Subdirectly irreducible algebras with hyperidentities of the variety of De Morgan algebras

The paper characterizes the class of subdirectly irreducible algebras satisfying hyperidentities of the variety of De Morgan algebras. Such algebras are called subdirectly irreducible De Morgan quasilattices. The suggested characterization is quite close to that of the classical case of subdirectly irreducible DeMorgan algebras.     

Super-De Morgan functions and free De Morgan quasilattices

A De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices). In this paper we give a functional representation of the free n-generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on nfree generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of nvariables.     

Congruence properties of pseudocomplemented De Morgan algebras

In this paper we first describe the Priestley duality for pseudocomplemented De Morgan algebras by combining the known dualities of distributive p‐algebras due to Priestley and for De Morgan algebras due to Cornish and Fowler. We then use it to characterize congruence‐permutability, principal join property, and the property of having only principal congruences for pseudocomplemented De Morgan algebras. The congruence‐uniform pseudocomplemented De Morgan algebras are also described.     

Boole–De Morgan Algebras and Quasi-De Morgan Functions

In this paper we establish a Stone-type and a Birkhoff-type representation theorems for Boole–De Morgan algebras and prove that the free Boole–De Morgan algebra on n free generators is isomorphic to the Boole–De Morgan algebra of quasi-De Morgan functions of n variables. Also we introduce the concept of Zhegalkin polynomials for quasi-De Morgan functions and consider the representation problem of those functions by polynomials.     

强De Morgan代数上的广义R0 t-模

在De Morgan代数上引入广义R0算子,举例说明了一般De Morgan代数中的广义R0算子不能构成t-模。引入强De Morgan代数的概念,讨论它的基本性质,证明强De Morgan代数L上的广义R0算子构成t-模(称为广义R0t-模)。给出若干重要反例,并证明强De Morgan代数上的广义R0t-模是左连续的。     

Unification and Projectivity in De Morgan and Kleene Algebras

We provide a complete classification of solvable instances of the equational unification problem over De Morgan and Kleene algebras with respect to unification type. The key tool is a combinatorial characterization of finitely generated projective De Morgan and Kleene algebras.     

Free Algebras in Certain Varieties of Distributive Pseudocomplemented De Morgan Algebras

In this paper we characterize the join irreducible elements of the free algebras on n free generators in the subvarieties of the variety V₀ of pseudocomplemented De Morgan algebras satisfying the identity xx′* = (xx′*)′*.     

弱射影与De Morgan代数的同余关系

本文借助弱射影和弱透视的概念刻划了Ｄｅ Ｍｏｒｇａｎ代数的同余关系,由此得到了Ｋａｌｍａｎ关于Ｄｅ Ｍｏｒｇａｎ代数次直不可约定理的一个新的证明并证明了一个完全分配的Ｄｅ Ｍｏｒｇａｎ代数的同余理想与同余关系一一对应的充要条件是Ｌ为弱可补的。     

G-De Morgan代数的G-同态与G-同余

将De Morgan代数的自同构群对De Morgan代数的作用,推广成抽象群对De Morgan代数的作用,引入了G-De Morgan代数的概念,讨论了G-De Morgan代数的G-同态、G-同余等性质,并研究了G-De Morgan代数的直积分解和次直不可约性.     

Relation algebras as expanded FL-algebras

This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.     

De Morgan Heyting algebras satisfying the identity xn(′*) ≈ x(n+1)(′*)

In this paper we investigate the sequence of subvarieties $ {\mathcal {SDH}_n} $of De Morgan Heyting algebras characterized by the identity x^n(′*) ≈ x^(n+1)(′*). We obtain necessary and sufficient conditions for a De Morgan Heyting algebra to be in $ {\mathcal {SDH}_1} $ by means of its space of prime filters, and we characterize subdirectly irreducible and simple algebras in $ {\mathcal {SDH}_1} $. We extend these results for finite algebras in the general case $ {\mathcal {SDH}_n} $. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Matrix Characterization of 4-Ary Algebraic Operations of Idempotent Algebras

Let (U; F) be an idempotent algebra. There is an r-ary essentially algebraic operation in F where there is not any (r + 3)-ary algebraic operation depending on at least r + 1 variables. In this paper, we prove that the set of all 4-ary algebraic operations of this algebras forms a finite De Morgan algebra, and then we characterize this De Morgan algebra.     

On symmetric extended MS-algebras whose congruences are permutable

Algebras whose congruences are permutable were investigated by a number of authors in the literature. In this paper, we study the symmetric extended MS-algebras whose congruences are permutable. Some results obtained by Jie Fang on symmetric extended De Morgan algebras are generalized.     

A special class of congruences on $${\kappa}$$-frames

In a paper published in 2012, the second author extended the well-known fact that Boolean algebras can be defined using only implication and a constant, to De Morgan algebras—this result led him to introduce, and investigate (in the same paper), the variety \({\mathcal{I}}\) of algebras, there called implication zroupoids (I-zroupoids) and here called implicator groupoids (\({\mathcal{I}}\)-groupoids), that generalize De Morgan algebras.The present paper is a continuation of the paper mentioned above and is devoted to investigating the structure of the lattice of subvarieties of \({\mathcal{I}}\), and also to making further contributions to the theory of implicator groupoids. Several new subvarieties of \({\mathcal{I}}\) are introduced and their relationship with each other, and with the subvarieties of \({\mathcal{I}}\) which were already investigated in the paper mentioned above, are explored.     

A Characterization of (2, n)-Semigroup of n-Place Functions and De Morgan (2, n)-Semigroup of n-Place Functions

In this article, by defining n Mann's compositions and one unary operation on the set of n-place functions over some set, we construct a De Morgan (2, n)-semigroup of n-place functions and so find an abstract characterization of this algebras.     

Topologies for intermediate logics

We investigate the problem of characterizing the classes of Grothendieck toposes whose internal logic satisfies a given assertion in the theory of Heyting algebras, and introduce natural analogues of the double negation and De Morgan topologies on an elementary topos for a wide class of intermediate logics.     

Hyperidentities of some generalizations of lattices

In the paper we present bases and hyperbases of hyperidentities of some generalizations of the variety L of all lattices and the variety D of distributive lattices. We describe the form of hyperidentities of some varieties with two binary operations. Received January 22, 1997; accepted in final form January 7, 1998.     

A normal form which preserves tautologies and contradictions in a class of fuzzy logics

Most of the normal forms for fuzzy logics are versions of conjunctive and disjunctive classical normal forms. Unfortunately, they do not always preserve tautologies and contradictions which is important, for example, for automated theorem provers based on refutation methods.De Morgan implicative systems are triples like the De Morgan systems, which consider fuzzy implications instead of t-conorms. These systems can be used to evaluate the formulas of a propositional language based on the logical connectives of negation, conjunction and implication. Therefore, they determine different fuzzy logics, called implicative De Morgan fuzzy logics.In this paper, we will introduce a normal form for implicative De Morgan systems and we will show that for implicative De Morgan fuzzy logics whose t-norms are strict, this normal form preserves contradictions as well as tautologies.     

Relating De Morgan triples with Atanassov’s intuitionistic De Morgan triples via automorphisms

In this paper the relation between De Morgan triples on the unit interval and Atanassov’s intuitionistic De Morgan triples is presented, showing how to obtain, in a canonical way, Atanassov’s intuitionistic De Morgan triples from De Morgan triples. Moreover, we also show that the automorphisms on the unit interval and on L∗ (the intuitionistic value lattice) are in one-to-one correspondence and how automorphisms on L∗ act on Atanassov’s intuitionistic De Morgan triples. It is also proved that the action of automorphisms and the canonical construction of De Morgan triples on L∗ commutes.