Pseudo BL-algebras Research Articles

Pseudo General Overlap Functions and Weak Inflationary Pseudo BL-Algebras

General overlap functions are generalized on the basis of overlap functions, which have better application effects in classification problems, and the (weak) inflationary BL-algebras as the related algebraic structure were also studied. However, general overlap functions are a class of aggregation operators, and their commutativity puts certain restrictions on them. In this article, we first propose the notion of pseudo general overlap functions as a non-commutative generalization of general overlap functions, so as to extend their application range, then illustrate their relationship with several other commonly used aggregation functions, and characterize some construction methods. Secondly, the residuated implications induced by inflationary pseudo general overlap functions are discussed, and some examples are given. Then, on this basis, we show the definitions of inflationary pseudo general residuated lattices (IPGRLs) and weak inflationary pseudo BL-algebras, and explain that the weak inflationary pseudo BL-algebras can be gained by the inflationary pseudo general overlap functions. Moreover, they are more extensive algebraic structures, thus enriching the content of existing non-classical logical algebra. Finally, their related properties and their relations with some algebraic structures such as non-commutative residuated lattice-ordered groupoids are investigated. The legend reveals IPGRLs include all non-commutative algebraic structures involved in the article.

ON DERIVATIONS OF PSEUDO-BL ALGEBRA

Pseudo-BL algebras are a natural generalization of BL-algebras and of pseudo-MV algebras.In this paper the notions of five different types of derivations on a pbl as generalizations of derivations of a BL-algebra are introduced. Moreover, as an extension of derivations of a pbl , the notions of $(varphi , psi)$-derivations are defined on these types. Finally, several related properties are discussed.

Results in L-algebras

We study some classes of L-algebras and we give characterizations of commutative KL-algebras and CL-algebras. It is proved that any commutative KL-algebra is a BCK-algebra and any self-similar CL-algebra is commutative. The commutative ideals are defined and studied, and it is proved that a CL-algebra is commutative if and only if all its ideals are commutative. We also present the L-algebras arising from other structures, such as BCK-algebras, pseudo MV-algebras, pseudo BL-algebras, pseudo-hoops, bounded R $$\ell $$ -monoids, BE-algebras, Hilbert algebras.

Filters and ideals in the generalization of pseudo-BL algebras

In this paper, we introduce the notion of quasi-pseudo-BL algebras as the generalization of pseudo-BL algebras and quasi-pseudo-MV algebras. First, we investigate the properties of quasi-pseudo-BL algebras and show the subdirect product composition of any quasi-pseudo-BL algebra. Especially, some properties of good quasi-pseudo-BL algebras are presented. Second, we discuss the filters of quasi-pseudo-BL algebras and prove that there exists a bijective correspondence between normal filters and filter congruences on a quasi-pseudo-BL algebra. The properties of some special filters are also discussed. Finally, we study the ideals of quasi-pseudo-BL algebras and investigate some connections between ideals and filters of a quasi-pseudo-BL algebra.

Strongly algebraically closed orthomodular near semirings

The notion of $$q^\prime $$ -compactness on a modular near ring is introduced and considered. Our aim is to show that if an induced lattice with an antitone involution on an orthogonal near semiring is complete and $$q^\prime $$ -compact, then the induced lattice is a strongly algebraically closed lattice. In particular, an open question proposed by A. Di-Nola, G. Georgescu and A. Iorgulescu about the connections of pseudo-BL algebras with other algebraic structures in Di Nola et al. (Mult Val Logic 8:717–750, 2002) is answered.

A representation for radicals in pseudo BL-algebras

A representation for radicals in pseudo BL-algebras

Classification of some elements in pseudo BL-algebras

Classification of some elements in pseudo BL-algebras

Classes of pseudo BL-algebras with right Boolean lifting property

In this paper, we define the right Boolean lifting property (left Boolean lifting property) RBLP (LBLP) for pseudo BL-algebra to be the property that all Boolean elements can be lifted modulo every right filter (left filter) and next we study the behavior of RBLP (LBLP) with respect to direct products of pseudo BL-algebra. We introduce some conditions, which turn out to be a strengthening and a weakling of RBLP (LBLP) respectively and which open new ways of approaching the study of the RBLP (LBLP) in pseudo BL-algebras.

On a new construction of pseudo BL-algebras

We present a new construction of a class of pseudo BL-algebras, called kite pseudo BL-algebras. We start with a basic pseudo hoop. Using two injective mappings from one set into the second one, and an identical copy of the basic pseudo hoop with the reverse order we construct a pseudo BL-algebra. We show that such a structure is a pseudo BL-algebra iff the basic pseudo hoop is cancellative. Starting with a commutative hoop we can obtain even a non-commutative pseudo BL-algebra or a pseudo MV-algebra, or an algebra with non-commuting negations. We describe the construction, subdirect irreducible kite pseudo BL-algebras and their classification.

State filters on fuzzy structures with internal states

The aim of this paper is the study of some properties of the state filters of a state pseudo BL-algebra $$\left( A,\sigma \right) $$ A , ? and also some properties of a state pseudo BL-algebra. By introducing the concepts of simple, semisimple and local state pseudo BL-algebra relative to its state filters set, we present some characterizations of a simple, semisimple and local state pseudo BL-algebras. Also, we adjust a result from universal algebra (Chinese remainder theorem) and, by introducing the concept of a primary state filter on a state BL-algebra, we present a characterization of a local state BL-algebra. We introduce the notion of extended state filter of a state filter associated to a subset of a state pseudo BL-algebra.

Kites and pseudo BL-algebras

We investigate a construction of a pseudo BL-algebra out of an l-group called a kite. We show that many well-known examples of algebras related to fuzzy logics can be obtained in that way. We describe subdirectly irreducible kites. As another application, we exhibit a new countably infinite family of varieties of pseudo BL-algebras covering the variety of Boolean algebras.

Fuzzy relation equations in pseudo BL-algebras

Bandler and Kohout investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions in pseudo BL-algebras.

On pseudo BL-algebras with internal state

The scope of this paper is to put in evidence some characteristics and properties of the operations with state filters of a pseudo BL-algebra with internal state. I define the concepts of maximal and prime state filter of a state pseudo BL-algebra, I characterize a maximal state filter and I prove a Prime state filter theorem. Also, I characterize the state-filters algebra of a state pseudo BL-algebra.

On decomposition of pseudo BL-algebras

Abstract We show that under some conditions, imposed on coatoms and maximal idempotents of a pseudo BL-algebra, we can decompose a pseudo BL-algebra M as an ordinal sum and we show that then M is linearly ordered. We investigate pseudo BL-algebras with a unique coatom a and with a maximal idempotent, and analyze two main situations: either a n = a n+1 holds for some n ≥ 1, or a n > a n+1 hold for any n ≥ 1. We note that there exist (subdirectly irreducible) algebras with two coatoms that are not linearly ordered, so the restriction to a single coatom is natural.

On fuzzy filters of pseudo BL-algebras

Abstract We introduce the concept of fuzzy filters of pseudo-BL-algebras and obtain some of their important properties. We further give several kinds of fuzzy filters, such as fuzzy Boolean filter, fuzzy normal filter, fuzzy prime filter, fuzzy ultrafilter and fuzzy obstinate filter and discuss the relations among the above fuzzy filters. We establish the Prime Fuzzy Filter Theorem in pseudo BL-algebras. By studying the relation between fuzzy Boolean filter and fuzzy normal filter in pseudo-BL-algebras, we solve an open problem that whether or not every Boolean filter is normal in pseudo-BL-algebras.

Bipartite pseudo-BL algebras

AbstractThe class of bipartite pseudo-BL algebras (denoted by

On the Structure of Pseudo BL-algebras and Pseudo Hoops in Quantum Logics

The main aim of the paper is to solve a problem posed in Di Nola et al. (Multiple Val. Logic 8:715–750, 2002) whether every pseudo BL-algebra with two negations is good, i.e. whether the two negations commute. This property is intimately connected with possessing a state, which in turn is essential in quantum logical applications. We approach the solution by describing the structure of pseudo BL-algebras and pseudo hoops as important families of quantum structures. We show when a pseudo hoop can be embedded into the negative cone of the reals. We give an equational base characterizing representable pseudo hoops. We also describe some subvarieties: normal-valued, and varieties where each maximal filter is normal. We produce some noncommutative covers and extend the area where each algebra is good. Finally, we show that there are uncountably many subvarieties of pseudo BL-algebras having members that are not good.

Classes of examples of pseudo-MV algebras, pseudo-BL algebras and divisible bounded non-commutative residuated lattices

We illustrate by classes of examples the close connections existing between pseudo-MV algebras, on the one hand, and pseudo-BL algebras and divisible bounded non-commutative residuated lattices, on the other hand. We use equivalent definitions of these algebras, as particular cases of pseudo-BCK algebras. We analyse the strongness, the pseudo-involutive center and the filters for each example.

Semisimple and semilocal pseudo BL-algebras

AbstractThe concepts of semisimple and semilocal pseudo BL-algebras are investigated. Many facts corresponding with them are considerated. Moreover, we give a negative answer to the question from [Di Nola, Georgescu and Iorgulescu (Multiplae Valued Logic 8: 715-750, 2002), Problem 1.33].

Fuzzy filters and fuzzy prime filters of bounded [formula omitted]-monoids and pseudo BL-algebras

The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. The sets of provable formulas in corresponding inference systems from the point of view of uncertain information can be described by fuzzy filters of those algebraic semantics. Bounded residuated lattice ordered monoids ( R ℓ -monoids) are a common generalization of pseudo BL-algebras (and consequently of pseudo MV-algebras) and Heyting algebras, i.e. algebras behind fuzzy and intuitionistic reasoning. In the paper, we introduce and investigate fuzzy filters of bounded R ℓ -monoids and fuzzy prime filters of pseudo BL-algebras.