Classification of Finite Commutative Semigroups with a Distinguished Idempotent

This study classifies finite commutative semigroups of orders two, three and four with a distinguished idempotent, where isomorphisms are required to preserve the distinguished element. For a commutative semigroup S with distinguished idempotent e, the set Se is shown to be a subsemigroup of S, with e acting as an identity on Se. The mapρ_e:S→Se, defined byρ_e (a)=ae, is also shown to be a surjective homomorphism and a retraction. To obtain the classification, all commutative multiplication tables satisfying ex = x were generated for orders two, three and four, and tested for associativity. We then reduced the associative tables under permutations of the non-distinguished elements, keeping e fixed. This produced 4, 37 and 621 associative labelled semigroups, which reduced to 4, 20 and 118 pointed isomorphism classes, respectively. We compared the resulting classes using structural properties, including the number of idempotents, the size and fibre pattern of Se, identity, zero, regularity, principal ideals, and power behaviour. The results show that specifying a distinguished idempotent provides useful structural information but does not uniquely determine the multiplication structure of a finite commutative semigroup.