Towards the enumeration of EL-semihypergroups

We investigate finite semigroups equipped with a preorder relation that is compatible with the semigroup operation and we develop an enumeration methodology based on automorphism group actions. For a fixed finite semigroup S, the automorphism group Aut(S) acts naturally on the set Pre(S) of compatible preorders via pushforward; the orbits of this action correspond exactly to isomorphism classes of preordered semigroups. Using known classifications of semigroups of small order, together with exhaustive generation of compatible preorders, our method yields a complete enumeration of preordered semigroups up to order five.  Finally, this framework opens a computational window toward the study of more complex algebraic systems, particularly EL-hyperstructures arising in hypercompositional algebra, where preordered semigroups serve as a fundamental tool for their construction.

ZOOM link: https://upr-si.zoom.us/j/65127426860?pwd=kYjS6yFt6yeUcYAGGPcnJShbb5UI7r.1