Annals of Communications in Mathematics

AbstractQuasi-ordered residuated systems as a generalization of both quasi-ordered commutative residuated lattices and hoop-algebras were developed in 2018 by Bonzio and Chajda. The ideas of the theory of hyper structures were applied to this algebraic structure by this author and, at the same time, developed the concepts of filters in a hyper quasi-ordered residuated system. In this paper, the conditions that determine the concept of implicative strong filters in it. Some equivalent conditions were found that also determine this concept.

AbstractIn this article, the concepts of interior, weak-interior and quasi-interior filters in a quasi-ordered Γ-semigroup are introduced and recognize some of their fundamental properties. In addition to the above, the relationships between these three classes of filters in quasi-ordered Γ-semigroups are considered. One of the specifics in this analysis, among others, is that the requirement that a filter in a quasi-ordered Γ-semigroup S has to be a sub-semigroup in S is omitted. Instead of this requirement in the determination of these three classes of filters the consistency requirement is incorporated

AbstractIn this article, we give a number of important properties of meet-commutative UP-algebras. In the class of these UP-algebras, we introduce and analyze the concepts of prime and irreducible UP-filters.

AbstractTo interact somebody with someone various almost ideals (shortly A -ideals), quasi A -ideals, bi quasi A -ideals, tri A -ideals and tri quasi A -ideals in ternary semiring and give some characterizations. We develop the implications ideal =⇒ quasi ideal =⇒ two sided bi quasi ideal =⇒ two sided tri quasi ideal =⇒ two sided tri quasi A -ideal =⇒ two sided bi quasi A -ideal =⇒ bi A -ideal =⇒ quasi A -ideal =⇒ A -ideal and reverse implications do not holds with examples. We show that the union of A -ideals (bi A -ideals, quasi A -ideals, bi quasi A -ideals) is a A -ideal (bi A -ideal, quasi A -ideal, bi quasi A -ideal) in ternary semiring.

AbstractThe concept of meet-commutative UP-algebras was introduced in 2016 by Sawika et al. Muhiuddin et al. introduced in 2021 the concept of prime UP-filter (of the first kind) and irreducible UP-filter in meet-commutative UP-algebras. Also, it has been shown that any prime UP-filter in such algebras is irreducible. In this paper, we introduce the concept of weakly irreducible UP-filters in such algebras and show that the prime UPfilter is between this and the irreducible UP-filter. Also, we show the possibility that each irreducible UP-filter is a weakly irreducible UP-filter

AbstractThe notion of Γ-semigroups has been introduced by M. K. Sen and N. K. Saha. The concept of (co-ordered) Γ-semigroups with apartness in Bishop’s constructive algebra was introduced by this author. Many classical notions and processes of semigroups and Γ-semigroups have been extended to (co-ordered) Γ-semigroups with apartness such as ideals, filters and the first theorem of isomorphism of this class of algebraic structures. In this paper, as a continuation of earlier research, the author designs a form of the third isomorphism theorems for Γ-semigroups and co-ordered Γ-semigroups with apartness which does not have its counterpart in the classical Γ-semigroup theory.

AbstractUP-algebra was introduced by Iampan 2017 as a generalization of KU-algebra. The concept of meet-commutative UP–algebras was introduced by Sawika et al. In such algebras, the notion of prime UP-filter of the first kind and the notion of prime UP-filter of the second kind are introduced. In this article, as a continuation of the previous two, the concept of prime UP–filters of the third kind was introduced and the relationship between these three types of prime UP-filters was discussed.

AbstractJU-algebra (or weak KU-algebra) is a generalization of a KU-algebra. The concepts of atoms in JU-algebras are introduced and their important basic properties are registered. In addition to the previous one, an extension of a (weak) KU-algebra A to the JU-algebra A ∪ {w} was designed.

AbstractBI-algebra was introduced in 2017 by A. Borumand Saeid, H. S. Kim and A. Rezaei. Then this class of logical algebras was the focus of many researchers. In this paper, we register an additional property of ideals in right distributive BI-algebras. Then, in this paper we discuss the following two things: the definition of the concept of atoms in right distributive BI-algebras and the registration of many properties of such a designed concept of atoms. In addition to the previous one, the paper designs an extension of the right distributive BI-algebra A = (A, ·, 0) by adding one element w /∈ A such that w is an atom in A.