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Arising in the study of Quantum Logics, PBZ ∗ -lattices are the paraorthomodular Brouwer–Zadeh lattices in which the pairs of elements with their Kleene complements satisfy the Strong De Morgan condition with respect to the Brouwer complement. They form a variety PBZL ∗ which includes that of orthomodular lattices considered with an extended signature (by endowing them with a Brouwer complement coinciding with their Kleene complement), as well as antiortholattices (whose Brouwer complements are trivial). The former turn out to have directly irreducible lattice reducts and, under distributivity, no nontrivial elements with bounded lattice complements, since the elements with bounded lattice complements coincide to the sharp elements in distributive PBZ ∗ -lattices. The variety DIST of distributive PBZ ∗ -lattices has an infinite ascending chain of subvarieties, and the variety generated by orthomodular lattices and antiortholattices has an infinity of pairwise disjoint infinite ascending chains of subvarieties. [ABSTRACT FROM AUTHOR]

We characterize all residuated lattices that have height equal to 3 and show that the variety they generate has continuum-many subvarieties. More generally, we study unilinear residuated lattices: their lattice is a union of disjoint incomparable chains, with bounds added. We we give two general constructions of unilinear residuated lattices, provide an axiomatization and a proof-theoretic calculus for the variety they generate, and prove the finite model property for various subvarieties. [ABSTRACT FROM AUTHOR]

In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák's theorem, in the sense that in the variety ℝ D ℙ ℂ ℍ of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák's formula. As applications of this result together with the above-mentioned Katriňák's theorem, we show that the varieties ℝ D B L ℙ , ℝ D ℙ ℂ ℍ , ℝ ℙ ℂ ℍ d and ℝ D B L ℍ of regular double p-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties ℝ D M ℙ , ℝ D M ℍ , ℝ D M D B L ℍ , ℝ D M D B L ℙ of regular De Morgan p-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double p-algebras, respectively, are also term-equivalent to each other. From these results and recent results of Adams, Sankappanavar and Vaz de Carvalho on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2 ℵ 0 . We then define new logics, ℛ D P C ℋ , ℛ P C ℋ d and ℛ D ℳ ℋ , and show that they are algebraizable with ℝ D ℙ ℂ ℍ , ℝ ℙ ℂ ℍ d and ℝ D M ℍ , respectively, as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality 2 ℵ 0 . [ABSTRACT FROM AUTHOR]

PBZ ∗ –lattices are bounded lattice–ordered structures endowed with two complements, called Kleene and Brouwer; by definition, they are the paraorthomodular Brouwer–Zadeh lattices in which the pairs of elements with their Kleene complements satisfy the Strong De Morgan condition. These algebras arise in the study of Quantum Logics and they form a variety PBZL ∗ which includes orthomodular lattices with an extended signature (with the two complements coinciding), as well as antiortholattices (whose Brouwer complements are trivial). We establish a lattice isomorphism between the lattice of subvarieties of the variety SAOL generated by the antiortholattices with the Strong De Morgan property and the ordinal sum of the three–element chain with the lattice of subvarieties of the variety PKA of pseudo–Kleene algebras, which proves that SAOL and thus PBZL ∗ has uncountably many subvarieties since PKA does and also gives us axiomatizations for all subvarieties of SAOL from those of the subvarieties of PKA ; furthermore, it proves that the variety PKA is generated by the class of the bounded involution lattice reducts of the members of SAOL and thus of those of any subvariety of PBZL ∗ that includes SAOL , hence neither of these classes is a variety. We also obtain an infinity of pairwise disjoint infinite ascending chains of subvarieties of SAOL . [ABSTRACT FROM AUTHOR]

A monoid is pseudo-complex if the semigroup variety it generates has uncountably many subvarieties, while the monoid variety it generates has only finitely many subvarieties. The smallest pseudo-complex monoid currently known is of order seven. The present article exhibits a pseudo-complex monoid of order six and shows that every smaller monoid is not pseudo-complex. Consequently, minimal pseudo-complex monoids are of order six. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

A generalized hypersubstitution of type τ = (n) is a function which takes the n-ary operation symbol f to the term of the same type σ(f ) which does not necessarily preserve the arity. Let HypG(n) be the set of all these generalized hypersubstitutions of type (n). The set HypG(n) with a binary operation and the identity generalized hypersubstitution forms a monoid. The objective of this paper is to study Green’s relations on the set of all regular elements of HypG(n).

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. [ABSTRACT FROM AUTHOR]

We introduce the notion of clone algebra (CA ), intended to found a one-sorted, purely algebraic theory of clones. CA s are defined by identities and thus form a variety in the sense of universal algebra. The most natural CA s, the ones the axioms are intended to characterise, are algebras of functions, called functional clone algebras (FCA ). The universe of a FCA , called ω -clone, is a set of infinitary operations on a given set, containing the projections and closed under finitary compositions. The main result of this paper is the general representation theorem, where it is shown that every CA is isomorphic to a FCA and that the variety CA is generated by the class of finite-dimensional CA s. This implies that every ω -clone is algebraically generated by a suitable family of clones by using direct products, subalgebras and homomorphic images. We conclude the paper with two applications. In the first one, we use clone algebras to give an answer to a classical question about the lattices of equational theories. The second application is to the study of the category of all varieties of algebras. [ABSTRACT FROM AUTHOR]

In this paper we study the variety of order Hilbert algebras, which is the equivalent algebraic semantics of the order implicational calculus of Bull (J Symb Logic, 29:33–34, 1964). [ABSTRACT FROM AUTHOR]

The set of all cancellable elements of the lattice of semigroup varieties has recently been shown to be countably infinite. But the description of all cancellable elements of the lattice MON of monoid varieties remains unknown. This problem is addressed in the present article. The first example of a monoid variety with modular but non-distributive subvariety lattice is first exhibited. Then a necessary condition of the modularity of an element in MON is established. These results play a crucial role in the complete description of all cancellable elements of the lattice MON . It turns out that there are precisely five such elements. [ABSTRACT FROM AUTHOR]

Let τn be a type of algebras in which all operation symbols have arity n, for a fixed n ≥ 1. For 0 < r ≤ n, this paper introduces a special kind of n-ary terms of type τn called K*(n, r)-full terms. The set of all K*(n, r)-full terms of type τn is closed under the superposition operation Sn; hence forms a clone denoted by cloneK*(n,r)(τn). We prove that cloneK* (n,r)(τn) is a Menger algebra of rank n. We study K*(n, r)-full hypersubstitutions and the related K*(n, r)-full closed identities and K*(n, r)-full closed varieties. A connection between identities in cloneK* (n,r)(τn) and K* (n, r)-full closed identities is established. The results obtained generalize the results of Denecke and Jampachon [K. Denecke and P. Jampachon, Clones of full terms, Algebra and Discrete Math. 4 (2004) 1–11].

A variety of semirings is said to be solid if each of its identities is satisfied as hyperidentity. There are precisely four solid varieties of semirings. Each of them contains every derived algebra, where the both fundamental operations are replaced by arbitrary binary term operations. If a variety contains all linear derived algebras, where the fundamental operations are replaced by term operations induced by linear terms, it is called linear-solid. We prove that a variety of semirings is solid if and only if it is linear-solid.

Let τn be a type of algebras with the n-ary operation symbols, for a fixed integer n ≥ 1. In this article, we introduce terms of type τn called order-decreasing full terms. We prove that the set of all order-decreasing full terms of type τn is closed under the superposition operation; and hence it forms an algebra denoted by M A O D n (τ n). Moreover, we prove that M A O D n (τ n) is a Menger algebra of rank n. Finally, we introduce and study order-decreasing full hypersubstitutions and the related order-decreasing full closed identities and order-decreasing full closed varieties. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the lattice of subvarieties of the ai-semiring variety defined by the additional identities

Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras and show that it generalizes Comer's double coset construction for relation algebras. Also, we explore how the double-division conucleus construction interacts with other class operators and in particular with variety generation. We focus on the fact that it preserves a special discriminator term, thus yielding interesting discriminator varieties of GBI-algebras, including RWkRA. To illustrate the generality of the variety of weakening relation algebras, we prove that all distributive lattice-ordered pregroups and hence all lattice-ordered groups embed, as residuated lattices, into representable weakening relation algebras on chains. Moreover, every representable weakening relation algebra is embedded in the algebra of all residuated maps on a doubly-algebraic distributive lattice. We give a number of other instructive examples that show how the double-division conucleus illuminates the structure of distributive involutive residuated lattices and GBI-algebras. [ABSTRACT FROM AUTHOR]

It is well known that the subvariety lattice of the variety of relation algebras has exactly three atoms. The (join-irreducible) covers of two of these atoms are known, but a complete classification of the (join-irreducible) covers of the remaining atom has not yet been found. These statements are also true of a related subvariety lattice, namely the subvariety lattice of the variety of semiassociative relation algebras. The present article shows that this atom has continuum many covers in this subvariety lattice (and in some related subvariety lattices) using a previously established term equivalence between a variety of tense algebras and a variety of semiassociative r -algebras. [ABSTRACT FROM AUTHOR]

Let A and B be idempotent varieties and suppose that the variety A ∨ B is congruence permutable. Then the Maltsev product A ∘ B is also congruence permutable. [ABSTRACT FROM AUTHOR]

Extending pioneering work by Weinberg, Conrad, McCleary, and others, we provide a systematic way of relating spaces of right orders on a partially ordered group, on the one hand, and spectral spaces of free lattice-ordered groups, on the other. The aim of the theory is to pave the way for further fruitful interactions between the study of right orders on groups and that of lattice-groups. Special attention is paid to the important case of orders on groups. [ABSTRACT FROM AUTHOR]

In this paper we consider mappings σ which map the binary operation symbol f to the term σ (f) which do not necessarily preserve the arity. These mapping are called generalized hypersubstitutions of type τ = (2) and we denote the set of all these generalized hypersubstitutions of type τ = (2) by HypG (2). The set HypG(2) together with a binary operation defined on this set and the identity generalized hypersubstitution which maps f to the term f(x1, x2) forms a monoid. In this paper, we determine all maximal completely regular submonoids of this monoid.

The lattice of all regular-solid varieties of semirings splits in two complete sublattices: the sublattice of all idempotent regular-solid varieties of semirings and the sublattice of all normal regular-solid varieties of semirings. In this paper, we discuss the idempotent part.

A rectangular band is defined as a direct product of a left zero semigroup and a right zero semigroup, and a rectangular group is defined as a direct product of a group and a rectangular band. In this paper, we give some equivalent conditions for Cayley graphs of a rectangular group to be isomorphic to each other. [ABSTRACT FROM AUTHOR]

We introduce a systematic method for densification, i.e., embedding a given chain into a dense one preserving certain identities, in the framework of FL algebras (pointed residuated lattices). Our method, based on residuated frames, offers a uniform proof for many of the known densification and standard completeness results in the literature. We propose a syntactic criterion for densification, called semianchoredness. We then prove that the semilinear varieties of integral FL algebras defined by semi-anchored equations admit densification, so that the corresponding fuzzy logics are standard complete. Our method also applies to (possibly non-integral) commutative FL chains. We prove that the semilinear varieties of commutative FL algebras defined by knotted axioms $${x^{m} \leq x^{n}}$$ (with $${m, n > 1}$$ ) admit densification. This provides a purely algebraic proof to the standard completeness of uninorm logic as well as its extensions by knotted axioms. [ABSTRACT FROM AUTHOR]

In this article, we study in detail the regular elements of a bounded, commutative and integral residuated lattice. We introduce the notion of a regular variety and explore its relationship with the Kolmogorov negative translation. In addition, we investigate the corresponding notions in the axiomatic extensions of the Full Lambek Calculus with exchange and weakening. [ABSTRACT FROM AUTHOR]

We prove that the following three conditions on a modular ortholattice L with respect to a given variety of modular ortholattices, $${\mathcal{V}}$$ , are equivalent: L is in the variety of modular ortholattices generated by the finite-dimensional members of $${\mathcal{V}}$$ ; L can be embedded in an atomisticmember of $${\mathcal{V}}$$ ; L has an orthogeometric representation in an anisotropic orthogeometry $${{(Q,\bot), \,\,{\rm where}\,\, [0, u] \in \mathcal{V} \,\,{\rm for\,\, all}\,\, u \in L_{\rm fin}(Q)}}$$ . [ABSTRACT FROM AUTHOR]

This paper is devoted to the semilattice ordered $\mathcal{V}$-algebras of the form ( A, Ω, + ), where + is a join-semilattice operation and ( A, Ω) is an algebra from some given variety $\mathcal{V}$. We characterize the free semilattice ordered algebras using the concept of extended power algebras. Next we apply the result to describe the lattice of subvarieties of the variety of semilattice ordered $\mathcal{V}$-algebras in relation to the lattice of subvarieties of the variety $\mathcal{V}$. [ABSTRACT FROM AUTHOR]

Within the lattice of varieties of pseudo MV-algebras, the variety $${\mathcal{B}}$$ of Boolean algebras is the least nontrivial variety. Komori identified all varieties of (commutative) MV-algebras that cover $${\mathcal{B}}$$ . The authors previously identified all solvable varieties of pseudo MV-algebras that cover $${\mathcal{B}}$$ . We will show the existence of continuum many nonsolvable varieties of pseudo MV-algebras that cover $${\mathcal{B}}$$ , show that periodically primitive u ℓ-groups cannot generate Boolean covers, and show that all noncommutative varieties that are Boolean covers must be Top Boolean. [ABSTRACT FROM AUTHOR]

Holland et al. (Algebra Univers 67:1-18, ) considered varieties ${\mathcal E}_n$ of lattice-ordered groups defined by partial metrics, and showed for all n that ${\mathcal E}_n$ is contained within the variety ${\mathcal L}_n$ defined by x y = y x. They also showed that if n were prime, then ${\mathcal E}_n = {\mathcal L}_n$. Letting ${\mathcal A}^2$ denote the metabelian variety (defined at the beginning of Section 2), this article continues their work, showing that for all n, ${\mathcal L}_n \cap {\mathcal A}^2 \subseteq {\mathcal E}_n$ while showing that if n is not prime, ${\mathcal L}_n \not\subseteq {\mathcal E}_n$. [ABSTRACT FROM AUTHOR]

In this paper we investigate the connections between quantifier elimination, decidability and Uniform Craig Interpolation in Δ-core fuzzy logics added with propositional quantifiers. As a consequence, we are able to prove that several propositional fuzzy logics have a conservative extension which is a Δ-core fuzzy logic and has Uniform Craig Interpolation. [ABSTRACT FROM AUTHOR]

We show that a large number of equations are preserved by Dedekind-MacNeille completions when applied to subdirectly irreducible FL-algebras/residuated lattices. These equations are identified in a systematic way, based on proof-theoretic ideas and techniques in substructural logics. It follows that many varieties of Heyting algebras and FL-algebras admit completions. [ABSTRACT FROM AUTHOR]

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [] and [] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH. We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH. Finally, we apply these results to give bases for 'small' subvarieties of BDQDSH, DQSSH, and DblSH. [ABSTRACT FROM AUTHOR]

The varieties of solvable lattice-ordered groups covering the abelian variety were shown independently by Gurchenkov, Reilly, and Darnel to be the Scrimger varieties of ℓ-groups and the three Medvedev representable covers. In this article, the authors give a parallel characterization of varieties of solvable unital ℓ-groups which cover the minimal nontrivial variety of boolean unital ℓ-groups. [ABSTRACT FROM AUTHOR]

We prove that every subvariety of BL-algebras which is not finitely generated is not σ-canonical. We also prove π-canonicity for an infinite family of subvarieties of BL-algebras that are not finitely generated. To do so we study the behavior of canonical extensions of ordered sums of posets. [ABSTRACT FROM AUTHOR]

An exclusion variety is a variety that is maximal with respect to not containing some semigroup. It is shown that if the bases of all exclusion varieties for some periodic semigroup S are known, then the bases of exclusion varieties for the monoid S1 can be computed. [ABSTRACT FROM AUTHOR]

Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to M-hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is an M-hyperidentity for a subset M of the set of all hypersubstitutions, the variety is called M-solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of M-solid varieties. In this paper, we study the order of each hypersubstitution of type (2, 2), i.e., the order of the cyclic subsemigroup generated by that hypersubstitution of the monoid of all hypersubstitutions of type (2, 2). The main result is that the order is 1, 2, 3, 4 or infinite. [ABSTRACT FROM AUTHOR]