On Ordered Fuzzy Points in Ordered Ternary Semigroups

Abstract

A ternary semigroup is a nonempty set $T$ together with a ternary operation $[\hspace{0.2cm}]:T\times T\times T \rightarrow T,$ written as $(a,b,c)\mapsto  [abc]$ satisfying the associative law $[[abc]uv] =[a[bcu]v]=[ab[cuv]]$ for all $a,b,c,u,v \in T.$ A partially ordered ternary semigroup $T$ is called an ordered ternary semigroup if for all $a, b, x, y \in T, a \leq b \Rightarrow [axy] \leq [bxy], [xay] \leq [xby],$ and $[xya] \leq [xyb].$ The concept of ordered ternary semigroup is a natural generalization of ordered semigroups and ternary semigroups. Let $\underline{T}$ be the set of all ordered fuzzy points in an ordered ternary semigroup $T.$ In this paper, we investigate some properties of ordered fuzzy points of ordered ternary semigroups.