Journal of Algebraic Hyperstructures and Logical Algebras
Volume:2 Issue: 2, Spring 2021

In this paper, we introduce nilpotent and solvable pair of polygroups. By a pair of polygroups $(P,N)$, we mean a polygroup $P$ with a normal sub-polygroup $N$ of $P$. In addition, we obtain a necessary and sufficient condition between nilpotency (solvability) of pair of polygroups and pair of groups. In particular, we extend the theory of groups to a pair of polygroups. Finally, we study the relationship between nilpotent and solvable pair of polygroups.

This paper introduces the novel concept of neutro-d-algebra. In neutro-d-algebra, the outcome of any given two elements under an underlying operation (neutro-sophication procedure) has three cases, such as: Appurtenance, non-appurtenance, or indeterminate and for an axiom: Equal, non-equal, or indeterminate. This study investigates the neutro-d-algebra and shows that neutro-d-algebra is different from d-algebra. The notation of neutro-d-algebra generates a new concept of neutro poset. Finally, we introduce a notation of neutro d-homomorphism and extend the structure of neutro-d-algebras.

Let R be a multiplicative hyperring and S be a subset of R be a multiplicatively closed subset of R. In this paper, we introduce and study the concept of S-prime hyperideals which is a generalization of prime hyperideals. Some properties of S-prime hyperideals in multiplicative hyperring are presented. Then we investigate the behaviour of S-prime hyperideals under homomorphism hyperrings, in factor hyperrings, Cartesian products of hyperrings, and the fundamental relation in the context of multiplicative hyperring.

We introduce a specific kind of equivalence relation ${xi^*}_n^{alpha}$ on a fuzzy hypergroup S such that the quotient $S/ {xi^*}_n^{alpha}$, the set of all equivalence classes, is an $alpha$-solvable group. This helps us to introduce the $alpha$-solvable fundamental relation $xi^{*alpha}$. In particular, we obtain an equivalent condition with transitivity of $xi^alpha$.

In this paper, we introduce the concept of fuzzy multigraphs and a type of isomorphism on fuzzy multi-graphs and study their properties. We show that how to assign a fuzzy multi-graph to the appropriate fuzzy multi-groups. We also prove that every fuzzy multi-group of group G can be embedded into the fuzzy multi-group of the group Aut(Γσ ∧), where Aut(Γσ ∧) is a group of automorphisms of the fuzzy multi-graph Γσ ∧.

Fundamental relations are one of the main tools in connection relation between hyperstructures theory and structures theory. In this paper, we introduce a general fundamental relation on any hypergroup in such a way that all fundamental relations are a special case of this relation. Also, this study considered the notation of the relation on the derivation of $k$-nilpotent groups from any hypergroups.

A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, the new concepts of (totally) irregular, strongly (totally) irregular, highly totally irregular, neighborly totally irregular, edge-irregular of vague graphs are introduced and generalized, and some properties and related results are investigated. Then, we present the concepts of homomorphism, weak isomorphism, co-weak isomorphism and isomorphism of irregular vague graphs and some results on (total) domination number, (total) 2-domination number, (total) cobondage number and (total) 2-cobondage number. Finally, one of their applications related to location map of Fire Stations and Emergency Medical centers in urban regions of ​​a metropolis is presented.