Showing papers in "Discussiones Mathematicae General Algebra and Applications in 2008"
TL;DR: In this paper, the authors study some interesting properties of regular ternary semigroups and characterize them by using various ideals of ternaries, e.g., the notion of semigroup idealism.
Abstract: In this paper we study some interesting properties of regular ternary semigroups, completely regular ternary semigroups, intra-regular ternary semigroups and characterize them by using various ideals of ternary semigroups.
21 citations
TL;DR: In this article, the homomorphic properties of fuzzy prime ideals of pseudo MV-algebras are investigated and a one-to-one correspondence between the maximal ideals and the set of fuzzy maximal ideals satisfying (0) = 1 and (1) = 0 is obtained.
Abstract: Fuzzy ideals of pseudo MV-algebras are investigated. The homomorphic properties of fuzzy prime ideals are given. A one-to-one correspondence between the set of maximal ideals and the set of fuzzy maximal ideals satisfying (0) = 1 and (1) = 0 is obtained.
15 citations
TL;DR: In this paper, it was shown that the sequences X k Tn k k and X k Un k k are two basis of the Q-vectorial space En [X], formed by the polynomials of Q [X] having the same parity as n and of degree n.
Abstract: Letting Tn (resp. Un) be the n-th Chebyshev polynomials of the rst (resp. second) kind, we prove that the sequences X k Tn k k and X k Un k k for n 2bn=2c k n bn=2c are two basis of the Q-vectorial space En [X] formed by the polynomials of Q [X] having the same parity as n and of degree n. Also Tn and Un admit remarkableness integer coordinates on each of the two basis.
9 citations
TL;DR: In this article, the semidirect product of a semigroup and a Γ-semigroup is studied and some interesting properties of this product are investigated. And the notion of wreath product is introduced.
Abstract: Let S = {a, b, c, . . .} and Γ = {α, β, γ, . . . } be two nonempty sets. S is called a Γ-semigroup if aαb ∈ S, for all α ∈ Γ and a, b ∈ S and (aαb)βc = aα(bβc), for all a, b, c ∈ S and for all α, β ∈ Γ. In this paper we study the semidirect product of a semigroup and a Γ-semigroup. We also introduce the notion of wreath product of a semigroup and a Γsemigroup and investigate some interesting properties of this product.
3 citations
TL;DR: In this article, the authors generalized topological Boolean algebras to closure and interior operators of MV-algebra which are an algebraic counterpart of the Lukasiewicz innite valued logic, and extended these kinds of operators to the wide class of bounded commutative R‘-monoids.
Abstract: Topological Boolean algebras are generalizations of topological spaces dened by means of topological closure and interior operators, respectively. The authors in [14] generalized topological Boolean algebras to closure and interior operators of MV-algebras which are an algebraic counterpart of the Lukasiewicz innite valued logic. In the paper, these kinds of operators are extended (and investigated) to the wide class of bounded commutative R‘-monoids that contains e.g. the classes of BL-algebras (i.e., algebras of the H ajek’s basic fuzzy logic) and Heyting algebras as proper subclasses.
3 citations
TL;DR: In this paper, necessary and sufficient conditions for the nonnegativity of the Moore-Penrose inverse of a real matrix in terms of certain spectral properties shared by all positive splittings of the given matrix were studied.
Abstract: In this short note we study necessary and sufficient conditions for the nonnegativity of the Moore-Penrose inverse of a real matrix in terms of certain spectral property shared by all positive splittings of the given matrix.
2 citations
TL;DR: In this paper, the authors distinguish (up to symmetry) eight cases and in each of these cases they present such a lattice of minimal cardinality, which is the case in which not both their join and their meet are complemented.
Abstract: Bounded lattices with an antitone involution the complemented elements of which do not form a sublattice must contain two complemented elements such that not both their join and their meet are complemented. We distinguish (up to symmetry) eight cases and in each of these cases we present such a lattice of minimal cardinality.
2 citations
TL;DR: In this paper, the notions of Noetherian pseudo MV-algebras and Artinian pseudoMV-algebra are introduced and their characterizations are established via fuzzy ideals.
Abstract: Institute of Mathematics and PhysicsUniversity of Podlasie3 Maja 54, 08{110 Siedlce, Polande-mail: gdymek@o2.plAbstractThe notionsof Noetherian pseudo MV-algebrasand Artinian pseudoMV-algebrasareintroduced and their characterizationsare established.Characterizations of them via fuzzy ideals are also given.Keywords: pseudo MV-algebra, (fuzzy) ideal, Noetherian (Artinian)pseudo MV-algebra.2000 Mathematics Subject Classi cation: 06D35.
2 citations
TL;DR: In this paper, it was shown that the varieties of entropic and distributive differential modals coincide, and the lattice of varieties of Entropic differential modal lattices was described.
Abstract: A differential modal is an algebra with two binary operations such that one of the reducts is a differential groupoid and the other is a semilattice, and with the groupoid operation distributing over the semilattice operation. The aim of this paper is to show that the varieties of entropic and distributive differential modals coincide, and to describe the lattice of varieties of entropic differential modals.
2 citations
TL;DR: In this paper, commutative directoids with the greatest element can be axiomatized as algebras with two binary operations satisfying four identities, and a minimal subvariety of this variety is described.
Abstract: We study commutative directoids with a greatest element, which can be equipped with antitone bijections in every principal filter. These can be axiomatized as algebras with two binary operations satisfying four identities. A minimal subvariety of this variety is described.
1 citations
TL;DR: In this article, it was shown that the analogous (k + 1)-level inflation can be used to characterize the algebras of Nk(V ) for any variety V having a unary term which satisfies two technical conditions.
Abstract: Let τ be a type of algebras. A common measurement of the complexity of terms of type τ is the depth of a term. For k ≥ 1, an identity s ≈ t of type τ is said to be k-normal (with respect to this depth complexity measurement) if either s = t or both s and t have depth ≥ k. A variety is called k-normal if all its identities are k-normal. Taking k = 1 with respect to the usual depth valuation of terms gives the wellknown property of normality of identities or varieties. For any variety V , there is a least k-normal variety Nk(V ) containing V , the variety determined by the set of all k-normal identities of V . The concept of k-normalization was introduced by K. Denecke and S.L. Wismath in [5], and an algebraic characterization of the elements of Nk(V ) in terms of the algebras in V was given in [4]. In [1] a simplified version of this characterization of Nk(V ) was given, in the special case of the 2-normalization of the variety V of all lattices, using a construction called the 3-level inflation of a lattice. In this paper we show that the analogous (k + 1)-level inflation can be used to characterize the algebras of Nk(V ) for any variety V having a unary term which satisfies two technical conditions. This includes any variety V which satisfies x ≈ t(x) for some unary term t of depth at least k, and in particular any variety, such as the variety of lattices, which satisfies an idempotent identity.
TL;DR: The DN-algebra as mentioned in this paper generalizes the concept of Newman algebras (introduced by H. Dobbertin) for a common axiomatization of Boolean algebra and Boolean rings.
Abstract: We introduce the so-called DN -algebra whose axiomatic system is a common axiomatization of directoids with an antitone involution and the so-called D-quasiring. It generalizes the concept of Newman algebras (introduced by H. Dobbertin) for a common axiomatization of Boolean algebras and Boolean rings.
TL;DR: In this paper, it was shown that a congruence on a semiring whose additive reduct (S, +) is an inverse semigroup is a Cliord congruent on the lattice C(S) of all congruences on S.
Abstract: Let S be a semiring whose additive reduct (S, +) is an inverse semigroup. The relations and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For 2 C(S), we have introduced four congruences min, max, min and max on S and showed that = [ min, max] and = [ min , max ]. Dieren t properties of and have been considered here. A congruence on S is a Cliord congruence if and only if max is a distributive lattice congruence and max is a skew-ring congruence on S. If ( ) is the least distributive lattice (resp. skew-ring) congruence on S then \ is the least Cliord congruence on S.
TL;DR: In this paper, a representation based on a q-lattice was proposed, i.e., the normalization of a lattice, for all normal identities of basic algebras.
Abstract: We consider algebras determined by all normal identities of basic algebras. For such algebras, we present a representation based on a q-lattice, i.e., the normalization of a lattice.
TL;DR: In this paper, it was shown that the covariety lattice LCV(K) of subcovarieties of a covarset K of F -coalgebras, where F : Set → Set preserves arbitrary intersections is isomorphic to the lattice of sub-covarsets of a Pκ-coalgebra for some cardinal κ.
Abstract: This paper shows basic properties of covariety lattices. Such lattices are shown to be infinitely distributive. The covariety lattice LCV(K) of subcovarieties of a covariety K of F -coalgebras, where F : Set → Set preserves arbitrary intersections is isomorphic to the lattice of subcoalgebras of a Pκ-coalgebra for some cardinal κ. A full description of the covariety lattice of Id-coalgebras is given. For any topology τ there exist a bounded functor F : Set → Set and a covariety K of F -coalgebras, such that LCV(K) is isomorphic to the lattice (τ,∪,∩) of open sets of τ .