Note on the number of doppelsemigroups of small order

Volodymyr M. Gavrylkiv

Note on the number of doppelsemigroups of small order

Group Theory 2026-06-27 v1

Authors: Volodymyr M. Gavrylkiv

Abstract

We study doppelsemigroups, i.e., algebraic structures equip\-ped 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 \texttt{GAP}, \texttt{Python}, and \texttt{C++}.

Keywords

doppelsemigroups classification duality isomorphism computer-assisted enumeration

Cite

@article{arxiv.2606.28908,
  title  = {Note on the number of doppelsemigroups of small order},
  author = {Volodymyr M. Gavrylkiv},
  journal= {arXiv preprint arXiv:2606.28908},
  year   = {2026}
}