Interassociativity and three-element doppelsemigroups

Abstract

In the paper we characterize all interassociates of some non-inverse semigroups and describe up to isomorphism all three-element (strong) doppelsemigroups and their automorphism groups. We prove that there exist 75 pairwise non-isomorphic three-element doppelsemigroups among which 41 doppelsemigroups are commutative. Non-commutative doppelsemigroups are divided into 17 pairs of dual doppelsemigroups. Also up to isomorphism there are 65 strong doppelsemigroups of order 3, and all non-strong doppelsemigroups are not commutative.