### Hypersemigroups and Fuzzy Hypersemigroups

#### Abstract

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 $f\wedge g\preceq g\circ f$ 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 $f\wedge g\preceq f\circ g$ 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