COMMUTATIVE FUNDAMENTAL RELATION IN FUZZY HYPERSEMIGROUPS
We introduce and study the construction of commutative fundamental relation in fuzzy hypersemigroups and investigate its basic properties. We will determine some necessary and sufficient conditions so that this relation is transitive in fuzzy hypersemigroups. Also we show that this relation is transitive in every fuzzy hypergroup.
On α-solvable fundamental groups
We introduce a specific kind of equivalence relation ξn on a fuzzy hypergroup S such that the quotient S/ξn, the set of all equivalence classes, is an α-solvable group. This helps us to introduce the α-solvable fundamental relation ξ∗α. In particular, we obtain an equivalent condition with transitivity of ξα. Article Information Corresponding Author: F. Mohammadzadeh; Received: April 2021; Revised: May 2021; Accepted: June 2021; Paper type: Original.
On Dimension of Hypervector Spaces
The purpose of this paper is the study of dimension of hypervector spaces. In this regard first we introduce the notions of linear independent (resp. dependent) and basis of hypervector spaces. Then we study the properties of hypervector spaces and prove that under certain conditions dimension for such spaces there exist. Finally, we use the fundamental relation on hypervector spaces to construct a functor from the category of hypervector spaces over a fixed field K and the category of classical vector spaces over K, and we will prove that this functor preserves dimension.