A characterization of regular, intra-regular, left quasi-regular and semisimple hypersemigroups in terms of fuzzy sets

We prove that an hypersemigroup H is regular if and only, for any fuzzy subset f of H, we have f f ◦ 1 ◦ f and it is intra-regular if and only if, for any fuzzy subset f of H, we have f 1 ◦ f ◦ f ◦ 1. An hypersemigroup H is left (resp. right) quasi-regular if and only if, for any fuzzy subset f of S we have f 1 ◦ f ◦ 1 ◦ f (resp. f f ◦ 1 ◦ f ◦ 1) and it is semisimple if and only if, for any fuzzy subset f of S we have f 1 ◦ f ◦ 1 ◦ f ◦ 1. The characterization of regular and intra-regular hypersemigroups in terms of fuzzy subsets are very useful for applications. 2010 AMS Subject classification. 20N99 (20M99, 08A72)