Discriminator Varieties Research Articles

Factor principal congruences and Boolean products in filtral varieties

Motivated by Haviar and Ploščica’s 2021 characterisation of Boolean products of simple De Morgan algebras, we investigate Boolean products of simple algebras in filtral varieties. We provide two main theorems. The first yields Werner’s Boolean-product representation of algebras in a discriminator variety as an immediate application. The second, which applies to algebras in which the top congruence is compact, yields a generalisation of the Haviar–Ploščica result to semisimple varieties of Ockham algebras. The property of having factor principal congruences is fundamental to both theorems. While major parts of our general theorems can be derived from results in the literature, we offer new, self-contained and essentially elementary proofs.

A Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions

The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.

Every minimal dual discriminator variety is minimal as a quasivariety

Let $$(\dagger )$$ denote the following property of a variety $$\mathcal V$$ : Every subquasivariety of $$\mathcal V$$ is a variety. In this paper, we prove that every idempotent dual discriminator variety has property $$(\dagger )$$ . Property $$(\dagger )$$ need not hold for nonidempotent dual discriminator varieties, but $$(\dagger )$$ does hold for minimal nonidempotent dual discriminator varieties. Combining the results for the idempotent and nonidempotent cases, we obtain that every minimal dual discriminator variety is minimal as a quasivariety

The structure of generalized BI-algebras and weakening relation algebras

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.

PBZ*-Lattices: Structure Theory and Subvarieties

We investigate the structure theory of the variety of PBZ*-lattices and some of its proper subvarieties. These lattices with additional structure originate in the foundations of quantum mechanics and can be viewed as a common generalisation of orthomodular lattices and Kleene algebras expanded by an extra unary operation. We lay down the basics of the theories of ideals and of central elements in PBZ*-lattices, we prove some structure theorems, and we explore some connections with the theories of subtractive and binary discriminator varieties.

Dual binary discriminator varieties

Left normal bands, strongly distributive skew lattices, implicative BCS-algebras, skew Boolean algebras, skew Boolean intersection algebras, and certain other non-commutative structures occur naturally as term reducts in the study of ternary discriminator algebras and the varieties that they generate, giving rise thereby to various classes of pointed discriminator varieties that generalise the class of pointed ternary discriminator varieties. For each such class of varieties there is a corresponding pointed discriminator function that generalises the ternary discriminator. In this paper some of the classes of pointed discriminator varieties that are contained in the class of dual binary discriminator varieties are characterised. A key unifying property is that the principal ideals of an algebra in a dual binary discriminator variety are entirely determined by the dual binary discriminator term for that variety.

Monadic Boolean algebras with an automorphism and their relation to $${\mathbf{Df}}_{\mathbf{2}}$$-algebras

In this work, we initiate an investigation of the class $${\mathcal {B}}_{T_km}$$ of monadic Boolean algebras endowed with a monadic automorphism of period k. These algebras constitute a generalization of monadic symmetric Boolean algebras. We determine the congruences on these algebras and we characterize the subdirectly irreducible algebras. This last result allows us to prove that $${\mathcal {B}}_{T_km}$$ is a discriminator variety and as a consequence, the principal congruences are characterized. Finally, we explore, in the finite case, the relationship between this class and the class $${\mathbf{Df}}_{\mathbf{2}}$$ of diagonal-free two-dimensional cylindric algebras.

Varieties generated by completions

We prove that persistently finite algebras are not created by completions of algebras, in any ordered discriminator variety. A persistently finite algebra is one without infinite simple extensions. We prove that finite measurable relation algebras are all persistently finite. An application of these theorems is that the variety generated by the completions of representable relation algebras does not contain all relation algebras. This answers Problem 1.1(1) from a paper by R. Maddux in the negative. At the same time, we confirm the suggestion in that paper that the finite maximal relation algebras constructed in a joint paper by M. Frias and R. Maddux are not in the variety generated by the completions of representable relation algebras. We prove that there are continuum many varieties between the variety generated by the completions of representable relation algebras and the variety of relation algebras.

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS

AbstractThe variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto which finitely subdirectly irreducible De Morgan monoids may be mapped by noninjective homomorphisms. The homomorphic preimages of C4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety (C4) within U are revealed here. There are just ten of them (all finitely generated). In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety—in fact, every finite subdirectly irreducible algebra is projective. Beyond U, all covers of (C4) [or of (D4)] within DMM are discriminator varieties. Of these, we identify infinitely many that are finitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.

Projective algebras and primitive subquasivarieties in varieties with factor congruences

We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra embedded in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regular involution, Heyting algebras with a dual pseudocomplement, and double-Heyting algebras.

Restricted Priestley Dualities and Discriminator Varieties

Anyone who has ever worked with a variety $$\varvec{\mathscr {A}}$$ of algebras with a reduct in the variety of bounded distributive lattices will know a restricted Priestley duality when they meet one—but until now there has been no abstract definition. Here we provide one. After deriving some basic properties of a restricted Priestley dual category $$\varvec{\mathscr {X}}$$ of such a variety, we give a characterisation, in terms of $$\varvec{\mathscr {X}}$$ , of finitely generated discriminator subvarieties of $$\varvec{\mathscr {A}}$$ . As an application of our characterisation, we give a new proof of Sankappanavar’s characterisation of finitely generated discriminator varieties of distributive double p-algebras. A substantial portion of the paper is devoted to the application of our results to Cornish algebras. A Cornish algebra is a bounded distributive lattice equipped with a family of unary operations each of which is either an endomorphism or a dual endomorphism of the bounded lattice. They are a natural generalisation of Ockham algebras, which have been extensively studied. We give an external necessary-and-sufficient condition and an easily applied, completely internal, sufficient condition for a finite set of finite Cornish algebras to share a common ternary discriminator term and so generate a discriminator variety. Our results give a characterisation of discriminator varieties of Ockham algebras as a special case, thereby yielding Davey, Nguyen and Pitkethly’s characterisation of quasi-primal Ockham algebras.

Expansions of Dually Pseudocomplemented Heyting Algebras

We investigate expansions of Heyting algebras (EHAs) in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many (dual) normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators (BAOs). We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally definable principal congruences using a single equation. Moreover, in the presence of a dual pseudocomplement operation, discriminator varieties are characterised by a pair of equations. We also prove that a variety of dually pseudocomplemented EHAs with a normal filter term is semisimple if and only if it is a discriminator variety. This generalises two known results, one by Kowalski and Kracht for finite-type varieties of BAOs, and the other by the present author for dually pseudocomplemented Heyting algebras without additional operations.

Discriminator varieties of double-Heyting algebras

Discriminator varieties of double-Heyting algebras

Factor varieties

The universal algebraic literature is rife with generalisations of discriminator varieties, whereby several investigators have tried to preserve in more general settings as much as possible of their structure theory. Here, we modify the definition of discriminator algebra by having the switching function project onto its third coordinate in case the ordered pair of its first two coordinates belongs to a designated relation (not necessarily the diagonal relation). We call these algebras factor algebras and the varieties they generate factor varieties. Among other things, we provide an equational description of these varieties and match equational conditions involving the factor term with properties of the associated factor relation. Factor varieties include, apart from discriminator varieties, several varieties of algebras from quantum and fuzzy logics.

The connection of skew Boolean algebras and discriminator varieties to Church algebras

We establish a connection between skew Boolean algebras and Church algebras. We prove that the set of all semicentral elements in a right Church algebra forms a right-handed skew Boolean algebra for the properly defined operations. The main result of this paper states that the variety of all semicentral right Church algebras of type \({\tau}\) is term equivalent to the variety of right-handed skew Boolean algebras with additional regular operations. As a corollary to this result, we show that a pointed variety is a discriminator variety if and only if it is a 0-regular variety of right-handed skew Boolean algebras.

On structural completeness versus almost structural completeness problem: A discriminator varieties case study

Fil: Campercholi, Miguel Alejandro Carlos. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Discriminator logics (Research announcement)

A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work.

Quasi-discriminator varieties

We generalize the notion of discriminator variety in such a way as to capture several varieties of algebras arising mainly from fuzzy logic. After investigating the extent to which this more general concept retains the basic properties of discriminator varieties, we give both an equational and a purely algebraic characterization of quasi-discriminator varieties. Finally, we completely describe the lattice of subvarieties of the pure pointed quasi-discriminator variety, providing an explicit equational base for each of its members.

Lattice-theoretic properties of algebras of logic

In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semi-linear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a,1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.

Boolean-like algebras

Using Vaggione’s concept of central element in a double-pointed algebra, we introduce the notion of Boolean-like variety as a generalisation of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean-like varieties, which still retain many of the pleasing properties of Boolean algebras. We prove that a double-pointed variety is discriminator if and only if it is semi-Boolean-like, idempotent, and 0-regular. This theorem yields a new Maltsev-style characterisation of double-pointed discriminator varieties.