Note on the number of doppelsemigroups of small order

We study doppelsemigroups, i.e., algebraic structures equipped with two associative binary operations satisfying a specified system of axioms. We investigate duality and isomorphisms of doppelsemigroups and examine the relationships between commutative, abelian, strong, and rectangular doppelsemigroups. Several examples are constructed, including nontrivial iso-opposite doppelsemigroups, noncommutative iso-dual doppelsemigroups, nonabelian iso-cross-dual doppelsemigroups, and nonstrong rectangular iso-opposite doppelsemigroups. Furthermore, we refine the complete classification of nonisomorphic doppelsemigroups of order 3. Finally, we present computer-assisted calculations yielding the numbers of all pairwise nonisomorphic doppelsemigroups and strong doppelsemigroups of orders up to 5, as well as all pairwise nonisomorphic commutative, abelian, and rectangular doppelsemigroups of orders up to 6, obtained using GAP, Python, and C++.