1 Informal introduction

The notion of structural completeness ( $\mathbf {SC}$ ) was introduced by W. A. Pogorzelski in [Reference Pogorzelski24]. A consequence relation is $\mathbf {SC}$ if all of its admissible rules are derivable. The distinction between the admissibility and derivability of a rule lies in the extent of its applicability within a given consequence relation. An admissible rule is one that “works” on the theorems of a given logic, meaning it guarantees that its conclusion is a theorem whenever all its premises are theorems. A derivable rule, on the other hand, is a rule in the “standard sense,” meaning it can be applied to an arbitrary set of formulas. Thus, roughly speaking, in a given consequence relation, $\varphi $ follows from a set of premises $\Sigma $ if there exists a finite subset $\Gamma $ of $\Sigma $ such that $\Gamma /\varphi $ is a derivable rule in the system. Clearly, if we are working within a fixed deductive system, every derivable rule is also admissible. However, numerous counterexamples demonstrate that the converse is not generally true. Two paradigmatic examples of non-derivable admissible rules are the Harrop’s rule

$$\begin{align*}\dfrac{\neg\varphi\to(\psi\lor\chi)}{(\neg\varphi\to\psi)\lor(\neg\varphi\to\chi)} \end{align*}$$

for intuitionistic logic $\mathbf {Int}$ and the disjunctive syllogism rule $\mathtt {DS}$

$$\begin{align*}\dfrac{\neg\varphi\lor\psi, \varphi}{\psi} \end{align*}$$

for the system of relevance logic $\mathbf {R}$ . Adding these rules to their respective systems does not affect the sets of theorems but alters the consequence relations. Thus, neither of these two logics is structurally complete ( $\mathbf {SC}$ ).

To emphasize the importance of the $\mathbf {SC}$ property, let us recall two major formal definitions of the notion of “a logic.” The first treats it as a mere set of theorems, usually generated from an initial set of axioms via certain rules. In this approach, there is no distinction between derivable and admissible rules, as these rules are applied only to theorems. The second definition views a logic as a consequence relation—a binary relation between sets of formulas (premises) and single formulas (conclusions). In a sense, the phenomenon of structural incompleteness underscores the superiority of the second approach, as two logics can share identical sets of theorems but differ when viewed as consequence relations.Footnote ¹ The distinction between derivable and admissible rules, which is only possible within the framework of consequence relations, is precisely why the second definition is more nuanced.

Hereditary structural completeness ( $\mathbf {HSC}$ ) is a stronger form of the standard structural completeness ( $\mathbf {SC}$ ). A consequence relation is said to be $\mathbf {HSC}$ if and only if it is $\mathbf {SC}$ and all of its extensions are also $\mathbf {SC}$ . In this paper, we will investigate the problem of hereditary structural completeness within extensions of the logic $\mathbf {R}$ -mingle ( $\mathbf {RM}$ ), which is the most well-known and widely studied extension of Belnap and Anderson’s system of relevance logic $\mathbf {R}$ .

The system $\mathbf {RM}$ is obtained by adding the “mingle” axiom $p \to (p \to p)$ to $\mathbf {R}$ . Dunn established in [Reference Dunn12] that Sugihara algebras provide an adequate algebraic semantics for $\mathbf {RM}$ . The theory of Sugihara algebras will be crucial to our investigation, as we approach the problem using the powerful framework of abstract algebraic logic. Our final theorem will demonstrate that the structure of the poset of all hereditarily structurally complete extensions of $\mathbf {RM}$ is the converse of the well-ordering $\omega ^+$ with an additional element adjoined above 1:

The problem of (hereditary) structural completeness has often been explored in the context of non-classical logics. So far, results regarding $\mathbf {SC}$ have been confined to axiomatic systems. A well-known theorem proven by Citkin in [Reference Citkin10] characterizes $\mathbf {HSC}$ superintuitionistic logics (axiomatic extensions of $\mathbf {Int}$ ) in the following manner: a given variety of Heyting algebras is $\mathbf {HSC}$ if and only if it omits five specific algebras; this is further equivalent to the statement that there are exactly five maximal non- $\mathbf {HSC}$ superintuitionistic logics. Similarly, Rybakov’s theorem from [Reference Rybakov27] indicates that, in the case of axiomatic extensions of modal logic $\mathbf {K4}$ , there are twenty such algebras.

While these characterizations are sophisticated and undoubtedly elegant, they can be seen as somewhat roundabout; they do not directly identify the $\mathbf {HSC}$ extensions themselves but rather highlight the algebras that must be omitted.Footnote ² In the current work, we will provide a direct characterization of the $\mathbf {HSC}$ extensions of $\mathbf {RM}$ : we will construct the poset of the respective quasivarieties and describe the algebras that generate them. In the final part of the paper, we will also examine a weaker version of $\mathbf {SC}$ —passive structural completeness—and obtain a Citkin-style characterization for passively structurally complete quasivarieties of Sugihara algebras. In this case, only two algebras will need to be omitted.

Structural completeness has been widely investigated among substructural logics [Reference Cintula and Metcalfe9, Reference Kowalski20, Reference Olson, Raftery and van Alten23]. In particular, the problem of $\mathbf {SC}$ has been addressed for certain fragments of $\mathbf {RM}$ with Ackermann’s constant $\mathbf {t}$ added to the original signature ( $\mathbf {RM}^{\mathbf {t}}$ ) in [Reference Olson and Raftery22, Reference Olson, Raftery and van Alten23]. The positive (negation-free) fragment of $\mathbf {RM}^{\mathbf {t}}$ is $\mathbf {HSC}$ [Reference Olson and Raftery22], while the purely implicational fragment of $\mathbf {RM}$ is not $\mathbf {SC}$ (see [Reference Olson, Raftery and van Alten23]). However, we will consider $\mathbf {RM}$ in its original signature.

The most pertinent result related to our inquiry is the theorem proven by Raftery and Świrydowicz in [Reference Świrydowicz and Raftery19], which states that there are no $\mathbf {SC}$ axiomatic extensions of $\mathbf {R}$ other than classical propositional logic ( $\mathbf {CPL}$ ) and inconsistent logic. Since $\mathbf {RM}$ is an axiomatic extension of $\mathbf {R}$ , it follows that neither $\mathbf {RM}$ nor any of its axiomatic extensions—excluding $\mathbf {CPL}$ and inconsistent logic—is $\mathbf {SC}$ . In this paper, we will tackle a more general problem: we will investigate arbitrary extensions of $\mathbf {RM}$ , not limited to axiomatic ones.

2 Preliminaries

We assume a countably infinite set of propositional variables, denoted by $\mathsf {Var} = \{p_1, p_2, p_3, \ldots \}$ (in practice, we will use letters $p, q, r, \ldots $ ). A language $\mathcal {L}$ is a set of symbols; specifically, $\mathcal {L} = \mathsf {Var} \cup \mathsf {Con}$ , where $\mathsf {Con}$ is the set of logical connectives $\{\neg , \lor , \wedge , \to , \leftrightarrow \}$ . As usual, $\neg $ is unary and the rest of the connectives are binary. The set of formulas $\mathcal {F}$ consists of properly constructed finite strings of symbols from $\mathcal {L}$ ; that is, $p, \neg \varphi , \varphi \ast \psi $ , where $\ast $ is any binary connective. A uniform substitution is a map from $\mathcal {F}$ to $\mathcal {F}$ that preserves logical operations, i.e., an endomorphism of the absolutely free formula algebra. We define $\vdash \subseteq 2^{\mathcal {F}} \times \mathcal {F}$ as a consequence relation if it satisfies the following properties:

• • reflexive: $\Sigma \vdash \sigma $ for any $\sigma \in \Sigma $ ;

• • monotonic: if $\Sigma \vdash \sigma $ , then $\Sigma \cup \Gamma \vdash \sigma $ ;

• • transitive: if $\Gamma \vdash \varphi $ and for any $\gamma \in \Gamma $ we have $\Sigma \vdash \gamma $ , then $\Sigma \vdash \varphi $ ;

• • structural: if $\Sigma \vdash \varphi $ , then $s(\Sigma ) \vdash s(\varphi )$ for any uniform substitution s;

• • finitary: $\Sigma \vdash \varphi $ if and only if $\Sigma ' \vdash \varphi $ for some finite subset $\Sigma '$ of $\Sigma $ .

Elements of $2^{\mathcal {F}} \times \mathcal {F}$ whose first component is finite will be called rules. Instead of writing $\langle \Gamma , \varphi \rangle $ , we will adopt the following notation: $\Gamma / \varphi $ , where $\Gamma \cup \{\varphi \}$ is a finite subset of $\mathcal {F}$ . If a given rule $\Gamma / \varphi $ is a member of a consequence relation $\vdash $ , then we say that $\Gamma / \varphi $ is a derivable rule of $\vdash $ . According to our definition, all consequence relations are finitary, so we can identify a given consequence relation with the set of its derivable rules. A derivable rule of a given consequence relation $\vdash $ of the form $\emptyset / \varphi $ is called a theorem of $\vdash $ and is shortly denoted by $\varphi $ . Each consequence relation is a set of rules, some of which may be theorems. For two consequence relations $\vdash _0,\,\vdash _1\, \subseteq 2^{\mathcal {F}} \times \mathcal {F}$ , we say that $\vdash _1$ is an extension of $\vdash _0$ if $\vdash _0 \,\,\subseteq \,\, \vdash _1$ . It is easy to see that the intersection of an arbitrary set of consequence relations is also a consequence relation: $\bigcap _{i \in I} \vdash _i$ is reflexive, monotonic, transitive, structural, and finitary, given that each $\vdash _i$ has these properties. Thus, for any set of rules X, there exists a least consequence relation $\vdash $ that contains X. In such cases, we will say that X defines $\vdash $ . A rule $\Gamma / \varphi $ is admissible in $\vdash $ if and only if, for any substitution $\sigma $ , $\sigma (\varphi )$ is a theorem of $\vdash $ whenever $\sigma (\Gamma )$ is a set of theorems of $\vdash $ . Furthermore, a rule $\Gamma / \varphi $ is passive in $\vdash $ iff there is no uniform substitution $\sigma $ , such that $\sigma (\Gamma )$ is a set of theorems of $\vdash $ . Thus, each passive rule of $\vdash $ is vacuously admissible. A given consequence relation $\vdash $ is structurally complete ( $\mathbf {SC}$ ) iff each admissible rule in $\vdash $ is derivable. Furthermore, a consequence relation is hereditarily structurally complete ( $\mathbf {HSC}$ ) when all of its extensions are $\mathbf {SC}$ . Consequence relations that are $\mathbf {SC}$ but not $\mathbf {HSC}$ (i.e., those that have at least one extension that is not $\mathbf {SC}$ ) will be called non-hereditarily structurally complete ( $\mathbf {nHSC}$ ).

It is natural to associate a closure operator C with a given consequence relation $\vdash $ in the following way: $\varphi \in C(X)$ if and only if $X \vdash \varphi $ for any $X \subseteq \mathcal {F}$ . It has also been common to first define a structural and finitary closure operator as a primitive notion and then treat the consequence operation as a secondary entity. However, it seems more natural to talk about derivable rules in the context of consequence relations (simply as their members). Since the notions of consequence operator and consequence relation are two sides of the same coin, we decide to stick to the latter, treating it as the fundamental notion that captures the meaning of the term ‘logic’.

A Horn formula, or quasi-identity as it is also called, is a first-order sentence of the form $\forall _{x_1, \ldots , x_l}(\varphi _1(x_1, \ldots , x_l) \approx \psi _1(x_1, \ldots , x_l) \& \ldots \& \varphi _n(x_1, \ldots , x_l) \approx \psi _n(x_1, \ldots , x_l) \Longrightarrow \varphi (x_1, \ldots , x_l) \approx \psi (x_1, \ldots , x_l)$ . In practice, we will omit the quantifiers. Identities are quasi-identities with an empty antecedent. Let $\Phi $ be a quasi-equation and $\mathfrak {A}$ be an algebra. $\mathfrak {A} \vDash \Phi $ means that $\Phi $ holds in $\mathfrak {A}$ . For a class of algebras, we write $\mathsf {K} \vDash \Phi $ to indicate that $\Phi $ holds in each member of $\mathsf {K}$ . We will use standard notation $\mathsf {H}, \mathsf {I}, \mathsf {S}, \mathsf {P}$ , and $\mathsf {P}_{\mathsf {U}}$ for the well-known algebraic closure operators defined on arbitrary classes of algebras. Thus, given a class of algebras $\mathsf {K}$ , $\mathsf {H}(\mathsf {K}), \mathsf {I}(\mathsf {K}), \mathsf {S}(\mathsf {K}), \mathsf {P}(\mathsf {K}), \mathsf {P}_{\mathsf {U}}(\mathsf {K})$ are $\mathsf {K}$ ’s closures under homomorphic images, isomorphic copies, subalgebras, direct products, and ultraproducts, respectively. For two algebras $\mathfrak {A}$ and $\mathfrak {B}$ , we will write $\mathfrak {A} \cong \mathfrak {B}$ when $\mathfrak {A} \in \mathsf {I}(\mathfrak {B})$ . Given a class of algebras $\mathsf {K}$ , $\mathbf {F}_{\mathsf {K}}(\kappa )$ will denote the $\kappa $ -generated algebra that is free in $\mathsf {I}\mathsf {S}\mathsf {P}(\mathsf {K})$ . We will also write $\mathsf {V}(\mathsf {K})$ for $\mathsf {H}\mathsf {S}\mathsf {P}(\mathsf {K})$ and $\mathsf {Q}(\mathsf {K})$ for $\mathsf {I}\mathsf {S}\mathsf {P}\mathsf {P}_{\mathsf {U}}(\mathsf {K})$ . We will use obvious notation for finite direct powers: $\mathfrak {A}^1 = \mathfrak {A}$ , and $\mathfrak {A}^{n+1} = \mathfrak {A}^n \times \mathfrak {A}$ . To indicate that an algebra $\mathfrak {A}$ is embeddable in (isomorphic to a subalgebra of) $\mathfrak {B}$ , i.e., $\mathfrak {A} \in \mathsf {I}\mathsf {S}(\mathfrak {B})$ , we will use the following notation: $\mathfrak {A} \preceq \mathfrak {B}$ . We will also write $\mathfrak {A} \prec \mathfrak {B}$ when $\mathfrak {A} \preceq \mathfrak {B}$ and $\mathfrak {A} \not \cong \mathfrak {B}$ . If $\mathsf {K} = \mathsf {V}(\mathsf {K})$ , then it is called a variety; if $\mathsf {K} = \mathsf {Q}(\mathsf {K})$ , then it is a quasivariety. Arbitrary varieties will be referred to as $\mathcal {V}$ and quasivarieties as $\mathcal {Q}$ .Footnote ³ Varieties are definable by identities, and quasivarieties by quasi-identities.

Now, we will recall some results concerning the algebraic counterpart of the structural completeness property, which were first formulated by Bergman in [Reference Bergman5].Footnote ⁴ For a consequence relation, being structurally complete is equivalent to having no proper extensions with the same set of theorems. To see that, let $\vdash _0\,\, \subsetneq \,\, \vdash _1$ be two consequence relations, where the second properly extends the first, with the additional property that they have the same theorems, i.e., $\vdash _0 \varphi $ if and only if $\vdash _1 \varphi $ for any $\varphi \in \mathcal {F}$ . Hence, there is a rule $\Gamma / \psi $ in $\vdash _1$ that does not belong to $\vdash _0$ . However, $\Gamma / \psi $ is admissible in $\vdash _0$ (otherwise, the set of theorems would be different). Hence, $\vdash _0$ is not $\mathbf {SC}$ . For the other direction, let $\vdash _0$ be such that each of its proper extensions also properly extends the set of its theorems. If there is a rule $\Gamma / \psi $ that is an admissible but non-derivable rule of $\vdash _0$ , then it is easy to see that $\vdash _1$ , defined by the set of rules $\vdash \cup \{\langle \Gamma , \psi \rangle \}$ , properly extends $\vdash _0$ and has exactly the same theorems, contradicting the initial assumption. Thus, in algebraic terms—assuming Blok–Pigozzi algebraizability, which ‘reverses the order’—we have the following:

Thus, the $\mathbf {SC}$ quasivarieties are precisely those that are generated by their $\omega $ -generated free algebras. A given quasivariety is $\mathbf {HSC}$ if and only if all of its subquasivarieties are $\mathbf {SC}$ —analogously, a consequence relation is $\mathbf {HSC}$ if and only if it has only $\mathbf {SC}$ extensions. Not surprisingly, a given $\mathbf {SC}$ quasivariety is $\mathbf {nHSC}$ when it has at least one subquasivariety that is not $\mathbf {SC}$ .

3 $\mathbf {RM}$ and Sugihara algebras

Let $\mathfrak {A}_\omega =\langle \mathbb {Z};\wedge ,\lor ,\neg ,\to \rangle $ , where $\mathbb {Z}$ is the set of integers, $x\wedge y=\min (x,y),x\lor y=\max (x,y)$ , $\neg k=-k$ and for $\to $ we have

$$ \begin{align*} k\to l &= \begin{cases} -k\lor l, & \text{if }k\leq l;\\ -k\wedge l, & \text{otherwise. } \end{cases} \ \end{align*} $$

It is evident that there are two types of finite subalgebras of $\mathfrak {A}_\omega $ : those that contain $0$ and those that do not. The former will be termed “odd,” while the latter will be referred to as “even.” Consequently, we introduce the following notation for $n\geq 1$ :

$$ \begin{align*} \mathfrak{A}_{2n} & =\langle\{-n,\dots,-1,1,\dots,n\};\wedge,\lor,\neg,\to\rangle,\\ \mathfrak{A}_{2n+1} & =\langle\{-n,\dots,-1,0,1,\dots,n\};\wedge,\lor,\neg,\to\rangle. \end{align*} $$

$\mathfrak {A}_1$ is the trivial one element algebra. $\mathfrak {A}_2$ is a two element boolean algebra. An infinite proper subalgebra of $\mathfrak {A}_\omega $ will be denoted by $\mathfrak {A}_{\omega \setminus \{0\}}=\langle \mathbb {Z}\setminus \{0\};\wedge ,\lor ,\neg ,\to \rangle $ . An algebra $\mathfrak {A}$ will be referred to as Sugihara algebra when $\mathfrak {A}\in \mathsf {V}(\mathfrak {A}_\omega )=\mathsf {Q}(\mathfrak {A}_\omega )$ .

Anderson and Belnap’s [Reference Anderson and Belnap4, p. 341] system of relevance logic $\mathbf {R}$ is defined by 13 axioms and two rules:

1. A1 $p\to p$

2. A2 $(p\to q)\to ((q\to r)\to (p\to r))$

3. A3 $p\to ((p\to q)\to q)$

4. A4 $(p\to (p\to q))\to (p\to q)$

5. A5 $p\wedge q\to p$

6. A6 $p\wedge q\to q$

7. A7 $((p\to q)\wedge (p\to r))\to (p\to q\wedge r)$

8. A8 $p\to p\lor q$

9. A9 $p\to q\lor p$

10. A10 $((q\to p)\wedge (r\to p))\to (q\lor r \to p)$

11. A11 $p\wedge (q\lor r)\to (p\wedge q)\lor r$

12. A12 $(p\to \neg q)\to (q\to \neg p)$

13. A13 $\neg \neg p\to p$

The two rules of the system is modus ponens MP $\{p,p\to q\}/q$ and the adjunction rule AD $\{p,q\}/p\wedge q$ .

The logic $\mathbf {R}$ –mingle ( $\mathbf {RM}$ ) is obtained by adding the “mingle axiom” $p\to (p\to p)$ to the axiomatic system of relevant logic $\mathbf {R}$ . We will consider $\mathbf {RM}$ as a consequence relation rather than merely as a set of theorems. The logic $\mathbf {RM}$ is algebraizable in the sense of [Reference Blok and Pigozzi7] with respect to the quasivariety of Sugihara algebras, using the set of formulas $\{p\to q, q\to p\}$ and the equation $p\approx p\to p$ . For a given finitary rule $R=\Gamma /\varphi $ , where $\Gamma =\{\gamma _1,\gamma _2,\dots ,\gamma _n\}$ we can say that Sugihara algebra $\mathfrak {A}$ satisfies R if $\mathfrak {A}\vDash \gamma _1\approx \gamma _1\to \gamma _1 \& \cdots \&\gamma _n\approx \gamma _n\to \gamma _n\Longrightarrow \varphi \approx \varphi \to \varphi $ . In such cases, we will use the abbreviated notation $\mathfrak {A}\vDash R$ and say that $\mathfrak {A}$ satisfies the rule R. It is straightforward to observe that, to check whether R holds in $\mathfrak {A}$ , we do not need to translate R into a quasi-equation; instead, we can treat $\mathfrak {A}$ as a logical matrix with the set of designated elements $\{a\in \mathsf {A}:a=a\to a\}$ . We naturally define an absolute value of an element $a\in \mathfrak {A}$ as $|a|:=a\to a$ . We define deductive filters $F\subseteq \mathsf {A}$ as standard lattice filters such that the set of designated values is included in F; that is, $\{a\in \mathsf {A}: a= |a|\}\subseteq F$ . Thus, as a consequence of algebraizability, we have an order isomorphism between the lattices of deductive filters and congruences for a given Sugihara algebra.

Now, we will gather results on Sugihara algebras from [Reference Blok and Dziobiak6, Reference Czelakowski and Dziobiak11, Reference Dunn12, Reference Krawczyk21], which will be crucial in the proof of the theorem.

Let us start with two important results that can be found in [Reference Blok and Dziobiak6, p. 275]

As a consequence of these, we can restate the theorem of Dunn [Reference Dunn12, Theorem 9, p. 9] in its purely algebraic version:

We can already observe that Sugihara algebras enjoy some nice properties. They are locally finite, their finite subdirectly irreducibles are chains, and the lattice of subvarieties of Sugihara algebras form an $\omega ^+$ well ordering. Consequently, due to the Blok–Pigozzi algebraizability, the lattice of axiomatic extensions of $\mathbf {RM}$ is the converse of $\omega ^+$ well-ordering. As will become evident below, there is also an elegant characterization of directly indecomposable Sugihara algebras.

Definition 3.4. Let $\mathfrak {A}=\langle \mathsf {A};\wedge ,\lor ,\neg ,\to \rangle $ be a Sugihara algebra. $\bot \mathfrak {A}\top =\langle \mathsf {A}\cup \{\bot ,\top \};\wedge ,\lor ,\neg ,\to \rangle $ , where $\bot $ and $\top $ are added as lower and upper bounds, $\wedge $ and $\lor $ are interpreted standardly in the resulting lattice, $\neg a=\neg ^{\mathfrak {A}} a$ for $a\in \mathsf {A}$ , $\neg \top =\bot $ , $\neg \bot =\top $ and:

$$ \begin{align*} a\to b = \begin{cases} a\to^{\mathfrak{A}} b, & \text{if }a,b\in\mathsf{A},\\ \top, & \text{if }a=\bot\text{ or }b=\top,\\ \bot, & \text{otherwise}. \end{cases} \end{align*} $$

We also write $\bot ^{n+1}\mathfrak {A}\top ^{n+1}$ to indicate $\bot \bot ^n\mathfrak {A}\top ^n\top $ . Note that ${\top , \bot }$ is assumed to be disjoint from $\mathsf {A}$ in $\bot \mathfrak {A}\top $ , so we must distinguish between the different ‘tops’ and ‘bottoms’ in algebras of the form $\bot ^n\mathfrak {A}\top ^n$ where $n \geq 2$ . To clarify, we will use $\bot _i$ and $\top _i$ to denote the ith bottom and top, respectively. Following this notation, $\bot ^n\mathfrak {A}\top ^n$ becomes $\bot _n \dots \bot _2\bot _1\mathfrak {A}\top _1\top _2 \dots \top _n$ , where $\top _i \leq \top _j$ and $\bot _j \leq \bot _i$ in terms of lattice ordering for $i \leq j$ .

It is immediate to see that for any subdirectly irreducible finite Sugihara algebra $\mathfrak {A}$ , $\bot \mathfrak {A}\top $ is also subdirectly irreducible, i.e., $\bot \mathfrak {A}_n\top \cong \mathfrak {A}_{n+2}$ . Next, we will recall a characterization of DI Sugihara algebras [Reference Blok and Dziobiak6, Corollary 2.5, p. 278].

Thus, DI Sugihara algebras are simply arbitrary Sugihara algebras extended with disjoint top and bottom elements. However, the most important tool for our investigations will be critical algebras. A finite algebra $\mathfrak {A}$ is said to be critical if and only if $\mathfrak {A} \notin \mathsf {I}\mathsf {S}\mathsf {P}({\mathfrak {B} : \mathfrak {B} \prec \mathfrak {A}})$ . It is well known that every locally finite quasivariety is generated by its critical algebras (cf. [Reference Hyndman and Nation18]). Since Sugihara algebras are locally finite, we will use critical algebras as fundamental building blocks for quasivarieties. A specific description of critical Sugihara algebras was provided in [Reference Czelakowski and Dziobiak11, p. 285], and this will serve as a crucial tool in our investigations.

It is easy to see that algebras of the third type take the form $\bot ^m\mathfrak {A}\top ^m$ , where $\mathfrak {A}$ is of the fourth type. To familiarize the reader with these types, we provide several examples: $\bot ^2\mathfrak {A}_4\times \mathfrak {A}_6\top ^2$ is an algebra of the third type, while $\mathfrak {A}_6\times \bot \mathfrak {A}_8\times \mathfrak {A}_5\top $ and $\mathfrak {A}_8\times \bot ^2\mathfrak {A}_6\times \mathfrak {A}_5\top ^2$ are of the fourth type. Additionally, $\bot ^{11}(\mathfrak {A}_6\times \bot \mathfrak {A}_8\times \mathfrak {A}_5\top )\top ^{11}$ is of the third type.

On the other hand, the following algebras do not satisfy this description: $\bot ^3\mathfrak {A}_3\times \mathfrak {A}_5\top ^3$ —at most one algebra in the construction can be odd; $\bot \mathfrak {A}_4\times \mathfrak {A}_4\top $ —algebras within the construction cannot be the same; and $\mathfrak {A}_8\times \bot ^2\mathfrak {A}_2\times \mathfrak {A}_3\top ^2$ —here, $4\not <1+2$ , contrary to the description.

Furthermore, in the context of nested product operations, the “horizontal” notation used in third- and fourth-type algebras quickly becomes difficult to read. It is hard to determine which algebras are extended by the respective tops and bottoms. Parentheses provide some clarification, but we argue that the vertical “quasi-Hasse” notation is easier to read. For example, consider the algebra $\bot (\mathfrak {A}_6\times (\bot ^3(\mathfrak {A}_4\times (\bot ^2\mathfrak {A}_2\times \mathfrak {A}_3\top ^2))\top ^3))\top $ . Let us present this in a vertical notation, somewhat resembling a Hasse diagram:

The scope of the extending tops and bottoms is indicated by the nearest ellipse. Fortunately, we will later introduce a method that simplifies these nested algebras, eliminating some notational problems in the subsequent sections.

One can observe that in algebras of the third and fourth type, for any $i<n$ , we have $\mathfrak {A}_{2k_i}\preceq \bot ^{p_{i+1}}\mathfrak {A}_{2k_{i+1}}\times \dots \bot ^{p_n}\mathfrak {A}_{2k_n}\times \mathfrak {A}_k\top ^{p_n}\ldots \top ^{p_{i+1}}$ . While this is straightforward to verify directly, the observation also follows from three key facts:

• • if $\bot \mathfrak {B}\top $ is critical, then $\mathfrak {B}$ is critical,

• • if $\mathfrak {A}\times \mathfrak {B}$ is critical, then both $\mathfrak {A}$ and $\mathfrak {B}$ are critical, and

• • the Lemma 3.4 from [Reference Czelakowski and Dziobiak11] which states that if $\mathfrak {A}\times \mathfrak {B}$ is critical, then it is isomorphic to $\mathfrak {A}_{2n}\times \bot \mathfrak {C}\top $ where $\mathfrak {A}_{2n}$ is embeddable in $\bot \mathfrak {C}\top $ under assumption that $\bot \mathfrak {C}\top $ is not subdirectly irreducible.

We will also use the following result from [Reference Krawczyk21, p. 1250]:

Thus, we know that the lattice of quasivarieties of Sugihara algebras is more complex than that of varieties: it does not form a chain, as it contains incomparable elements. As the theorem shows, such elements already appear at the very bottom of the lattice.

4 The theorem and its proof

Our ultimate goal is to isolate the poset of all $\mathbf {HSC}$ consequence relations extending $\mathbf {RM}$ . To achieve this, we begin by describing all structurally complete quasivarieties of Sugihara algebras. Next, within the set of $\mathbf {SC}$ quasivarieties, we distinguish the $\mathbf {SC}$ quasivarieties that possess the additional property of hereditariness from those that do not. The fundamental theorem we aim to prove takes the following form:

Theorem 4.1. The set of all $\mathbf {HSC}$ subquasivarieties of Sugihara algebras is

$$ \begin{align*} \{\mathsf{Q}(\mathfrak{A}_1),\mathsf{Q}(\mathfrak{A}_2), \mathsf{Q}(\mathfrak{A}_2\times\mathfrak{A}_3)\}\cup\{\mathsf{Q}(\mathfrak{A}_2\times\mathfrak{A}_{2k}):k\geq 2\}\cup\{\mathsf{Q}(\mathfrak{A}_2\times\mathfrak{A}_{\omega\setminus\{0\}})\}. \end{align*} $$

In the general algebraic setting, an approach to the problem of (hereditary) structural completeness based on projectivity has proven to be fruitful [Reference Aglianò and Citkin2, Reference Aglianò and Ugolini3]. However, as has already become apparent, we have a rather rich theory of Sugihara algebras at our disposal, and thus our strategy will differ. In one way or another, most of the proofs that follow will rely on the algebras from Theorem 3.6, since locally finite quasivarieties are known to be generated by their critical members [Reference Hyndman and Nation18]. However, the algebras of the third and fourth types in Theorem 3.6 are rather complex and difficult to handle. To ease the process, we will simplify certain instances of these algebras by providing their $\mathcal {Q}$ -equivalent descriptions. As will become apparent later, this simplification reveals that Theorem 3.6 cannot be strengthened into a full characterization (note that it presents only a necessary condition for criticality, as it is framed as an implication). $\mathcal {Q}$ -equivalence, or Horn equivalence, as we will also refer to it, is a weaker version of the standard model-theoretic notion of elementary equivalence. As the reader likely knows, or suspects, two algebras are Horn equivalent if they satisfy the same quasi-identities, rendering them indistinguishable from one another from the perspective of quasivarieties. To make things precise: by a quasi-identical theory of a given algebra $\mathfrak {A}$ , we understand the set $\mathsf {Th}_{\mathsf {q}}(\mathfrak {A}):=\{\varphi :\mathfrak {A}\vDash \varphi ,\,\,\varphi \text { is a Horn formula}\}$ . The definition of Horn-equivalence has the following form:

We start with a simple observation.

The equivalence of the three statements follows directly from the fact that quasivarieties are defined by quasi-identities. In the case where both algebras are finite, we can modify the second statement to $\mathsf {I}\mathsf {S}\mathsf {P}(\mathfrak {A}) = \mathsf {I}\mathsf {S}\mathsf {P}(\mathfrak {B})$ . We are now ready to prove the key “simplification” lemma. This lemma will be used to show that certain algebras of the third type are Horn-equivalent to a product of two Sugihara chains.

Proof. Without the loss of generality, we assume that $\mathsf {A}\subseteq \mathsf {B}\subseteq \mathsf {C}$ . We will show that $\mathsf {Q}(\mathfrak {A}\times \bot ^n\mathfrak {C}\top ^n)=\mathsf {Q}(\mathfrak {A}\times \bot ^n\mathfrak {B}\times \mathfrak {C}\top ^n)$ .

For the left-to-right inclusion we prove that $\mathfrak {A}\times \bot ^n\mathfrak {C}\top ^n\preceq \mathfrak {A}\times \bot ^n\mathfrak {B}\times \mathfrak {C}\top ^n$ . The embedding e is given simply by

$$ \begin{align*} (a,c) &\mapsto (a,(a,c)) &\text{ for } (a,c)\in\mathsf{A}\times\mathsf{C};\\ (a,x_i) &\mapsto (a,x_i) &\text{ for } (a,x_i)\in\mathsf{A}\times\{\bot_i,\top_i\}, 1\leq i\leq n. \end{align*} $$

Is is easy to see that such a function is indeed an injective homomorphism. Thus $\mathfrak {A}\times \bot ^n\mathfrak {C}\top ^n\in \mathsf {S}(\mathfrak {A}\times \bot ^n\mathfrak {B}\times \mathfrak {C}\top ^n)\subseteq \mathsf {Q}(\mathfrak {A}\times \bot ^n\mathfrak {B}\times \mathfrak {C}\top ^n)$ , which implies $\mathsf {Q}(\mathfrak {A}\times \bot ^n\mathfrak {C}\top ^n)\subseteq \mathsf {Q}(\mathfrak {A}\times \bot ^n\mathfrak {B}\times \mathfrak {C}\top ^n)$ .

For the opposite inclusion, we will show that $\mathfrak {A}\times \bot ^n\mathfrak {B}\times \mathfrak {C}\top ^n\preceq \mathfrak {A}^2\times (\bot ^n\mathfrak {C}\top ^n)^2\cong (\mathfrak {A}\times \bot ^n\mathfrak {C}\top ^n)^2$ . Now the embedding e is defined by

$$ \begin{align*} (a,(b,c)) &\mapsto ((a,a),(b,c)) &\text{ for } (a,b,c)\in\mathsf{A}\times\mathsf{B}\times\mathsf{C};\\ (a,x_i) &\mapsto ((a,a),(x_i,x_i)) &\text{ for } (a,x_i)\in\mathsf{A}\times\{\bot_i,\top_i\}, 1\leq i\leq n. \end{align*} $$

It is obvious that e is injective. To see that e preserves $\neg $ , let $(a,b,c)\in \mathsf {A}\times \mathsf {B}\times \mathsf {C}$ . $e(\neg (a,{\kern1.5pt}(b,c))){\kern1.5pt}={\kern1.5pt}e((\neg a,{\kern1.5pt}(\neg b, \neg c))){\kern1.5pt}={\kern1.5pt}((\neg a,\neg a),{\kern1.5pt} (\neg b,\neg c)){\kern1.5pt}={\kern1.5pt} \neg ((a,a),{\kern1.5pt}(b,c)){\kern1.5pt}=\neg e((a,(b,c)))$ . The case when $(a,x_i)\in \mathsf {A}\times \{\bot _i,\top _i\}$ is equally trivial by the fact that $\neg \bot _i=\top _i$ and $\neg \top _i=\bot _i$ . For binary operations we check only $\to $ since $\lor ,\wedge $ are trivial. Let $(a,b,c), (a',b',c')\in \mathsf {A}\times \mathsf {B}\times \mathsf {C}$ . We will show two cases as examples and leave the rest for the reader since they can be shown in the same manner.

For the first example let: $e((a,(b,c)))\to e((a',(b',c')))=((a,a),(b,c))\to ((a',a'),{\kern1pt}(b',c')){\kern1pt}={\kern1pt}((a\to a',a\to a'),{\kern1pt}(b{\kern1pt}\to{\kern1pt} b',c{\kern1pt}\to{\kern1pt} c')){\kern1pt}={\kern1pt}e((a,(b,c))){\kern1pt}\to{\kern1pt} (a', (b', c')))$ .

To prove one more case: $e((a',\bot _i))\to e((a,(b,c)))=((a',a'),(\bot _i,\bot _i))\to ((a,a),(b,c))=((a'\to a,a'\to a),(\top _i,\top _i))\ =\ e((a'\to a,\top _i))\ =\ e((a',\bot _i)\to (a,(b,c)))$ .

Thus, we have shown $\mathfrak {A}\times \bot ^n\mathfrak {B}\times \mathfrak {C}\top ^n\in \mathsf {S}\mathsf {P}(\mathfrak {A}\times \bot ^n\mathfrak {C}\top ^n)\subseteq \mathsf {Q}(\mathfrak {A}\times \bot ^n\mathfrak {C}\top ^n)$ . This proves the right-to-left inclusion.

As we have proven the equality of the two quasivarieties, the statement from the lemma follows immediately by the Fact 4.3.

As a consequence of the above lemma, we can assert that Theorem 3.6 cannot be strengthened to an equivalence. Given the assumption of Lemma 4.4, we have $\mathfrak {A} \times \bot ^n \mathfrak {B} \times \mathfrak {C} \top ^n \in \mathsf {Q}(\mathfrak {A} \times \bot ^n \mathfrak {C} \top ^n)$ , which implies that $\mathfrak {A} \times \bot ^n \mathfrak {B} \times \mathfrak {C} \top ^n$ is not critical. For a concrete example, consider $\mathfrak {A}_4 \times \bot \mathfrak {A}_6 \times \mathfrak {A}_8 \top $ , which is clearly an algebra of the fourth type. Yet, by Lemma 4.4, we have $\mathfrak {A}_4 \times \bot \mathfrak {A}_6 \times \mathfrak {A}_8 \top \in \mathsf {Q}(\mathfrak {A}_4 \times \mathfrak {A}_{10})$ , and $\mathfrak {A}_4 \times \mathfrak {A}_{10}$ is a proper subalgebra of $\mathfrak {A}_4 \times \bot \mathfrak {A}_6 \times \mathfrak {A}_8 \top $ . This shows that $\mathfrak {A}_4 \times \bot \mathfrak {A}_6 \times \mathfrak {A}_8 \top $ is not critical.

However, the authors seem to acknowledge the potential incompleteness of their description, referring to it as “a certain, satisfactory-for-our-purpose description” [Reference Czelakowski and Dziobiak11, p. 281]. This description remains satisfactory for our purposes as well, as it will suffice to prove the main theorem.

It is known that for every $\mathbf {SC}$ quasivariety $\mathcal {Q}$ , we have $\mathcal {Q} = \mathsf {Q}(\mathbf {F}_{\mathcal {Q}}(\omega ))$ . However, free algebras on finite generators are usually complicated, and $\omega $ -generated ones are even more so. Therefore, we seek an alternative description of $\mathbf {SC}$ Sugihara quasivarieties—specifically, simpler, possibly finite algebras that generate them. By Theorem 3.3, there are countably infinitely many varieties of Sugihara algebras, each generated by a single algebra of the form $\mathfrak {A}_\alpha $ , where $\alpha \leq \omega $ . Thus, we can partition the lattice of subquasivarieties of Sugihara algebras into a countable set of equivalence classes, with each class corresponding to the algebras that generate the same variety. In this framework, $\mathbf {SC}$ quasivarieties are precisely the minimal elements of each equivalence class, represented by $\mathsf {V}(\mathfrak {A}_\alpha )$ for $\alpha \leq \omega $ .

In practice, the case where $\alpha = \omega $ turns out to be special, so we will first address the finite cases, where $n < \omega $ , and then move on to the remaining infinite case.

In order to determine the equivalence class of a given quasivariety, we introduce the notion of the degree of a given critical Sugihara algebra.

Definition 4.6. Let $\mathfrak {A}$ be isomorphic to one of the four types of algebras from Theorem 3.6. We define the degree of $\mathfrak {A}$ , $deg(\mathfrak {A})$ to be:

$$ \begin{align*} deg(\mathfrak{A}) = \begin{cases} n, & \text{if }\mathfrak{A}\cong\mathfrak{A}_n,\\ max(\{2i,k\}), & \text{if }\mathfrak{A}\cong\mathfrak{A}_{2i}\times\mathfrak{A}_k,\\ max(\{2k_n,k\}) + 2(p_n +\cdots +p_1), & \text{if }\mathfrak{A}\cong\mathfrak{B},\\ max(\{2k_n,k\}) + 2(p_n +\cdots +p_1), & \text{if }\mathfrak{A}\cong\mathfrak{A}_{2k_0}\times\mathfrak{B}, \end{cases} \end{align*} $$

where $\mathfrak {B}=\bot ^{p_1}\mathfrak {A}_{2k_1}\times \dots \bot ^{p_n}\mathfrak {A}_{2k_n}\times \mathfrak {A}_k\top ^{p_n}\ldots \top ^{p_1}$ .

Observe that the notion of the degree of a given Sugihara algebra does not coincide with the standard lattice-theoretic notion of height (i.e., the cardinality of the longest chain in a lattice). To illustrate this, consider $\mathfrak {B}=\bot \mathfrak {A}_2\times \mathfrak {A}_3\top $ . By definition 4.6, $deg(\mathfrak {B})=5$ . However, the set $\{\bot ,(-1,0),(0,0),(1,0),(1,1),\top \}\subseteq \mathsf {B}$ forms a six-element subchain of $\mathfrak {B}$ .

By local finiteness of Sugihara algebras, we have the following:

In Lemma 4.7, we addressed all finite cases, i.e., the members of the equivalence classes $[\mathsf {V}(\mathfrak {A}_m)]{/\sim }$ for $m \in \omega $ . We now turn to the infinite case, involving the members of $[\mathfrak {A}_\omega ]{/\sim }$ .

Lemmas 4.7 and 4.8 establish an equivalent condition for membership in a given equivalence class, which ultimately depends on the degree of critical algebras. Based on this result, we will characterize each $\mathbf {SC}$ quasivariety as being generated by a single (in most cases finite) algebra.

Once again, we address the infinite case separately.

Proof. First we prove equality, then we will move on to structural completeness. Inclusion from the left to right is trivial. To see that $\mathsf {Q}(\mathfrak {A}_2\times \mathfrak {A}_{\omega \setminus \{0\}})\subseteq \mathsf {Q}(\{\mathfrak {A}_2\times \mathfrak {A}_{2m}:m\in \omega \})$ let U be a non-principal ultrafilter over $\omega $ . It can be proven that $\mathfrak {A}_2\times \mathfrak {A}_{\omega \setminus \{0\}}$ is embeddable in $(\Pi _{n\in \omega }(\mathfrak {A}_2\times \mathfrak {A}_{2n}))_{/_U}$ . To see that, first define a function f from $\{-1,1\}\times \mathbb {Z}\setminus \{0\}$ into the direct product: $\Pi _{n\in \omega }(\{-1,1\}\times \{-n,\dots ,-1,1,\dots ,n\})$ by

$$ \begin{align*} f((x,n))(m)= \begin{cases} (1,1), &\text{ if }m<n,\\ (x,n), &\text{ otherwise.} \end{cases} \end{align*} $$

Now e given by $e((x,n))=[f((x,n))]_{/_U}$ can be easily seen to be an embedding of $\mathfrak {A}_2\times \mathfrak {A}_{\omega \setminus \{0\}}$ into $\Pi _{n\in \omega }(\mathfrak {A}_2\times \mathfrak {A}_{2n})_{/_U}$ .

Let $\mathsf {Q}(\mathsf {K})\in [\mathsf {V}(\mathfrak {A}_\omega )]_{/\sim }$ , where $\mathsf {K}$ is a class of critical algebras. By Lemma 4.8 for each n there is $\mathfrak {A}\in \mathsf {K}$ such that $deg(\mathfrak {A})\geq n$ . By the reasoning from 4.9 we know that there is $\mathfrak {A}_2\times \mathfrak {A}_m\in \mathsf {Q}(\mathfrak {A})$ , where $n\leq m=deg(\mathfrak {A})$ . Thus $\mathfrak {A}_2\times \mathfrak {A}_m\in \mathsf {Q}(\mathsf {K})$ . Also for any $2j=l\leq m$ , $\mathfrak {A}_2\times \mathfrak {A}_l\preceq \mathfrak {A}_2\times \mathfrak {A}_m$ , which means that $\mathsf {Q}(\{\mathfrak {A}_2\times \mathfrak {A}_{2m}:m\in \omega \})\subseteq \mathsf {Q}(\mathsf {K})$ .

By combining Lemmas 4.9 and 4.10, we can now state that the set of all $\mathbf {SC}$ quasivarieties of Sugihara algebras is $\{\mathsf {Q}(\mathfrak {A}_1), \mathsf {Q}(\mathfrak {A}_2 \times \mathfrak {A}_{\omega \setminus \{0\}})\} \cup \{\mathsf {Q}(\mathfrak {A}_2 \times \mathfrak {A}_n) : 2 \leq n \in \omega \}$ . As will become apparent later, this subposet of the entire lattice of quasivarieties of Sugihara algebras is illustrated in Figure 1. We are now ready to eliminate some of the $\mathbf {SC}$ quasivarieties from Figure 1 as non-hereditary.

Figure 1

Structurally complete quasivarieties of Sugihara algebras.

We have spotted infinitely many quasivarieties which are $\mathbf {nHSC}$ . To prove that the remaining $\mathbf {SC}$ extensions are hereditary, let us define three special rules.

(NP) $$\begin{align} \{p,\neg p\}/q&\end{align}$$

(DS) $$ \begin{align} \{p,\neg p\lor q\}/q \end{align} $$

(R) $$ \begin{align} \{p,q,\neg(p\to q)\}/r \end{align} $$

We have a non-paraconsistency rule $\mathtt {NP}$ , disjunctive syllogism $\mathtt {DS}$ and a special rule $\mathtt {R}$ . Due to algebraizability, these rules are equivalent to respective quasi-identities—here, we chose the ‘logical notation’ over the ‘algebraic’ one for the sake of readability.

Lemma 4.12. It is the case that:

$$ \begin{align*} \mathfrak{A}_2\times\mathfrak{A}_{\omega\setminus\{0\}}&\vDash\mathtt{NP},\mathtt{DS},\mathtt{R}\\ \mathfrak{A}_3&\nvDash\mathtt{NP}\\ \mathfrak{A}_4&\nvDash\mathtt{R}\\ \mathfrak{A}_2\times\mathfrak{A}_3&\nvDash\mathtt{DS} \end{align*} $$

The only remaining step for the main result to follow is to prove the following:

The fundamental Theorem 4.1 follows from Lemmas 4.9–4.11 and 4.13–4.15. Additionally, we have an immediate corollary stating that the quasivarieties from Lemma 4.11 are the only ones that are $\mathbf {nHSC}$ . Formally, from Lemmas 4.9–4.11, and Theorem 4.1, we derive the following corollary:

The content of Theorem 4.1, along with Corollary 4.16, is depicted in Figure 2.

Figure 2

Structurally complete quasivarieties of Sugihara algebras.

5 Passive structural completeness

Finally, let us note that there is a weaker version of the $\mathbf {SC}$ property discussed in the literature. This is the concept of overflow, or passive structural completeness, as formulated in [Reference Wroński30]. A logical system (or a quasivariety) is said to be overflow complete if all of its passive admissible rules are derivable. It has been shown that a quasivariety is overflow complete if and only if all of its non-trivial members satisfy exactly the same positive sentences [Reference Wroński30, Fact  2, p. 68]. Recall that a positive first-order sentence is one constructed using the existential quantifier $\exists $ , the falsum constant $\mathbf {f}$ , and the connectives of conjunction $\&$ and disjunction $\veebar $ . Using the previously proven lemmas, along with some new ones, we can provide a Citkin-style characterization of overflow complete extensions of $\mathbf {RM}$ in terms of two ‘omitting’ algebras.

In order to prove the theorem, we shall first recall the original Wroński characterization of overflow completeness, and then modify it for our purposes. Standardly, by $\mathsf {Th}(\mathsf {K})$ we will understand a first-order theory of a class of structures  $\mathsf {K}$ .

We can slightly weaken the assumption of the above characterization.

In the first formulation, proving overflow completeness comes down to showing positive equivalence of all non-trivial members of a given quasivariety. In the second variant, we can restrict ourselves to the generating class plus the free algebra on infinitely many free generators.

Now we can move on to the proof of the theorem:

On the basis of our main Theorems 4.1 and 5.5, we can characterize those quasivarieties of Sugihara algebras which are overflow complete but not $\mathbf {SC}$ .

6 Active structural completeness

Another variant of $\mathbf {SC}$ is active or almost structural completeness ( $\mathbf {ASC}$ ), a concept recently introduced and algebraically developed in [Reference Dzik and Stronkowski13]. This notion can be seen as complementary to $\mathbf {PSC}$ . A consequence relation is said to be $\mathbf {ASC}$ if its only admissible non-derivable rules are passive.

We will adopt a strategy similar to the one used previously: first, we will modify the original algebraic characterization of $\mathbf {ASC}$ to align with the techniques established in the proof of the main theorem. Then, we will apply these methods to characterize all $\mathbf {ASC}$ subquasivarieties of Sugihara algebras and, subsequently, to isolate those that are $\mathbf {ASC}$ but not $\mathbf {SC}$ .

Among the many equivalent algebraic properties for a quasivariety to be $\mathbf {ASC}$ , we recall the condition most useful for our purposes [Reference Dzik and Stronkowski13, Theorem 3.1, p. 532]:

Once again, we will modify the theorem in a manner similar to the approach taken for $\mathbf {PSC}$ : restricting $\mathfrak {A}$ to be a critical algebra and replacing the free algebra with the product of a two-element Boolean algebra and a characteristic chain. This leads to the following modified version:

We have an immediate corollary: