De Morgan Algebras Research Articles

Adding an implication to logics of perfect paradefinite algebras

Abstract Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion () order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix. We studied these logics extensively in Gomes et al. ((2022). Electronic Proceedings in Theoretical Computer Science357 56–76.) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and order-preserving and $\top$ -assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil’s “symmetric modal logic.” For the order-preserving Set-Set logic, in particular, we obtain a Set-Set axiomatization that is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.

Antitone involutions on tensor products of complete lattices and complete distributivity

The purpose of this paper is to study antitone involutions on tensor products of complete lattices. A lattice with antitone involution is called an involution lattice. We show that if M is a completely distributive involution lattice, then for each complete involution lattice L there exists a unique antitone involution on the tensor product M⊗L such that the natural embeddings of M and L into M⊗L are involution-preserving. This is best possible, since the described property characterizes complete distributivity in the class of complete involution lattices. When M and L are completely distributive involution lattices, M⊗L with the aforementioned antitone involution is the codomain of a universal bimorphism in the sense of the category of all completely distributive de Morgan algebras and their join- and involution-preserving maps. The case that M and L are orthocomplemented is explored too.

New Type of Extended Soft Set Operation: Complementary Extended Union Operation

Soft set theory was proposed by Molodtsov in 1999 to model some problems involving uncertainty. It has a wide range of theoretical and practical applications. Soft set operations constitute the basic building blocks of soft set theory. Many kinds of soft set operations have been described and applied in various ways since the inception of the theory. In this paper, to contribute to the theory, a new soft set operation, called complementary extended union operation, is defined, its properties are discussed in detail to obtain the relationship of each operation with other soft set operations, and the distributions of these operations over other soft set operations are examined. We obtain that the complementary extended union operation along with other certain types of soft set operations construct some well-known algebraic structure such as Boolean Algebra, De Morgan Algebra, semiring, and hemiring in the set of soft sets with a fixed parameter set. Since Boolean Algebra is fundamental in digital logic design, computer science, information retrieval, set theory and probability; De Morgan Algebra in logic and set theory, computer science, artificial intelligence, circuit design; semirings in theoretical computer science, optimization problems, economics, cryptography and coding theory, and hemirings in combinatorics, mathematical economics, theoretical computer science, these algebraic structures provide essential tools for various applications, facilitating the analysis, design, and optimization of systems across many disciplines, and thus this study is expected to contribute to decision-making methods and cryptography based on soft sets.

Decidability of topological quasi-Boolean algebras

A sequent calculus S for the variety tqBa of all topological quasi-Boolean algebras is established. Using a construction of syntactic finite algebraic model, the finite model property of S is shown, and thus the decidability of S is obtained. We also introduce two non-distributive variants of topological quasi-Boolean algebras. For the variety TDM 5 of all topological De Morgan lattices with the axiom 5, we establish a sequent calculus S 5 and prove that the cut elimination holds for it. Consequently the decidability of S 5 is established. Furthermore, this proof-theoretic method is applied to some subvarieties of TDM 5 .

A topological duality for tense modal pseudocomplemented De Morgan algebras

Abstract In this paper, we define and study the variety of tense modal pseudocomplemented De Morgan algebras. This variety is a proper subvariety of the variety of tense tetravalent modal algebras. A tense modal pseudocomplemented De Morgan algebra is a modal pseudocomplemented De Morgan algebra endowed with two tense operators G and H satisfying additional conditions. Also, the variety of tense modal pseudocomplemented De Morgan algebras is intimately connected with some well-known varieties of De Morgan algebras with tense operators.

A study of rough inclusion on algebras with quasi-Boolean base

Pre-rough algebra has emerged from the rough set theory, which is a quasi-Boolean algebra with a few additional axioms. Four types of abstract rough inclusion are defined in pre-rough algebra which give rise to four different implication operators within this algebra. Properties of these four implications are studied from the angle of residuation. Logics of pre-rough algebra with respect to different implications are presented.

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.

Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications

ABSTRACT 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 \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] 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 \[\mathbb{R}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{P}\] , ℝ D ℙ ℂ ℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] , ℝ ℙ ℂ ℍ d \[\mathbb{R}\mathbb{P}\mathbb{C}{{\mathbb{H}}^{d}}\] and ℝ D B L ℍ \[\mathbb{R}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{H}\] 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 \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{P}\] , ℝ D M ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{H}\] , ℝ D M D B L ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{H}\] , ℝ D M D B L ℙ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{P}\] 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 \[{{2}^{{{\aleph }_{0}}}}\] . We then define new logics, ℛ D P C ℋ \[\mathcal{R}\mathcal{D}\mathcal{P}\mathcal{C}\mathcal{H}\] , ℛ P C ℋ d \[\mathcal{R}\mathcal{P}\mathcal{C}{{\mathcal{H}}^{d}}\] and ℛ D ℳ ℋ \[\mathcal{R}\mathcal{D}\mathcal{M}\mathcal{H}\] , and show that they are algebraizable with ℝ D ℙ ℂ ℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] , ℝ ℙ ℂ ℍ d \[\mathbb{R}\mathbb{P}\mathbb{C}{{\mathbb{H}}^{d}}\] and ℝ D M ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{H}\] , 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 \[{{2}^{{{\aleph }_{0}}}}\] .

Quantale-valued convex structures as lax algebras

Based on a unital and commutative quantale (Q,⁎), a Q-valued lax extension of the nonempty finite powerset monad and a Q-valued finitary closure space (also called algebraic Q-valued closure space) are introduced. It is proved that the category of (Pf,Q)-categories with respect to the Q-valued lax extension of the nonempty finite powerset monad Pf is isomorphic to that of Q-valued finitary closure spaces. Considering the Q-valued finitary closure spaces as linkages, it is shown that balanced Q-convex structures can be treated as (Pf,Q)-categories when Q is required to be a frame and Q-fuzzifying convex structures can be treated as (Pf,Q)-categories when Q is required to be a completely distributive De Morgan algebra.

Fuzzy Propositional Configuration Logics

We introduce and investigate a weighted propositional configuration logic over De Morgan algebras. This logic is able to describe software architectures with quantitative features especially the uncertainty of the interactions that occur in the architecture. We deal with the equivalence problem of formulas in our logic by showing that every formula can be written in a specific form. Surprisingly, there are formulas which are equivalent only over specific De Morgan algebras. We provide examples of formulas in our logic which describe well-known software architectures equipped with quantitative features such as the uncertainty and reliability of their interactions.

Modelling Uncertainty in Architectures of Parametric Component-Based Systems

In this paper, we propose a logic-based characterization of uncertainty in architectures of parametric component-based systems, where the parameter is the number of instances of each component type. For this, we firstly introduce an extended propositional interaction logic over De Morgan algebras and we show that its formulas can encode the uncertainty of several architectures applied in systems with a finite number of components. In turn, we introduce a first-order extended interaction logic over De Morgan algebras which is applied for modelling uncertainty in the interactions of well-known parametric architectures. Moreover, we prove that the equivalence problem for a large class of formulas of that logic is decidable in doubly exponential time by providing an effective translation to fuzzy recognizable series. For any such formula over a totally ordered De Morgan algebra, we further prove that we can compute in exponential time the set of sequences of parametric fuzzy interactions which ensure the trustworthiness of the formula according to a particular threshold.

A note on k-cyclic modal pseudocomplemented De Morgan algebras

Symmetric and k-cyclic structure of modal pseudocomplemented De Morgan algebras was introduced previously. In this paper, we first present the construction of epimorphisms between finite symmetric (or 2-cyclic) modal pseudocomplemented De Morgan algebras. Furthermore, we compute the cardinality of the set of all epimorphism between finite structures. Secondly, we present the construction of finite free algebras on the variety of k-cyclic modal pseudocomplemented De Morgan algebras and display how our computations are in fact generalizations to others in the literature. Our work is strongly based on the properties of epimorphisms and automorphisms and the fact that the variety is finitely generated.

Rough set models of some abstract algebras close to pre-rough algebra

Rough set theory has already been algebraically investigated for decades and quasi-Boolean algebra has formed a basis for a number of structures emerging out of rough sets. Pre-rough algebra is one such algebra amongst them. A number of structures based on quasi-Boolean algebra but weaker than pre-rough algebra already exist. In this paper some algebras and their logics are added. Rough set models of the newly created algebras and some of the existing algebras are presented.

The spectra of quasi-Boolean algebras

Abstract In the present paper, we introduce the notions of quasi-Boolean algebras as the generalization of Boolean algebras. First we discuss the related properties of quasi-Boolean algebras. Second we define filters of quasi-Boolean algebras and investigate some properties of filters in quasi-Boolean algebras. We also show that there is a one-to-one correspondence between the set of filters and the set of filter congruences on a quasi-Boolean algebra. Then we investigate the prime filters and maximal filters of quasi-Boolean algebras, showing that the prime filters of a quasi-Boolean algebra are precisely the maximal filters and the prime spectrum of a quasi-Boolean algebra is a compact Hausdorff topological space. Finally, we define and study the reticulation of a quasi-Boolean algebra.

Some implicative topological quasi-Boolean algebras and rough set models

In this paper a number of implicative topological algebras have been developed which are based on the implicative quasi-Boolean algebra with operator (IqBaO) [10]. Their independence is established by several examples. Logics corresponding to these algebras are presented. Two new pairs of lower-upper approximations of a set have been introduced in order to develop the notion of duality with respect to the quasi-complementation. Set theoretic rough set models of some of the algebras are constructed using these lower-upper approximations and quasi-complementation.

Paraconsistent and Paracomplete Logics Based on k-Cyclic Modal Pseudocomplemented De Morgan Algebras

Paraconsistent and Paracomplete Logics Based on k-Cyclic Modal Pseudocomplemented De Morgan Algebras

On Logics of Perfect Paradefinite Algebras

The present study shows how to enrich De Morgan algebras with a perfection operator that allows one to express the Boolean properties of negation-consistency and negation-determinedness. The variety of perfect paradefinite algebras thus obtained (PP-algebras) is shown to be term-equivalent to the variety of involutive Stone algebras, introduced by R. Cignoli and M. Sagastume, and more recently studied from a logical perspective by M. Figallo-L. Cant\'u and by S. Marcelino-U. Rivieccio. This equivalence plays an important role in the investigation of the 1-assertional logic and of the order-preserving logic associated to PP-algebras. The latter logic (here called PP<=) is characterized by a single 6-valued matrix and is shown to be a Logic of Formal Inconsistency and Formal Undeterminedness. We axiomatize PP<= by means of an analytic finite Hilbert-style calculus, and we present an axiomatization procedure that covers the logics corresponding to other classes of De Morgan algebras enriched by a perfection operator.

Logics of involutive Stone algebras

An involutive Stone algebra (IS-algebra) is simultaneously a De Morgan algebra and a Stone algebra (i.e., a pseudo-complemented distributive lattice satisfying the Stone identity \(\mathop {\sim }x \vee \mathop {\sim }\mathop {\sim }x \approx 1\)). IS-algebras have been studied algebraically and topologically since the 1980s, but a corresponding logic (here denoted \(\mathcal {IS}_{\le }\)) has been introduced only very recently. This logic is the departing point of the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that \(\mathcal {IS}_{\le }\) is a conservative expansion of the Belnap-Dunn four-valued logic (i.e., the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic (i.e., every strengthening of the Belnap-Dunn logic) so as to obtain an extension of \(\mathcal {IS}_{\le }\). We show that every logic thus defined can be axiomatized by adding a fixed finite set of multiple-conclusion rule schemata to the corresponding super-Belnap base logic. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of \(\mathcal {IS}_{\le }\). In fact, as in the super-Belnap case, we establish that the finitary extensions of \(\mathcal {IS}_{\le }\) are already uncountably many. When the base super-Belnap logic possesses a disjunction, we show that we can reduce the multiple-conclusion calculus to a traditional one; some of the multiple-conclusion axiomatizations so introduced are analytic and are thus of independent interest from a proof-theoretic standpoint. We also consider a few extensions of \(\mathcal {IS}_{\le }\) that cannot be obtained in the above-described way, but can nevertheless be axiomatized finitely by other methods.

A FUNCTIONAL REPRESENTATION OF FREE DE MORGAN ALGEBRAS

It is well known that free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free distributive lattice on n free generators is isomorphic to the lattice of monotone Boolean functions of n variables. In this paper we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. Keywords: Antichain, monotone Boolean function, De Morgan function, free De Morgan algebra.

Semidistributivity and Whitman Property in implication zroupoids

AbstractIn 2012, the second author introduced, and initiated the investigations into, the variety 𝓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebraA= 〈A, →, 0 〉, where → is binary and 0 is a constant, is called animplication zroupoid(𝓘-zroupoid, for short) ifAsatisfies: (x→y) →z≈ [(z′ →x) → (y→z)′]′, wherex′ :=x→ 0, and 0″ ≈ 0. Let 𝓘 denote the variety of implication zroupoids andA∈ 𝓘. Forx,y∈A, letx∧y:= (x→y′)′ andx∨y:= (x′ ∧y′)′. In an earlier paper, we had proved that ifA∈ 𝓘, then the algebraAmj= 〈A, ∨, ∧〉 is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for everyA∈ 𝓘, the bisemigroupAmjis semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety 𝓜𝓔𝓙 of 𝓘, defined by the identity:x∧y≈x∨y, satisfies the Whitman Property. We conclude the paper with two open problems.