On Rough Sets and Hyperlattices - Ali Akbar Estaji; Fereshteh Bayati (Ratio Mathematica, 2018)
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 $θ$ be a hyper congruence relation on $L$. We show that if $μ$ is a fuzzy subset of $L$, then $\overline{θ}(<μ>)=\overline{θ}(<\overline{θ}(μ)>)$ and $\overline{θ}(μ^*) =\overline{θ}((\overline{θ}(μ))^*)$, where $<μ>$ is the least hyper fuzzy ideal of $L$ containing $μ$ and $$μ^*(x) = sup\{α \in [0, 1]: x \in I( μ_{α} )\}$$ for all $x \in L$. Next, we prove that if $μ $ is a hyper fuzzy ideal of $L$, then $μ$ is an upper rough fuzzy ideal. Also, if $θ$ is a $\wedge-$complete on $L$ and $μ $ is a hyper fuzzy prime ideal of $L$ such that $\overline{θ}(μ)$ is a proper fuzzy subset of $L$, then $μ$ is an upper rough fuzzy prime ideal. Furthermore, let $θ$ be a $\vee$-complete congruence relation on $L$. If $μ $ is a hyper fuzzy ideal, then $μ$ is a lower rough fuzzy ideal and if $μ $ is a hyper fuzzy prime ideal such that $\underline{θ}(μ)$ is a proper fuzzy subset of $L$, then $μ$ is a lower rough fuzzy prime ideal.