonds.

For an ordinal $\alpha $ , a map $s\in {}^{\alpha }{\mathsf {M}}$ , and $\lozenge \in {\mathsf {M}}$ , we write $s\hat \ \lozenge \in {}^{\alpha +1}{\mathsf {M}}$ for the map whose restriction to $\alpha $ is $s $ and whose value on $\alpha $ is $\lozenge $ . Thinking of $s $ as a sequence, $s\hat \ \lozenge $ is the same sequence with $\lozenge $ appended.

For each $s\in {}^{<\omega }{\mathsf {M}}$ , we define the first-order formula $R_s xy$ of the signature of frames, by induction on the ordinal ${ \mathrm{dom}}(s)$ :

• • $R_{\emptyset } xy$ is $x=y$ , where $\emptyset $ is the empty map in $^0{\mathsf {M}}$ .

• • $R_{s\hat \ \lozenge }xy$ is $\exists z(R_s xz\wedge R_{\lozenge } zy)$ .

Finally, let ${\cal F}$ be a frame and $w_0\in {\cal F}$ . The inner subframe of ${\cal F}$ generated by $w_0$ is the subframe ${\cal F}(w_0)$ of ${\cal F}$ with domain

$$\begin{align*}\{w\in{\cal F}:{\cal F}\models R_s w_0w\mbox{ for some }s\in{}^{<\omega}{\mathsf{M}} \}. \end{align*}$$

Taking $s=\emptyset $ here, we see that ${\cal F}(w_0)$ contains $w_0$ . It is the smallest inner subframe of ${\cal F}$ to do so.

2.6 ${\mathsf {M}}$ -BAOs and translations

It is now well known that modal logic can be ‘algebraised’ using boolean algebras with operators (BAOs). We will just give some definitions; for more information, see, e.g., [Reference Blackburn, de Rijke and Venema2, chap. 5]. We will pass fairly freely between modal logic and algebra.

An ${\mathsf {M}}$ -BAO is an algebra ${\cal A}$ with signature $\{+,-,0,1\}\cup {\mathsf {M}}$ , where the $\{+,-,0,1\}$ -reduct of ${\cal A}$ is a boolean algebra and each $\lozenge \in {\mathsf {M}}$ is a unary function symbol interpreted in ${\cal A}$ as a normal additive operator (that is, ${\cal A}\models \lozenge 0=0$ and ${\cal A}\models \lozenge (a+b)=\lozenge a+\lozenge b$ for each $a,b\in {\cal A}$ ). This is as per [Reference Blackburn, de Rijke and Venema2, definition 5.19]. The double use of $\lozenge $ for a modal operator and a function symbol should not cause problems in practice.

Standardly, each modal ${\mathsf {M}}$ -formula $\varphi $ can be translated to a term $\tau _{\varphi }$ of this signature (an ‘ ${\mathsf {M}}$ -BAO term’). Introduce pairwise distinct variables $v_i$ ( $i<\omega $ ). We let $\tau _{\top }=1$ , $\tau _{q_i}=v_i$ for $i<\omega $ , $\tau _{\neg \varphi }=\,-\tau _{\varphi }$ , $\tau _{\varphi \vee \psi }=\tau _{\varphi }+\tau _{\psi }$ , and $\tau _{\lozenge \varphi }=\lozenge \tau _{\varphi }$ , for $\lozenge \in {\mathsf {M}}$ . We can now say that an ${\mathsf {M}}$ -BAO ${\cal A}$ validates a modal formula $\varphi $ if ${\cal A}\models \forall \bar v(\tau _{\varphi }=1)$ , where $\bar v$ is a tuple enumerating the variables occurring in $\tau _{\varphi }$ . As usual, a class ${\cal C}$ of ${\mathsf {M}}$ -BAOs validates a modal formula $\varphi $ if every ${\cal A}\in {\cal C}$ does, and ${\cal C}$ validates a set of modal formulas if it validates every formula in the set.

A variety of ${\mathsf {M}}$ -BAOs is a class V of ${\mathsf {M}}$ -BAOs defined by a set of equations. The set $L_V$ of modal ${\mathsf {M}}$ -formulas validated by V is then a normal modal ${\mathsf {M}}$ -logic as defined in Section 2.2. If L is a normal modal ${\mathsf {M}}$ -logic, $V_L$ denotes the variety of all ${\mathsf {M}}$ -BAOs defined by the set $\{\tau _{\varphi }=1:\varphi \in L\}$ of equations. We have $V_{L_V}=V$ and $L_{V_L}=L$ . See, e.g., [Reference Blackburn, de Rijke and Venema2, sec. 5.2] for more.

Conversely, each ${\mathsf {M}}$ -BAO term $\tau $ written with the variables $v_i$ can be translated to a modal ${\mathsf {M}}$ -formula $\varphi _{\tau }$ . We let $\varphi _1=\top $ , $\varphi _0=\bot $ , $\varphi _{v_i}=q_i$ , $\varphi _{-\tau }=\neg \varphi _{\tau }$ , $\varphi _{\tau +\upsilon }=\varphi _{\tau }\vee \varphi _{\upsilon }$ , and $\varphi _{\lozenge \tau }=\lozenge \varphi _{\tau }$ for $\lozenge \in {\mathsf {M}}$ . We extend this translation to equations: given ${\mathsf {M}}$ -BAO terms $\tau ,\upsilon $ , we let $\varphi _{\tau =\upsilon }$ be the modal formula $\varphi _{\tau }\leftrightarrow \varphi _{\upsilon }$ .

2.7 Canonical extensions

For an ${\mathsf {M}}$ -BAO ${\cal A}$ , we can form, in the usual way:

• • Its canonical frame ${\cal A}_+$ , whose domain comprises the ultrafilters of ${\cal A}$ , and which is regarded as a frame in the sense of Definition 2.2, with interpretations given as usual by ${\cal A}_+\models R_{\lozenge }\mu \nu $ iff $a\in \nu \,{\Rightarrow }\, \lozenge a\in \mu $ for all $a\in {\cal A}$ , where $\mu ,\nu $ are any ultrafilters of ${\cal A}$ and $\lozenge \in {\mathsf {M}}$ .

• • Its canonical extension ${\cal A}^{\sigma }$ (which is also an ${\mathsf {M}}$ -BAO). It is also known as a ‘perfect extension’ or the ‘canonical embedding algebra’, and written $\mathfrak {Em}{\cal A}$ .

For details, see, e.g., [Reference Blackburn, de Rijke and Venema2, definition 5.40] or [Reference Jónsson and Tarski14].

2.8 Sahlqvist formulas

Sahlqvist theory is now a basic and well-known part of modal logic, so we will be very brief: see, e.g., [Reference Blackburn, de Rijke and Venema2, sec. 3.6] or [Reference Chagrov and Zakharyaschev4, sec. 10.3] for details. Sahlqvist formulas are modal ${\mathsf {M}}$ -formulas of a certain syntactic form, and they have some useful properties. Every Sahlqvist formula $\varphi $ has a first-order correspondent: an easily-computable first-order sentence $\chi _{\varphi }$ of the signature of frames (Definition 2.2), such that for an arbitrary frame ${\cal F}$ we have ${\cal F}\models \chi _{\varphi }$ iff $\varphi $ is valid in ${\cal F}$ . As an example, if $\lozenge \in {\mathsf {M}}$ and $q\in {\mathsf {Q}}$ then $\varphi =\lozenge \lozenge q\to \lozenge q$ is a Sahlqvist formula, and $\chi _{\varphi }$ is $\forall xyz(R_{\lozenge } xy\wedge R_{\lozenge } yz\to R_{\lozenge } xz)$ , saying that $R_{\lozenge }$ is transitive. Moreover, every Sahlqvist formula $\varphi $ is canonical: an ${\mathsf {M}}$ -BAO ${\cal A}$ validates $\varphi $ iff $\varphi $ is valid in the frame ${\cal A}_+$ (iff ${\cal A}_+\models \chi _{\varphi }$ , iff ${\cal A}^{\sigma }$ validates $\varphi $ ). The logic axiomatised by a set $\Phi $ of Sahlqvist formulas (i.e., the smallest normal ${\mathsf {M}}$ -modal logic containing $\Phi $ ) is the logic of the class of frames defined by $\{\chi _{\varphi }:\varphi \in \Phi \}$ , and hence is Kripke complete.

2.9 Canonicity in power

This definition of $\kappa $ -canonicity, or canonicity in the power $\kappa $ , is a slight variant of ones in [Reference Fine and Kanger8, p. 30] and [Reference Goldblatt12, sec. 6]. The connection of these notions to canonicity of modal logics is again well known [Reference Blackburn, de Rijke and Venema2, sec. 5.3]: V is canonical iff $L_V$ is canonical—i.e., every canonical frame of $L_V$ validates $L_V$ —and V is $\kappa $ -canonical iff every canonical frame of $L_V$ defined with $\leq \kappa $ atoms validates $L_V$ .

Let us consider resolvedness in the case ${\mathsf {M}}=\{\lozenge \}$ , where $|{\mathsf {M}}|+\omega =\omega $ . [Reference Goldblatt12, theorem 6.1] gives an example of a variety of $\{\lozenge \}$ -BAOs that is finitely canonical (n-canonical for all finite n) but not $\omega $ -canonical. Nonetheless, as far as we are aware, all known $\omega $ -canonical varieties of $\{\lozenge \}$ -BAOs are totally canonical. No nonresolved varieties of $\{\lozenge \}$ -BAOs are known. Fine [Reference Fine and Kanger8, p. 30] asked if there are any (his question was for modal logics and we have read it across for varieties). This is asking whether there is a dichotomy between totally canonical and non- $\omega $ -canonical varieties of $\{\lozenge \}$ -BAOs, with no other kinds existing.

Let us unpack this a little more. For arbitrary ${\mathsf {M}}$ now, let E be the set of all ${\mathsf {M}}$ -BAO equations, and let $\Xi =\wp (E)$ . Varieties are defined by members of $\Xi $ . A subset $\Sigma \subseteq \Xi $ therefore defines a particular kind of variety: a $\Sigma $ -variety is a variety of ${\mathsf {M}}$ -BAOs defined by a set of equations in $\Sigma $ . Since $\Sigma $ is a set, the ZFC axiom of replacement ensures that there exists a cardinal $\kappa $ such that every non-canonical $\Sigma $ -variety V contains a $\kappa $ -generated algebra ${\cal A}$ with ${\cal A}^{\sigma }\notin V$ . Hence, every $\kappa $ -canonical $\Sigma $ -variety is totally canonical. Let $\kappa _{\Sigma }$ be the least such $\kappa $ . Then $\Sigma \subseteq \Sigma '\subseteq \Xi $ implies $\kappa _{\Sigma }\leq \kappa _{\Sigma '}$ ; if every $\Sigma $ -variety is canonical then $\kappa _{\Sigma }=0$ ; and every $\Sigma $ -variety is resolved iff $\kappa _{\Sigma }\leq |{\mathsf {M}}|+\omega $ .

As far as we are aware, all known varieties of ${\mathsf {M}}$ -BAOs are resolved, and it is an open question whether every variety of ${\mathsf {M}}$ -BAOs is resolved — that is, $\kappa _{\Xi }\leq |{\mathsf {M}}|+\omega $ . In the absence of an answer, we can still try to identify large sets $\Sigma $ for which every $\Sigma $ -variety is resolved. That is what we will do in the current paper.

3.1 The variety V

We now fix a variety V of ${\mathsf {M}}$ -BAOs. Further conditions on V will be imposed later.

3.2 The two-sorted structure ${\mathfrak B}$

Next we fix a ‘big’ two-sorted structure

$$ \begin{align*}{\mathfrak B}=({\cal B},{\cal B}_+), \end{align*} $$

where ${\cal B}\in V$ . We say that ${\cal B}$ comprises the elements of ${\mathfrak B}$ of algebra sort, and ${\cal B}_+$ comprises the elements of point sort. Further conditions on ${\mathfrak B}$ will be imposed later. It may be confusing but has to be accepted that each ultrafilter of ${\cal B}$ is both a subset of ${\cal B}$ and an element of ${\cal B}_+$ .

The two-sorted signature ${\cal L}$ of ${\mathfrak B}$ comprises the following symbols with the following interpretations in ${\mathfrak B}$ :

1. 1. The algebra operations $+,-,0,1,\lozenge $ of ${\cal B}$ for each $\lozenge \in {\mathsf {M}}$ , all taking algebra elements to algebra elements. Their interpretation in ${\mathfrak B}$ is copied from ${\cal B}$ . As abbreviations, we write $x\cdot y=\,-({-}x+\,{-}y)$ and $\Box x=\,{-}\lozenge\, {-}x$ for each $\lozenge \in {\mathsf {M}}$ .

2. 2. A binary relation symbol ${\,{\mathrel {\underline {\in }}}\,}:{\cal B}\times {\cal B}_+$ , with ${\mathfrak B}\models b\,{\mathrel {\underline {\in }}}\, \mu $ iff b is a member of the ultrafilter $\mu $ .

3. 3. The binary relation symbols $R_{\lozenge }:{\cal B}_+\times {\cal B}_+$ (for each $\lozenge \in {\mathsf {M}}$ ) introduced for frames in Definition 2.2, with interpretations copied from the canonical frame ${\cal B}_+$ . So for each $\lozenge \in {\mathsf {M}}$ ,

   $$\begin{align*}{\mathfrak B}\models\forall xy(R_{\lozenge} xy\leftrightarrow\forall z(z\mathrel{\underline{\in}} y\to\lozenge z\mathrel{\underline{\in}} x)). \end{align*}$$

   Note that formulas of the signature of frames, such as $R_sxy$ ( $s\in {}^{<\omega }{\mathsf {M}}$ ) from Definition 2.4, are also ${\cal L}$ -formulas.

4. 4. Unary relation symbols $Q_i$ ( $i<\omega $ ) of point sort. They can be forgotten about until Section 3.9. For now, bear in mind that they are indeed in ${\cal L}$ and get interpretations in ${\cal L}$ -structures.

We leave the sorts of variables and structure elements to be determined by context.

3.3 Countable ${\mathsf {M}}$

From now on, we assume that ${\mathsf {M}}$ is countable. This is the most common case encountered in practice, although there are some exceptions, such as the signatures of some infinite-dimensional cylindric and polyadic algebras. For countable ${\mathsf {M}}$ , saying that a variety is resolved says that it is canonical iff it is $\omega $ -canonical.

We make this countability restriction purely for simplicity of exposition. Suitably formulated, our results readily generalise to uncountable ${\mathsf {M}}$ (see Section 7.4).

3.4 The structure ${\mathfrak A}\preceq {\mathfrak B}$

We fix a countable elementary substructure

$$\begin{align*}{\mathfrak A}=({\cal A},W)\preceq{\mathfrak B}. \end{align*}$$

Such a structure exists because ${\mathsf {M}}$ , and hence ${\cal L}$ , are countable (and this is the only time we use the countability of ${\mathsf {M}}$ ). We write ${\cal L}({\mathfrak A})$ for ${\cal L}$ expanded by a new constant $\underline {a}$ for each element a of ${\mathfrak A}$ . From now on, $\mathfrak {A,B}$ will denote the expansions of these respective structures to ${\cal L}({\mathfrak A})$ -structures in which each $\underline {a}$ names a. Note that these expansions continue to satisfy ${\mathfrak A}\preceq {\mathfrak B}$ .

Easily, ${\mathfrak A}\preceq {\mathfrak B}$ implies that ${\cal A}\preceq {\cal B}$ as algebras, so ${\cal A}$ is an ${\mathsf {M}}$ -BAO and ${\cal A}\in V$ (since V is an elementary class). Although ${ \mathrm{dom}}{\mathfrak A}\subseteq { \mathrm{dom}} {\mathfrak B}$ , for clarity we will sometimes use the inclusion map

$$\begin{align*}\iota:{\mathrm{dom}}{\mathfrak A}\to{\mathrm{dom}} {\mathfrak B}. \end{align*}$$

It is an ${\cal L}({\mathfrak A})$ -elementary embedding.

To save time, we will use the easy half of the Tarski–Vaught criterion (see [Reference Chang and Keisler5, proposition 3.1.2] or [Reference Hodges13, theorem 2.5.1]): since ${\mathfrak A}\preceq {\mathfrak B}$ , if $\varphi (x)$ is an ${\cal L}({\mathfrak A})$ -formula and ${\mathfrak B}\models \exists x\varphi (x)$ , then there is $a\in {\mathfrak A}$ with ${\mathfrak B}\models \varphi (a)$ .

3.5 The ${\cal L}'$ -expansions $\mathfrak {A',B'}$

Since ${\mathfrak A}$ and ${\mathfrak B}$ are elementarily equivalent ${\cal L}({\mathfrak A})$ -structures, by Frayne’s theorem [Reference Chang and Keisler5, corollary 4.3.13] there exist an ultrapower ${\mathfrak A}^*$ and an ${\cal L}({\mathfrak A})$ -elementary embedding $\sigma :{\mathfrak B}\to {\mathfrak A}^*$ . (The ultrapower ${\mathfrak M}^*$ , using the same ultrafilter, is defined for every structure ${\mathfrak M}$ .) Let $\delta :{\mathfrak A}\to {\mathfrak A}^*$ be the diagonal embedding (called the ‘natural embedding’ in [Reference Chang and Keisler5, p. 221]). Because each element of ${\mathfrak A}$ is named by a constant of ${\cal L}({\mathfrak A})$ , we have $\delta =\sigma \circ \iota $ (see Figure 1).

Fig. 1

The ${\cal L}({\mathfrak A})$ -elementary maps $\delta ,\iota ,\sigma $ . The diagram commutes.

Of course, $\delta :{\mathfrak A}'\to {\mathfrak A}^{\prime *}$ is still the diagonal embedding, so is ${\cal L}'$ -elementary. The maps $\iota :{\mathfrak A}'\to {\mathfrak B}'$ and $\sigma :{\mathfrak B}'\to {\mathfrak A}^{\prime *}$ are ${\cal L}'$ -embeddings, but as Example 3.8 shows, they are not always ${\cal L}'$ -elementary.

3.6 Simple formulas

The following is a weak (but still useful) preservation result.

Proof. (1) is proved by induction on the structure of $\theta $ as a simple formula. If $\theta $ is an ${\cal L}({\mathfrak A})$ -formula, the result holds because $\sigma $ is ${\cal L}({\mathfrak A})$ -elementary. Suppose that $\theta =\underline {\mu }(\tau ({\bar x}))$ , where $\mu \in {\cal A}_+$ and $\tau ({\bar x})$ is an algebra-sorted ${\cal L}({\mathfrak A})$ -term. Let ${\bar b}\in {\mathfrak B}$ have sorts matching those of ${\bar x}$ ,Footnote ² and write $b=\tau ^{{\mathfrak B}}({\bar b})\in {\cal B}$ . Then

$$\begin{align*}\begin{array}{r@{\ \ }cl@{\ \ }l} {\mathfrak B}'\models\theta({\bar b}) &\iff& {\mathfrak B}'\models\underline{\mu}(\tau({\bar b})) &\mbox{as }\theta=\underline{\mu}(\tau({\bar x})) \\ &\iff& {\mathfrak B}'\models\underline{\mu}(b) &\mbox{by definition of }b \\ &\iff& {\mathfrak A}^{\prime *}\models\underline{\mu}(\sigma(b)) &\mbox{by definition of the expansion }{\mathfrak B}' \\ &\iff& {\mathfrak A}^{\prime *}\models\underline{\mu}(\tau(\sigma({\bar b}))) &\mbox{as } \sigma \mbox{ is } {\cal L}({\mathfrak A})\mbox{-elementary, so } \sigma(b)=\tau^{{\mathfrak A}^*}(\sigma({\bar b})) \\ &\iff& {\mathfrak A}^{\prime *}\models\theta(\sigma({\bar b})) &\mbox{as }\theta=\underline{\mu}(\tau({\bar x})). \end{array} \end{align*}$$

The boolean cases are standard.

It is enough to prove (2) for ${\cal L}'$ -sentences of the form $\forall {\bar y}\theta $ , where $\theta ({\bar y})$ is simple. For, given this much, if ${\mathfrak A}'\models \exists {\bar x}\forall {\bar y}\theta ({\bar x},{\bar y})$ , then choose ${\bar a}\in {\mathfrak A}'$ with ${\mathfrak A}'\models \forall {\bar y}\theta ({\bar a},{\bar y})$ . Let $\underline {{\bar a}}$ be the tuple of constants in ${\cal L}({\mathfrak A})$ corresponding to ${\bar a}$ . Clearly, $\theta (\underline {{\bar a}},{\bar y})$ is also a simple ${\cal L}'$ -formula, and ${\mathfrak A}'\models \forall {\bar y}\theta (\underline {{\bar a}},{\bar y})$ . So by assumption, ${\mathfrak B}'\models \forall {\bar y}\theta (\underline {{\bar a}},{\bar y})$ . Then certainly, ${\mathfrak B}'\models \exists {\bar x}\forall {\bar y}\theta ({\bar x},{\bar y})$ .

So let $\theta ({\bar y})$ be simple and suppose that ${\mathfrak A}'\models \forall {\bar y}\theta $ . As $\delta :{\mathfrak A}'\to {\mathfrak A}^{\prime *}$ is ${\cal L}'$ -elementary, ${\mathfrak A}^{\prime *}\models \forall {\bar y}\theta $ as well. Let ${\bar b}\in {\mathfrak B}$ be arbitrary subject to matching the sorts of ${\bar y}$ . Since ${\mathfrak A}^{\prime *}\models \forall {\bar y}\theta $ , certainly ${\mathfrak A}^{\prime *}\models \theta (\sigma ({\bar b}))$ . Part 1 now gives us ${\mathfrak B}'\models \theta ({\bar b})$ . Since ${\bar b}$ was arbitrary, we obtain ${\mathfrak B}'\models \forall {\bar y}\theta $ as required.

3.7 Some applications

Corollary 3.4 will be used several times, beginning with the following:

So as well as $\underline {\mu }^{{\mathfrak B}'}\subseteq {\cal B}$ , we also have $\underline {\mu }^{{\mathfrak B}'}\in {\cal B}_+$ .

Since ${\mathfrak A}=({\cal A},W)\preceq {\mathfrak B}=({\cal B},{\cal B}_+)$ , we have $W\subseteq {\cal B}_+$ , so each $w\in W$ is an ultrafilter of ${\cal B}$ , and $w=\{b\in {{\cal B}}:{\mathfrak B}\models b\,{\mathrel {\underline {\in }}}\, w\}$ . However, taking ${\mathfrak A}$ in isolation, the elements of W are just ‘abstract points’. Unless ${\cal A}={\cal B}$ , elements of W are not subsets of ${\cal A}$ , and for $w\in W$ we do not have $w=\hat w$ . Nonetheless, we have the following.

In fact, $\hat w$ is the intersection with ${\cal A}$ of the ultrafilter $w$ of ${\cal B}$ , for each $w\in W$ . In general, the inclusion ${\widehat W}\subseteq {\cal A}_+$ is strict. For example, this happens when ${\cal A}_+$ is uncountable, since W and hence ${\widehat W}$ are countable.

Example 3.8. The converse implication in Proposition 3.3(2) can fail. For example, suppose that ${\cal A}_+$ is uncountable. So there is $\mu \in {\cal A}_+\setminus {\widehat W}$ . Consider the simple ${\cal L}'$ -formula

$$\begin{align*}\theta(x,y)\;{\buildrel \mathrm{def} \over =}\; y\mathrel{\underline{\in}} x\leftrightarrow\underline{\mu}(y). \end{align*}$$

By Corollary 3.5 we have $\underline {\mu }^{{\mathfrak B}'}\in {\cal B}_+$ , and evidently ${\mathfrak B}'\models \forall y(y\,{\mathrel {\underline {\in }}}\, \underline {\mu }^{{\mathfrak B}'}\leftrightarrow \underline {\mu }(y))$ . Hence, $\underline {\mu }^{{\mathfrak B}'}$ witnesses ${\mathfrak B}'\models \exists x \forall y\theta $ . But $\mu \notin {\widehat W}$ , so there is no $w\in W$ with ${\mathfrak A}'\models \forall y(y\,{\mathrel {\underline {\in }}}\, w\leftrightarrow \underline {\mu }(y))$ . Hence, ${\mathfrak A}'\not \models \exists x \forall y\theta $ .

So the elementarily-equivalent ${\cal L}'$ -structures ${\mathfrak A}'$ and ${\mathfrak A}^{\prime *}$ need not be elementarily equivalent to ${\mathfrak B}'$ , since they may disagree on the ${\cal L}'$ -sentence $\exists x\forall y\theta $ . It follows that the embeddings $\iota :{\mathfrak A}'\to {\mathfrak B}'$ and $\sigma :{\mathfrak B}'\to {\mathfrak A}^{\prime *}$ are not ${\cal L}'$ -elementary in general.

3.8 An embedding $f:{\cal A}_+\to {\cal B}_+$

By Corollary 3.5, f is well defined. We plainly have

(1) $$ \begin{align} {\mathfrak B}'\models\forall x(\underline{\mu}(x)\leftrightarrow x\mathrel{\underline{\in}} f(\mu)) \end{align} $$

for each $\mu \in {\cal A}_+$ . A similar map was given by Surendonk [Reference Surendonk18] (see [Reference Surendonk, Kracht, de Rijke, Wansing and Zakharyaschev19, Reference Surendonk20] for more on this topic).

If the expression ‘ $f(\hat w)=w$ ’ appears confusing, recall from the discussion after Definition 3.6 that $w$ and $\hat w$ are quite different objects and cannot be identified. While $\hat w$ is an ultrafilter of ${\cal A}$ , $w$ is ‘actually’ an ultrafilter of ${\cal B}$ , and so is $f(\hat w)$ .

We remark that f need not be a bounded morphism (V would be easily seen to be resolved if it were). For if it were, then $R_{\lozenge }^{{\cal B}_+}(w)\subseteq f({\cal A}_+)$ for every $\lozenge \in {\mathsf {M}}$ and $w\in W$ ; yet it can be that $|R_{\lozenge }^{{\cal B}_+}(w)|>|{\cal A}_+|$ . See [Reference Surendonk18, sec. 6] for details and related discussion.

A positive answer would imply that the canonical frames of all free V-algebras of infinite rank are elementarily equivalent. For discussion around this, see, e.g., [Reference Surendonk18] and [Reference Surendonk20, question 3.47].

3.9 A strategy for proving total canonicity

We will now describe a method, based on embeddings $f:{\cal A}_+\to {\cal B}_+$ as above, that will be used to show that certain varieties are resolved.

We assume that $\phi _0$ is a modal ${\mathsf {M}}$ -formula true at some point in ${\cal B}_+$ under the valuation $g:{\mathsf {Q}}\to \wp ({\cal B}_+)$ given by $g(q_i)=Q_i^{\mathfrak B}$ for each $i<\omega $ .

Recall that each modal ${\mathsf {M}}$ -formula $\psi $ has a standard translation: an ${\cal L}$ -formula $\psi ^x(x)$ for each point-sorted variable x. We define $\psi ^x$ by induction on $\psi $ as usual: $\top ^x=\top $ , $q_i^x=Q_i(x)$ for $i<\omega $ , $(\neg \psi )^x=\neg (\psi ^x)$ , $(\psi \vee \chi )^x=\psi ^x\vee \chi ^x$ , and $(\lozenge \psi )^x=\exists y(R_{\lozenge } xy\wedge \psi ^y)$ for each $\lozenge \in {\mathsf {M}}$ , where y is some point-sorted variable other than $ x$ . A standard induction on $\psi $ now shows that for each $t\in {\cal B}_+$ (and any x), we have

(2) $$ \begin{align} ({\cal B}_+,g),t\models\psi \iff {\mathfrak B}\models\psi^x(t). \end{align} $$

Since $\phi _0$ is satisfied in the Kripke model $({\cal B}_+,g)$ , we have ${\mathfrak B}\models \exists x\phi _0^x$ . By the Tarski–Vaught criterion (Section 3.4), there is $w_0\in W$ with ${\mathfrak B}\models \phi _0^x(w_0)$ , and so

(3) $$ \begin{align} ({\cal B}_+,g),w_0\models\phi_0. \end{align} $$

We fix such a $w_0$ . Let ${\cal F}={\cal A}_+(\widehat {w_0})$ be the inner subframe of ${\cal A}_+$ generated by $\widehat {w_0}$ (see Definitions 2.4 and 3.6), and let ${\cal M}$ be the submodel of $({\cal B}_+,g)$ with domain $f({\cal F})$ . By Corollary 3.10, $w_0=f(\widehat {w_0})\in f({\cal F})={\cal M}$ . See Figure 2 for a rough illustration.

Fig. 2

Schematic diagram of elements of the strategy.

For countable ${\mathsf {M}}$ , this result reduces the problem of showing that a variety is resolved to the problem of showing that the Truth Lemma holds in it.

The main challenge in establishing the Truth Lemma is the following. Suppose that $m\in {\cal M}$ and $({\cal B}_+,g),m\models \lozenge \psi $ , so there is $z\in {\cal B}_+$ with ${\mathfrak B}\models R_{\lozenge } mz$ and $({\cal B}_+,g),z\models \psi $ . We need to find such a z in ${\cal M}$ . It suffices to find one in W, and our whole strategy relies upon doing so. In favourable circumstances, we can do it using the Tarski–Vaught criterion from Section 3.4, even when $m\notin W$ . The proof of Proposition 4.2 illustrates the process.

4.1 Canonicity of extensions of $K5_2$

We now turn to $5_2$ .

Lemma 4.1. A frame validates $5_2$ iff for all $n<\omega $ it satisfies

(5) $$ \begin{align} \text{for all } x,y,z, \ R_{\lozenge}{}^nxy \text{ and } R_{\lozenge}{}^{n+1}xz \text{ implies } R_{\lozenge} yz. \end{align} $$

Proof. Suppose we have an algebra ${\cal B}\in V$ , a countable elementary substructure ${\mathfrak A}=({\cal A},W)$ of ${\mathfrak B}=({\cal B},{\cal B}_+)$ , an embedding $f:{\cal A}_+\to {\cal B}_+$ such that $f(\hat w)=w$ for every $w\in W$ (Corollary 3.10), and a valuation g on ${\cal B}_+$ ; with ${\cal M}$ being the submodel of $({\cal B}_+,g)$ with domain $f({\cal F})$ , where ${\cal F}$ is the inner subframe ${\cal A}_+(\widehat {w_0})$ of ${\cal A}_+$ generated by $\widehat {w_0}$ for some $w_0\in W$ . Then we have to show that for each $m\in {\cal M}$ and modal formula $\psi $ , we have ${\cal M},m\models \psi $ iff $({\cal B}_+,g),m\models \psi $ .

The proof is by induction on the formation of $\psi $ . The significant case is $\lozenge \psi $ , assuming inductively the result for $\psi $ . If ${\cal M},m\models \lozenge \psi $ , then there is some $z_0$ in ${\cal M}$ with $R_{\lozenge }^{{\cal M}} m z_0$ and ${\cal M},z_0\models \psi $ . But then $({\cal B}_+,g),z_0\models \psi $ by the induction hypothesis on $\psi $ , and $R_{\lozenge }^{{\mathfrak B}} m z_0$ as ${\cal M}$ is a submodel of ${\cal B}_+$ . This shows that $({\cal B}_+,g),m\models \lozenge \psi $ .

Conversely, suppose that $({\cal B}_+,g),m\models \lozenge \psi $ with $m\in {\cal M}$ . Then there is some $z_0\in {\cal B}_+$ with $R_{\lozenge }^{{\mathfrak B}} m z_0$ and $({\cal B}_+,g),z_0\models \psi $ . Hence ${\mathfrak B}\models \psi ^x(z_0)$ by (2). Since $m\in {\cal M}$ , we have $(R_{\lozenge }^{{\mathfrak B}})^n w_0 m$ for some $n\geq 0$ , hence $(R_{\lozenge }^{{\mathfrak B}})^{n+1}w_0 z_0$ . Therefore,

$$ \begin{align*} {\mathfrak B}\models \exists z_1\cdots\exists z_n\exists x(R_{\lozenge} \underline{w_0}z_1 \land R_{\lozenge} z_1z_2\land\cdots\land R_{\lozenge} z_nx\land\psi^x). \end{align*} $$

Now $w_0\in W$ , so $n+1$ applications of the Tarski–Vaught criterion to this last fact yield elements $w_1,\dots ,w_n,w\in W$ such that

$$ \begin{align*}{\mathfrak B}\models R_{\lozenge} w_0 w_1\land R_{\lozenge} w_1w_2\land\cdots\land R_{\lozenge} w_nw\land\psi^x(w). \end{align*} $$

So $(R_{\lozenge }^{{\mathfrak B}})^{n+1}w_0w$ . Now $W\not \subseteq {\cal M}$ in general. But as $f:{\cal A}_+\to {\cal B}_+$ is an embedding, and $\hat u\in {\cal A}_+$ and $f(\hat u)=u$ for each $u\in W$ , we see that $(R^{{\cal A}_+}_{\lozenge })^{n+1}\widehat {w_0}\hat w$ . Since ${\cal F}={\cal A}_+(\widehat {w_0})$ , we obtain $\hat w\in {\cal F}$ , and hence $w=f(\hat w)\in {\cal M}$ . By (2), $({\cal B}_+,g),w\models \psi $ . Hence ${\cal M},w\models \psi $ by the induction hypothesis. Also, we now have $(R_{\lozenge }^{{\mathfrak B}})^n w_0 m$ and $(R_{\lozenge }^{{\mathfrak B}})^{n+1} w_0 w$ , so (5) gives $R^{{\mathfrak B}}_{\lozenge } mw$ . This is because ${\cal B}\in V$ , so ${\cal B}$ validates $5_2$ , hence so does the frame ${\cal B}_+$ , and so Lemma 4.1 applies to ${\cal B}_+$ .

Since $R^{{\mathfrak B}}_{\lozenge } mw$ we get $R^{{\cal M}}_{\lozenge } mw$ as m and $w$ are in ${\cal M}$ . Together with ${\cal M},w\models \psi $ , this implies the desired ${\cal M},m\models \lozenge \psi $ , and completes the inductive case for $\lozenge $ .

$K5_n$ is the smallest normal logic to contain $5_n$ . $K5_2$ is a sublogic of $K5$ , as can be seen model-theoretically by deriving the case $n=2$ of (4) from its $n=1$ case. (A proof-theoretic demonstration is also straightforward.) It follows from the result just proved and Theorem 3.13 that:

4.2 Continuum-many logics between $K5_2$ and $K5$

Theorem 4.3 applies to all normal extensions of $K5$ . But whereas $K5$ has only countably many extensions [Reference Nagle and Thomason15], we will now show that $K5_2$ has continuum many. Indeed there are continuum many between $K5_2$ and $K5$ .

Fig. 3

The intransitive frame ${\cal D}_j$ validating $5_2$ .

The proof adapts a construction from [Reference Chagrov and Zakharyaschev4, p. 162]. For each $j\in \omega $ , let ${\cal D}_j$ be the ‘diamond-shaped’ frame depicted in Figure 3. We take this as an intransitive irreflexive frame in which the only relations that hold are those given by the displayed arrows. Thus the relation $R_{\lozenge }$ of ${\cal D}_j$ is

$$ \begin{align*}\begin{array}{@{}r@{\ }l} \{ (j+1,j'),(j+1,j''), (j',j-1), (j'',j-1)\} \cup\{(i,i-1): j-1\geq i\geq 1\},&\mbox{if }j>0, \\[2pt] \{ (1,0'),(1,0'')\},&\mbox{if }j=0. \end{array} \end{align*} $$

By inspection, this relation satisfies (4) with $n=2$ , so ${\cal D}_j$ validates 5 $_2$ . (If we replaced $R_{\lozenge }$ by its transitive closure, $5_2$ would not be valid when $j>0$ .)

For each $i\in \omega $ , let $\alpha _i$ be the constant formula $\lozenge ^i\top \land \neg \lozenge ^{i+1}\top $ . In ${\cal D}_j$ , if $i<j$ or $i=j+1$ , then $\alpha _i$ holds at i and nowhere else. If $i>j+1$ , then $\alpha _i$ does not hold at any point. On the other hand, $\alpha _j$ holds at both $j'$ and $j''$ . Now let $\varphi _i$ be the formula

$$ \begin{align*}\Box(\alpha_i \to p) \lor \Box(\alpha_i \to \neg p). \end{align*} $$

If $i\ne j$ , then $\alpha _i$ is true at no more than one point in ${\cal D}_j$ , so ${\cal D}_j\models \varphi _i$ . But ${\cal D}_j\not \models \varphi _j$ , for making p true just at $j'$ gives a model on ${\cal D}_j$ at which $\alpha _j \to p$ is false at $j''$ and $\alpha _j \to \neg p$ is false at $j'$ , hence $\varphi _j$ is false at $j+1$ .

For each non-empty subset $I\subseteq \omega $ , let $L_I$ be the smallest normal modal logic containing $K5_2\cup \{\varphi _i:i\in I\}$ . Suppose that $I\ne J\subseteq \omega $ with, say, some $j\in J\setminus I$ . As $j\notin I$ we have ${\cal D}_j\models \varphi _i$ for all $i\in I$ . Since also ${\cal D}_j\models K5_2$ , the logic determined by ${\cal D}_j$ includes $K5_2\cup \{\varphi _i:i\in I\}$ , so it includes $L_I$ as the smallest such. Thus ${\cal D}_j\models L_I$ . But ${\cal D}_j\not \models \varphi _j$ , so then $\varphi _j\notin L_I$ . On the other hand $j\in J $ , so $\varphi _j\in L_J$ by definition. Hence $L_I\ne L_J$ . The case $I\setminus J\ne \emptyset $ is likewise.

Thus there are $2^{\aleph _0}$ logics $L_I$ , one for each subset I of $\omega $ , and they all include K5 $_2$ . But they are all included in $K5$ , because every formula $\varphi _i$ is a $K5$ -theorem. To show this, it is enough to show that $\varphi _i$ is valid in every point-generated $K5$ -frame, since these frames determine the logic. By an analysis originally due to Segerberg [Reference Segerberg17], each such frame either (i) is a single irreflexive point, or (ii) has the property that every point has a reflexive point in its future. In case (i), every formula of the form $\Box \psi $ is valid in the frame, hence the $\varphi _i$ ’s are. In case (ii), for any $i\in I$ the formula $\lozenge ^{i+1}\top $ is true at every point, hence $\alpha _i$ is false everywhere and so $\varphi _i$ again is valid.

5.1 Achronal sets and sequences

Sets with at most one element are $\lozenge $ -achronal. Subsets of $\lozenge $ -achronal sets are $\lozenge $ -achronal, and similarly for sequences. The following lemma is elementary but worth nailing down.

The condition ‘there is $Y\subseteq S$ of cardinality $\kappa $ such that $R^{\cal F}_{\lozenge }[Y]$ is an $\subseteq $ -antichain’ clearly follows from condition 3 in the lemma, but is strictly weaker, as we may have $|R^{\cal F}_{\lozenge }[Y]|<\kappa $ . As an informal example, let ${\cal F}=(\omega ,R)$ , where $R=\emptyset $ . Here, ${\cal F}$ has no achronal subset with more than one element, but $R[\omega ]=\{\emptyset \}$ is trivially an $\subseteq $ -antichain.

5.2 The logic $KU_n^{\mathsf {M}}$

Definition 5.3. Let $0<n<\omega $ . We write $KU_n^{\mathsf {M}}$ for the smallest normal modal ${\mathsf {M}}$ -logic containing the following set $U^{\mathsf {M}}_n$ of axioms:

$$\begin{align*}U_n^{\mathsf{M}}=\Big\{ \bigvee_{i\leq n}\Box\big(\blacksquare q_i\to\bigvee_{j\leq n,\,j\neq i}\blacksquare q_j\big) \;:\; \lozenge,\blacklozenge\in{\mathsf{M}} \Big\}. \end{align*}$$

For avoidance of doubt, $i$ and $j$ here denote ordinals in $\{0,\ldots ,n\}$ .

A normal modal ${\mathsf {M}}$ -logic is said to be of finite achronal width if it contains $U^{\mathsf {M}}_n$ for some $n$ with $0<n<\omega $ .

5.3 First-order correspondents of the $U_n^{\mathsf {M}}$

The $U_n^{\mathsf {M}}$ axiom shown above is equivalent to $(\bigwedge _{i}\lozenge (\blacksquare q_i\wedge \neg \bigvee _{j\neq i}\blacksquare q_j))\to \bot $ , a Sahlqvist formula according to [Reference Blackburn, de Rijke and Venema2, definition 3.51]. Its first-order correspondent [Reference Blackburn, de Rijke and Venema2, theorem 3.54] is

$$\begin{align*}\forall xy_0\ldots y_n\bigvee_{i\leq n} \big[ R_{\lozenge} xy_i\to\bigvee_{j\leq n,\,j\neq i}\forall z(R_{\blacklozenge} y_iz\to R_{\blacklozenge} y_jz)\big]. \end{align*}$$

The part following the $\forall xy_0\ldots y_n$ is the disjunction of the formulas $\neg R_{\lozenge } xy_i$ , for each $i\leq n$ , and $\forall z(R_{\blacklozenge } y_iz\to R_{\blacklozenge } y_jz)$ , for each distinct $i,j\leq n$ . Reordering these to put the $\neg R_{\lozenge } xy_i$ first, we see that the whole correspondent is logically equivalent to

(6) $$ \begin{align} \forall xy_0\ldots y_n \Big( \Big( \bigwedge_{i\leq n}R_{\lozenge} xy_i \Big) \to \bigvee_{i,j\leq n,\>i\neq j} \forall z(R_{\blacklozenge} y_iz\to R_{\blacklozenge} y_jz) \Big). \end{align} $$

This is the form that we will use. A frame validates $U_n^{\mathsf {M}}$ iff (6) holds in the frame for every $\lozenge ,\blacklozenge \in {\mathsf {M}}$ .

Informally, (6) says that for any $R_{\lozenge }$ -successors ${y_0,\ldots {},y_{n}}$ of some given world, we have $R_{\blacklozenge }(y_i)\subseteq R_{\blacklozenge }(y_j)$ for some distinct $i,j\leq n$ . That is, the sequence $(y_i:i\leq n)$ is not $\blacklozenge $ -achronal. So by Lemma 5.2, (6) is equivalent to each of:

• • Every $\blacklozenge $ -achronal set contained in some $R_{\lozenge }(x)$ has at most n elements.

• • For each x, the $\subseteq $ -width of $R_{\blacklozenge }[R_{\lozenge }(x)]$ is at most n.

Consequently, $U_n^{\mathsf {M}}$ is valid in precisely those frames ${\cal F}$ such that for each $x\in {\cal F}$ and $\lozenge \in {\mathsf {M}}$ , every achronal subset of $R^{\cal F}_{\lozenge }(x)$ has at most n elements.

5.4 Alternative formulation

The axiom shown in Definition 5.3 can be reformulated more in the style of [Reference Fine7] and [Reference Xu24], as

$$\begin{align*}\Big(\bigwedge_{i\leq n}\lozenge (p_i\wedge\blacksquare q_i)\Big) \to\bigvee_{i,j\leq n,\,i\neq j}\lozenge(p_i\wedge\blacksquare q_j), \end{align*}$$

where ${p_0,\ldots {},p_{n}},{q_0,\ldots {},q_{n}}\in {\mathsf {Q}}$ are pairwise distinct atoms. This form is also Sahlqvist, with first-order correspondent (6). So while the two forms are not equivalent formulas, they lie in the same normal modal ${\mathsf {M}}$ -logics. As axioms, they are interchangeable. We give the alternative form in case it is more appealing.

5.5 $U_1^{\mathsf {M}}$

To gain some familiarity with the $U_n^{\mathsf {M}}$ , let us quickly discuss the simplest case, when $M=\{\lozenge \}$ and $n=1$ . Then $U_1^{\mathsf {M}}=\{\Box (\Box q_0\to \Box q_1)\vee \Box (\Box q_1\to \Box q_0)\}$ . This is not a new axiom. For example, Van Benthem [Reference Van Benthem1, lemma 3.2] used $\Box (\Box p\to \Box q)\vee \Box (\Box q\to \Box p)$ as part of a Kripke-incompleteness proof, and stated an equivalent form of the correspondent (6) for it. We will say a little more about this in Section 6.

Plainly, every frame ${\cal F}$ such that

(7) $$ \begin{align} (R^{\cal F}_{\lozenge}[{\cal F}],\subseteq) \mbox{ is a chain---i.e.},\ R^{\cal F}_{\lozenge}(x)\subseteq R^{\cal F}_{\lozenge}(y) \mbox{ or } R^{\cal F}_{\lozenge}(y)\subseteq R^{\cal F}_{\lozenge}(x) \mbox{ for every } x,y\in{\cal F} \end{align} $$

has no achronal subset with more than one element, so satisfies (6) for $n=1$ and validates $U_1^{\mathsf {M}}$ . Indeed, it validates $U^{\mathsf {M}}_n$ for all $n\geq 1$ .

This provides a source of simple examples. Any frame forming a transitive chain, such as $(\omega ,<)$ , satisfies (7), but other frames do too. For example, the irreflexive transitive frame ${\cal G}_j$ shown in Figure 5 below satisfies (7) and validates $U_1^{\mathsf {M}}$ , but has an infinite R-antichain.

5.6 Comparing the $U_n^{\mathsf {M}}$ with each other

Let $K^{\mathsf {M}}$ denote the smallest normal modal ${\mathsf {M}}$ -logic. Plainly, (6) for n implies (6) for $n+1$ , and it follows that $KU_1^{\mathsf {M}}\supseteq KU_2^{\mathsf {M}}\supseteq \cdots \supseteq K^{\mathsf {M}}$ .

We show that the inclusions are strict. For any set X, let ${\cal F}_X$ be the ‘lawn rake-like’ frame with domain $\{a\}\cup X$ , for some point $a\notin X$ , and with

(8) $$ \begin{align} R_{\lozenge}^{{\cal F}_X}= R=\{(a,x),(x,x):x\in X\} \quad\mbox{for each }\lozenge\in{\mathsf{M}}. \end{align} $$

The achronal subsets of ${\cal F}_X$ are $\{a\}$ and the subsets of X. These all have size at most $|X|$ , if $X\neq \emptyset $ . So for $0<n<\omega $ , the frame ${\cal F}_{n+1}$ validates $U^{\mathsf {M}}_{n+1}$ . But ${\cal F}_{n+1}$ does not validate $U^{\mathsf {M}}_n$ , since $n+1$ is an achronal set with $>n$ elements contained in $R(a)$ . So the inclusion $KU_n^{\mathsf {M}}\supset KU_{n+1}^{\mathsf {M}}$ is strict. We will show more in Section 5.9.3.

Given any modal ${\mathsf {M}}$ -formula $\varphi \notin K^{\mathsf {M}}$ , filtrating the canonical model for $K^{\mathsf {M}}$ provides a finite Kripke model ${\cal M}$ satisfying $\neg \varphi $ . Let $|{\cal M}|=n$ , say. Then evidently, the frame of ${\cal M}$ validates $U_n^{\mathsf {M}}$ . Hence, $\varphi \notin KU_n^{\mathsf {M}}$ . We deduce that $\bigcap _{0<n<\omega }KU_n^{\mathsf {M}}=K^{\mathsf {M}}$ .

Of course there are many normal ${\mathsf {M}}$ -logics (containing $K^{\mathsf {M}}$ but) not containing any $KU_n^{\mathsf {M}}$ . For instance, the logic of the lawn-rake frame ${\cal F}_{\omega }$ does not contain any $KU^{\mathsf {M}}_n$ , since $\omega $ is an infinite achronal subset of ${\cal F}_{\omega }$ .

5.7 Comparing the $U_n^{\mathsf {M}}$ with other logics

Let us take a look at the monomodal case, where ${\mathsf {M}}=\{\lozenge \}$ . We will write $U^{\{\lozenge \}}_n$ as simply $U_n$ , and $R_{\lozenge }$ as just R. For suitable n, the logic $KU_n$ weakens some known logics.

1. 1. $KU_1\subset K5_2\subset K5$ , where $K5_2$ and $K5$ are as described in Section 4.

   As we said at the start of Section 5, the correspondent of $5_2$ (see (4) and (5)) says that all successors of a given world have the same R-future, and this implies (6) with $n=1$ . So every frame that validates $5_2$ also validates $U_1$ .

   Since $K5_2$ is Sahlqvist axiomatised, it is Kripke complete [Reference Blackburn, de Rijke and Venema2, theorem 4.42]. So for each modal formula $\varphi $ , if $\varphi \notin K5_2$ then $\neg \varphi $ is satisfied in some Kripke model whose frame validates $5_2$ . By the above, this frame also validates $U_1$ , and consequently $\varphi \notin KU_1$ . We deduce that $KU_1\subseteq K5_2$ as stated. The inclusion $K5_2\subseteq K5$ was indicated in Section 4.1.

   The frame $(\omega ,<)$ validates $U_1$ , as we saw in Section 5.5, but any two distinct points in it have different futures, so it does not validate $5_2$ . Hence, the inclusion $KU_1\subset K5_2$ is strict. We saw in Section 4.2 that the inclusion $K5_2\subset K5$ is (very) strict.

2. 2. $KU_n\subset K4I_n$ for every n, where the latter is as in [Reference Fine7] and is the smallest normal modal logic containing the Sahlqvist axioms

   (9) $$ \begin{align} \begin{array}{rcll} 4&=&\lozenge\lozenge p\to\lozenge p &\mbox{(transitivity),} \\ I_n&=&(\bigwedge_{i\leq n}\lozenge q_i)\to\bigvee_{i,j\leq n,\,i\neq j}\lozenge(q_i\wedge(q_j\vee\lozenge q_j)). \end{array} \end{align} $$

   The first-order correspondent of $I_n$ says that no $R(x)$ contains an R-antichain with $>n$ points. Given this and transitivity, if ${y_0,\ldots {},y_{n}}\in R(x)$ , there are $i\neq j$ with $y_i=y_j$ or $Ry_iy_j$ , and hence (by transitivity) with $R(y_j)\subseteq R(y_i)$ . So (6) holds for n, and hence, every frame validating $4I_n$ also validates $U_n$ . Again, $K4I_n$ is Sahlqvist and so Kripke complete, and it follows that $KU_n\subseteq K4I_n$ .

   Fig. 4

   The logics $KU_n$ , $K4I_n$ , $K5_2$ , and $K5$ (monomodal case).

   

   The inclusion $KU_n\subset K4I_n$ is proper for each n, since frames such as $ (\omega ,\{(i,i+1):i<\omega \}) $ validate $U_n$ but not even $K4$ . We will see transitive examples in Section 5.9.1.

3. 3. The lawn-rake frame ${\cal F}_{n+1}$ from Section 5.6 validates $K4I_{n+1}$ , since it is transitive and its only subset $\{a\}\cup (n+1)$ with $>n+1$ points is not an R-antichain—see (8). But as we said, it does not validate $U_{n}$ . Hence, $KU_n\not \subseteq K4I_{n+1}$ .

4. 4. Rather trivially, $K4\not \subseteq K5$ , since the frame $(\{0,1,2\},\{(0,1)\}\cup \{1,2\}^2)$ validates $K5$ but is not transitive. Also, $K5_2\not \subseteq K4I_1$ , since $(\omega ,<)$ validates $K4I_1$ but (as we saw in item 1) not $5_2$ .

The situation is therefore as shown in Figure 4. No inclusions can be added that do not already follow from the ones shown by transitivity of inclusion.

1. 5. $U_n$ is somewhat related to Xu’s axiom $\mathsf {Wid}_n^*$ [Reference Xu24, p. 1178], whose correspondent says that for every x, every R-antichain with $>n$ points in $R(x)$ contains distinct points $y,y'$ with the same proper future—that is, $\forall z(R^{\bullet } yz \leftrightarrow R^{\bullet } y'z)$ , where $R^{\bullet } yz$ abbreviates $R yz\wedge \neg R zy$ . This clearly bears some similarity to (6), but the two are independent even with transitivity. The lawn-rake frame ${\cal F}_{\omega }$ from Section 5.6 is transitive and validates all $K4\mathsf {Wid}_n^*$ , but no $U_n$ . Now let $K4U_n$ be the smallest normal modal logic containing $U_n$ and the transitivity axiom 4. The transitive frame $(W,R)$ , where $W=\{a\}\cup (\omega \times 2)$ , $a\notin \omega \times 2$ , and $R=\{(a,(i,k)),((i,k),(j,1)):j\leq i<\omega ,\; k<2\}$ , validates $K4U_1$ (by virtue of (7)), and so all $K4U_n$ , but no $\mathsf {Wid}_n^*$ , since no two points in the R-antichain $\omega \times \{0\}$ have the same proper successors.

5.8 $S4U_n=S4I_n$

There is another connection between $U_n$ and $I_n$ . For a binary relation R on a set W, let be the relation on W defined by iff $R(y)\subseteq R(x)$ . It is an exercise to verify that (i) an R-achronal set (defined in the obvious way) is the same thing as an -antichain; (ii) is reflexive and transitive; (iii) iff R is reflexive; (iv) iff R is transitive. By (ii)–(iv), we get (v) .

However, (vi) is not monotonic with respect to inclusion, even on transitive relations. For suppose $|W|\geq 2$ and $R=\{(x,x):x\in W\}$ (identity). Then $\emptyset $ and R are transitive and $\emptyset \subseteq R$ . But R is also reflexive, so by (iii) and (iv), . So .

Fig. 5

Irreflexive transitive frame ${\cal G}_j$ for Sections 5.9.1 and 5.9.2.

Lastly, and getting to the point, (vii) if ${\cal F}=(W,R)$ is rootedFootnote ³ and transitive, and , then ${\cal F}$ validates $U_n$ iff validates $I_n$ , for each $0<n<\omega $ .

It follows from (iii), (iv), and (vii) that $S4U_n=S4I_n$ , where these are the smallest normal modal logics containing 4, the reflexivity axiom $ p\to \lozenge p$ , and $U_n$ (respectively,  $I_n$ ).

5.9 Continuum-many logics in intervals in Figure 4

Here we remain in the monomodal case: ${\mathsf {M}}=\{\lozenge \}$ . We still write $U^{\{\lozenge \}}_n$ as $U_n$ and $R_{\lozenge }$ as R.