Classification of Finite Preordered Semigroups via Group Action induced by Pushforward

We investigate finite semigroups equipped with a preorder relation that is compatible with the semigroup operation. We introduce the concept of a compatible preorder and define the associated structure of a preordered semigroup, which captures the interaction between the semigroup operation and order-theoretic properties. To classify such structures, 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, and the orbits of this action correspond exactly to isomorphism classes of preordered semigroups. To address anti-isomorphism, we incorporate the converse relation, which allows us to identify preorders equivalent up to inversion, effectively reducing the computational complexity. By considering the quotient of Pre(S) modulo, the converse relation, and extending the action via pushforward to the equivalence classes, we obtain a systematic approach to enumerate preordered semigroups up to both isomorphism and anti-isomorphism. Using known classifications of semigroups of a small order, together with exhaustive generation of compatible preorders, our method provides a complete enumeration of preordered semigroups up to an order of five. Finally, this framework opens a computational window toward the study of more complex algebraic systems, notably EL-hyperstructures arising in hypercompositional algebra, where preordered semigroups serve as a fundamental tool for their construction.