A class of semihypergroups connected to preordered weak Γ-semigroups

AbstractWe introduce the concept of weak Γ-semigroups as a generalization of Γ-semigroups. Using preordered weak Γ-semigroups, we obtain a class of semihypergroups and we analyze them in this paper. A connection between morphisms of semihypergroups associated with preordered Γ-semigroups and morphisms of preordered structures is also investigated

On Rough Sets and Hyperlattices

In this paper, we introduce the concepts of upper and lower rough hyper fuzzy ideals (filters) in a hyperlattice and their basic properties are discussed. Let $\theta$ be a hyper congruence relation on $L$. We show that if $\mu$ is a fuzzy subset of $L$, then $\overline{\theta}()=\overline{\theta}()$ and $\overline{\theta}(\mu^*) =\overline{\theta}((\overline{\theta}(\mu))^*)$, where is the least hyper fuzzy ideal of $L$ containing $\mu$ and \mu^*(x) = sup\{\alpha \in [0, 1]: x \in I( \mu_{\alpha} )\}$$ for all $x \in L$. Next, we prove that if $\mu$ is a hyper fuzzy ideal of $L$, then $\mu$ is an upper rough fuzzy ideal. Also, if $\theta$ is a $\wedge-$complete on $L$ and $\mu$ is a hyper fuzzy prime ideal of $L$ such that $\overline{\theta}(\mu)$ is a proper fuzzy subset of $L$, then $\mu$ is an upper rough fuzzy prime ideal. Furthermore, let $\theta$ be a $\vee$-complete congruence relation on $L$. If $\mu$ is a hyper fuzzy ideal, then $\mu$ is a lower rough fuzzy ideal and if $\mu$ is a hyper fuzzy prime ideal such that $\underline{\theta}(\mu)$ is a proper fuzzy subset of $L$, then $\mu$ is a lower rough fuzzy prime ideal