Hypersemigroups and Fuzzy Hypersemigroups (European Journal of Pure and Applied Mathematics, 2017)
The aim is to show that the theory of hypersemigroups and the theory of fuzzy hypersemigroups are parallel to each other, in the following sense: An hypersemigroup$H$ is intra-regular, for example, if and only if $A\cap B\subseteq B*A$ for every right ideal $A$ and every left ideal $B$ of $H$. And an hypersemigroup$H$ is intra-regular if and only if $f\wedge g\preceq g° f$ for every fuzzy right ideal $f$ and every fuzzy left ideal $g$ of $H$. An hypersemigroup$H$ is left quasi-regular if and only if $A\cap B\subseteq A*B$ for every ideal $A$ and every nonempty subset $B$ of $H$. And an hypersemigroup$H$ is left quasi-regular if and only if $f\wedge g\preceq f° g$ for every fuzzy ideal $f$ and every fuzzy subset $g$ of $H$.
Keywords: Hypersemigroup; right (left) ideal; bi-ideal; fuzzy right (left) ideal; fuzzy bi-ideal; regular; intra-regular; left (right) quasi-regular; semisimple