Journal of Algebraic Hyperstructures and Logical Algebras
Volume:3 Issue: 3, Summer 2022

The notions of a prime (strongly prime, semiprime, irreducible, and strongly irreducible) double-framed soft bi-ideals (briefly, prime, (strongly prime, semiprime, irreducible and strongly irreducible) DFS bi-ideals) in ordered semigroups are introduced and related properties are investigated. Several examples of these notions are provided. The relationship between prime and strongly prime, irreducible and strongly irreducible DFS bi-ideals are considered and  haracterizations of these concepts are established. The Characterizations of regular and intraregular ordered semigroups in terms of these notions are studied.

Logic gives a technique for the articial intelligence to make the computers simulate human being in dealing with certainty and uncertainty in information. Various logical algebras have been proposed and researched as the semantical  systems of non-classical logical systems. In this paper using the concept of commutator, we introduce the Engel algebra and then study a condition on infinite subsets of infinite algebras. We also show that some logical algebras satisfy to this condition but do not have the properties associated with that condition.

A Dokdo structure is used to study implicative ideals in BCK-algebras. The notion of Dokdo implicative ideals in BCK-algebras is introduced, and the relevant properties are investigated. The relationship between Dokdo subalgebras, Dokdo ideals, and Dokdo implicative ideals are discussed, and conditions that allow a Dokdo ideal and a Dokdo subalgebra to be a Dokdo implicative ideal are provided.

This paper aims is to introduce states, Bosbach states and state-morphism operators on BI-algebras. We define state ideals on BI-algebras and give a characterization of the least state ideal of a BI-algebra. It is proved that the kernel of a Bosbach state on a BI-algebra X is an ideal of X. Further, by these concepts, we introduce the notions of state BI-algebras and state-morphism BI-algebras. The notion of complement pairs of a BI-algebra X is defined, and proves that under suitable conditions, there is a one-to-one correspondence between complement pairs of BI-algebras and state-morphism operators on BI-algebras.

Let G be a group (monoid) with identity e and R be a commutative Krasner hyperring. In this paper, we introduce the concepts of graded absorbing hyperideals of a graded Krasner hyperring such as, graded 2-absorbing hyperideals, graded n-absorbing hyperideals and graded 2-absorbing subhypermodules. Some basic properties of these structures  and characterizations of these graded absorbing hyperideals and homogeneous components are proved.

In this research, we apply the notations of the kernel and relative measure of an interval-valued grey to introduce grey groups (groups are based on interval-valued grey) and grey hypergroups (hypergroups are based on interval-valued grey). The primary method used in this research is based on linear inequalities related to elements of grey (hyper)groups and (hyper)groups. It found a relation between grey hypergroups and grey groups via the fundamental relation and proves that the identity element of any given group plays a main role in the grey groups and show that its measure is greater than or equal to its degree of greyness and less than or equal to its kernel, respectively. We show that any given grey group is a generalization of a group and analyze that interval-valued grey groups are different from the interval-valued fuzzy group.