The concept of fuzzy multiset is well established in dealing with many real life problems. It is possible to find various applications of algebraic hypercompositional structures in natural, technical and social sciences, where symmetry, or the lack of symmetry, is clearly specified and laid out. In this paper, we use fuzzy multisets to introduce the concept of fuzzy multi- $H_v$ -ideals as a generalization of fuzzy $H_v$ -ideals. Moreover, we introduce the concept of generalized fuzzy multi- $H_v$ -ideals as a generalization of generalized fuzzy $H_v$ -ideals. Finally, we investigate the properties of these new concepts and present different examples.

Neutrosophy, the study of neutralities, is a new branch of Philosophy that has applications in many different fields of science. Inspired by the idea of Neutrosophy, Smarandache introduced NeutroAlgebraicStructures (or NeutroAlgebras) by allowing the partiality and indeterminacy to be included in the structures’ operations and/or axioms. The aim of this paper is to combine the concept of Neutrosophy with hyperstructures theory. In this regard, we introduce NeutroSemihypergroups as well as NeutroH_v-Semigroups and study their properties by providing several illustrative examples.

A fuzzy multiset is a generalization of a fuzzy set. This paper aims to combine the innovative notion of fuzzy multisets and hypergroups. In particular, we use fuzzy multisets to introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroups. Different operations on fuzzy multi-hypergroups are defined and discussed and some results known for fuzzy hypergroups are generalized to fuzzy multi-hypergroups.

Ordered semigroups are understood through their subsets. The aim of this article is to study ordered semigroups through some new substructures. In this regard, quasi-filters and $(m,n)$-quasi-filters of ordered semigroups are introduced as new types of filters. Some properties of the new concepts are investigated, different examples are constructed, and the relations between quasi-filters and quasi-ideals as well as between $(m,n)$-quasi-filters and $(m,n)$-quasi-ideals are discussed.

In this paper, we apply the concept of linear Diophantine fuzzy sets in $BCK/BCI$-algebras. In this respect, the notions of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are introduced and some vital properties are discussed. Additionally, characterizations of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are considered. Moreover, the associated results for linear Diophantine fuzzy subalgebras, linear Diophantine fuzzy ideals and linear Diophantine fuzzy commutative ideals are obtained.