On Commutative Be-Algebras

Pan-American journal of mathematics, 2024

Then this class of logical algebras was the focus of many researchers. In this paper, the concept of atoms in BE-algebras is introduced and analyzed, and, in addition, it is directly connected to two-membered BE-filters. A criterion was found for determining the existence of atoms in these algebras. In addition to the previous one, the paper designs two types of BE-algebra extensions by adding one element so that the additional element is an atom in it. In addition to the previous one, two new types of filters in BE-algebras are designed.

In this paper we use the regular congruence relation ∼ I to construct a quotient B-algebra X∖I from a self distributive BE-algebra X. Then we study the notion of homomorphisms on BE-algebras and the properties are studied. Finally we stated and prove the first, second and third isomorphism theorems in self distributive BE-algebras.

Soft Computing, 2018

We consider thirty generalizations of BCK algebras (RM, RML, BCH, BCC, BZ, BCI algebras and many others). We investigate the property of commutativity for these algebras. We also give 10 examples of proper commutative finite algebras. Moreover, we review some natural classes of commutative RML algebras and prove that they are equationally definable.

Open Journal of Mathematical Sciences (OMS) 2523-0212 (online) 2616-4906 (Print), 2020

In this article, we revisit the axioms of JU-algebras previously recognizable as 'pseudo KU-algebras', which we may call as 'weak KU-algebras' and discussed the definitions of some of their substructures. We also associate this class of algebras with the classes of BE-algebras and UP-algebras. In addition, we introduce and analyze some new classes of ideals in this class of algebras. Keywords: JU-algebras, ideal and filter in JU-algebras, closed ideal, ag-ideal, t-ideal, ()-ideal and associative ideal. MSC: 03G25.

AIP Conference Proceedings

In this paper we want to introduce the idea of Cartesian product of BE-algebras. The concept plays a significant role in the study of BE-algebras. Furthermore we continue to study the Cartesian product of filters, ideals and essences in BE-algebra with some characteristic properties.

Applications and Applied Mathematics: An International Journal (AAM), 2013

Hilbert algebras are introduced for investigations in intuitionistic and other non - classical logics and BE -algebra is a generalization of dual BCK -algebra. In this paper, we investigate the relationship between Hilbert algebras and BE -algebras. In fact, we show that a commutative implicative BE -algebra is equivalent to the commutative self distributive BE -algebra, therefore Hilbert algebras and commutative self distributive BE -algebras are equivalent.

Zenodo (CERN European Organization for Nuclear Research), 2008

Since all the algebras connected to logic have, more or less explicitely, an associated order relation, it follows that they have two presentations, dual to each other. We classify these dual presentations in "left" and "right" ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as "left" algebras or as "right" algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the "left" presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the properties satisfied. In this work (Parts I-V) we make an exhaustive study of these algebras-with two bounds and with one bound-and we present classes of finite examples, in bounded case. In this Part I, divided in two because of its length, after surveying chronologically several algebras related to logic, as residuated lattices, Hilbert algebras, MV algebras, divisible residuated lattices, BCK algebras, Wajsberg algebras, BL algebras, MTL algebras, WNM algebras, IMTL algebras, NM algebras, we propose a methodology in two steps for the simultaneous work with them (the first part of Part I). We then apply the methodology, redefining those algebras as particular cases of reversed left-BCK algebras. We analyse among others the properties Weak Nilpotent Minimum and Double Negation of a bounded BCK(P) lattice, we introduce new corresponding algebras and we establish hierarchies (the subsequent part of Part I).

Analele Universitatii "Ovidius" Constanta - Seria Matematica

In this paper, we introduce a new algebra, called a BI-algebra, which is a generalization of a (dual) implication algebra and we discuss the basic properties of BI-algebras, and investigate ideals and congruence relations.

Journal of Universal Computer Science, 2007

Since all the algebras connected to logic have, more or less explicitely, an associated order relation, it follows that they have two presentations, dual to each other. We classify these dual presentations in "left" and "right" ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as "left" algebras or as "right" algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the "left" presentation and several algebras of logic have been redefined as particular cases of BCK algebras.