We obtain new characterizations of the Hobby–McKenzie varieties via compatible reflexive ternary structures. Based on our findings, we prove that in the lattice of interpretability types of varieties, the filter of the interpretability types of Hobby–McKenzie varieties is prime.

We study the cardinality of classes of equational theories (varieties) and logics by applying descriptive set theory. We affirmatively solve open problems raised by Jackson and Lee (2018, Transactions of the American Mathematical Society, 370, 4785–4812) regarding the cardinalities of subvariety lattices, and by Bezhanishvili et al. (2025, Journal of Mathematical Logic, in press) regarding the degrees of the finite model property (fmp). By coding equations and formulas by natural numbers, and theories and logics by real numbers, we examine their position in the Borel hierarchy. We prove that every interval of equational theories in a countable language corresponds to a $\boldsymbol {\Pi }^0_1$ set, and every fmp span of a normal modal logic to a $\boldsymbol {\Pi }^0_2$ set. It follows that they have cardinality either $\leq \aleph _0$ or $2^{\aleph _0}$, provably in ZFC. In the same manner, we observe that the set of pretabular extensions of a tense logic is a $\boldsymbol {\Pi }^0_2$ set, so its cardinality is either $\leq \aleph _0$ or $2^{\aleph _0}$. We also point out a negative solution to another open problem raised by Jackson and Lee, op. cit., regarding the existence of independent systems, which relies on Ježek et al. (1990, Bulletin of the Australian Mathematical Society, 42, 57–70).

A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of J-trivial monoids is Cross if and only if it excludes 14 specific almost Cross varieties. Consequently, these 14 varieties exhaust all almost Cross varieties of J-trivial monoids.

In this paper, we describe étale Boolean right restriction monoids in terms of Boolean inverse monoids. We show that the Thompson groups arise naturally in this context.

It was proved by Maksimova in 1977 that exactly eight varieties of Heyting algebras have the amalgamation property, and hence exactly eight axiomatic extensions of intuitionistic propositional logic have the deductive interpolation property. The prevalence of these properties for substructural logics and varieties of pointed residuated lattices (their algebraic semantics) is far less well understood. Taking as our starting point a formulation of intuitionistic propositional logic as the full Lambek calculus with exchange, weakening, and contraction, we investigate the role of the exchange rule—algebraically, the commutativity law—in determining the scope of these properties. First, we show that there are continuum-many varieties of idempotent semilinear residuated lattices that have the amalgamation property and contain non-commutative members, and hence continuum-many axiomatic extensions of the corresponding logic that have the deductive interpolation property in which exchange is not derivable. We then show that, in contrast, exactly 60 varieties of commutative idempotent semilinear residuated lattices have the amalgamation property, and hence exactly 60 axiomatic extensions of the corresponding logic with exchange have the deductive interpolation property. From this latter result, it follows also that there are exactly 60 varieties of commutative idempotent semilinear residuated lattices whose first-order theories have a model completion.

Gödel algebras are the Heyting algebras satisfying the axiom $(x \to y) \vee (y \to x)=1$. We utilize Priestley and Esakia dualities to dually describe free Gödel algebras and coproducts of Gödel algebras. In particular, we realize the Esakia space dual to a Gödel algebra free over a distributive lattice as the, suitably topologized and ordered, collection of all nonempty closed chains of the Priestley dual of the lattice. This provides a tangible dual description of free Gödel algebras without any restriction on the number of free generators, which generalizes known results for the finitely generated case. A similar approach allows us to characterize the Esakia spaces dual to coproducts of arbitrary families of Gödel algebras. We also establish analogous dual descriptions of free algebras and coproducts in every variety of Gödel algebras. As consequences of these results, we obtain a formula to compute the depth of coproducts of Gödel algebras and show that all free Gödel algebras are bi-Heyting algebras.

An element x of a lattice L is modular if L has no five-element sublattice isomorphic to the pentagon in which x would correspond to the lonely midpoint. We classify all modular elements of the lattice of all monoid varieties.

This article is the second in a series investigating cartesian closed varieties. In first of these, we showed that every non-degenerate finitary cartesian variety is a variety of sets equipped with an action by a Boolean algebra B and a monoid M which interact to form what we call a matched pair ${\left[\smash{{B} \mathbin{\mid}{M} }\right]}$. In this article, we show that such pairs ${\left[\smash{{B} \mathbin{\mid}{M} }\right]}$ are equivalent to Boolean restriction monoids and also to ample source-étale topological categories; these are generalizations of the Boolean inverse monoids and ample étale topological groupoids used to encode self-similar structures such as Cuntz and Cuntz–Krieger $C^\ast$-algebras, Leavitt path algebras, and the $C^\ast$-algebras associated with self-similar group actions. We explain and illustrate these links and begin the programme of understanding how topological and algebraic properties of such groupoids can be understood from the logical perspective of the associated varieties.

We examine the consequences of having a total division operation $\frac {x}{y}$ on commutative rings. We consider two forms of binary division, one derived from a unary inverse, the other defined directly as a general operation; each are made total by setting $1/0$ equal to an error value $\bot $, which is added to the ring. Such totalised divisions we call common divisions. In a field the two forms are equivalent and we have a finite equational axiomatisation E that is complete for the equational theory of fields equipped with common division, which are called common meadows. These equational axioms E turn out to be true of commutative rings with common division but only when defined via inverses. We explore these axioms E and their role in seeking a completeness theorem for the conditional equational theory of common meadows. We prove they are complete for the conditional equational theory of commutative rings with inverse based common division. By adding a new proof rule, we can prove a completeness theorem for the conditional equational theory of common meadows. Although, the equational axioms E fail with common division defined directly, we observe that the direct division does satisfy the equations in E under a new congruence for partial terms called eager equality.

For any $n<\omega $ we construct an infinite $(n+1)$-generated Heyting algebra whose n-generated subalgebras are of cardinality $\leq m_n$ for some positive integer $m_n$. From this we conclude that for every $n<\omega $ there exists a variety of Heyting algebras which contains an infinite $(n+1)$-generated algebra, but which contains only finite n-generated algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [4].

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. We show that the 6-element Brandt monoid generates a finitely universal variety of monoids and, by the previous results, it is the smallest generator for a monoid variety with this property. It is also deduced that the join of two Cross varieties of monoids can be finitely universal. In particular, we exhibit a finitely universal variety of monoids with uncountably many subvarieties which is the join of two Cross varieties of monoids whose lattices of subvarieties are the 6-element and the 7-element chains, respectively.

For every group G, the set $\mathcal {P}(G)$ of its subsets forms a semiring under set-theoretical union $\cup $ and element-wise multiplication $\cdot $, and forms an involution semigroup under $\cdot $ and element-wise inversion ${}^{-1}$. We show that if the group G is finite, non-Dedekind, and solvable, neither the semiring $(\mathcal {P}(G),\cup ,\cdot )$ nor the involution semigroup $(\mathcal {P}(G),\cdot ,{}^{-1})$ admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.

This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$-varieties. We prove that the lattice of $\mathtt {DNA}$-logics is dually isomorphic to the lattice of $\mathtt {DNA}$-varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite $\mathtt {DNA}$-varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$-formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1

We show that some results of L. Makar-Limanov, P. Malcolmson and Z. Reichstein on the existence of free-associative algebras are valid in the more general context of varieties of algebras.

A logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence relative to some class of algebras. We characterize logics admitting an equational completeness theorem that are either locally tabular or have some tautology. In particular, it is shown that a protoalgebraic logic admits an equational completeness theorem precisely when it has two distinct logically equivalent formulas. While the problem of determining whether a logic admits an equational completeness theorem is shown to be decidable both for logics presented by a finite set of finite matrices and for locally tabular logics presented by a finite Hilbert calculus, it becomes undecidable for arbitrary logics presented by finite Hilbert calculi.

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.

All known structural extensions of the substructural logic $\textbf{FL}_{\textbf{e}}$, the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee , \cdot , 1\}$-equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.

A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ of all logics ordered under interpretability is studied. It is shown that in $\mathsf {Log}$ infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between $\mathsf {Log}$ and the lattice of interpretability types of varieties are investigated.

Meet semidistributive varieties are in a sense the last of the most important classes in universal algebra for which it is unknown whether it can be characterized by a strong Maltsev condition. We present a new, relatively simple Maltsev condition characterizing the meet-semidistributive varieties, and provide a candidate for a strong Maltsev condition.

For every n, we evaluate the smallest k such that the congruence inclusion $\alpha (\beta \circ _n \gamma ) \subseteq \alpha \beta \circ _{k} \alpha \gamma $ holds in a variety of reducts of lattices introduced by K. Baker. We also study varieties with a near-unanimity term and discuss identities dealing with reflexive and admissible relations.