On dimonoids of small order

This paper is devoted to the study of dimonoids, i.e., algebraic structures equipped with two associative binary operations satisfying a given system of axioms.

We introduce several new classes of dimonoids and determine their automorphism groups and halos.

In particular, we construct examples of iso-dual nonabelian dimonoids, as well as nonabelian dimonoids with nonempty halo.

Using these constructions, we obtain a complete classification, up to isomorphism, of all nonabelian noncommutative nontrivial dimonoids of order 3, thereby resolving the problem concerning this classification.

Consequently, all three-element dimonoids are classified up to isomorphism, yielding exactly 52 pairwise nonisomorphic dimonoids of order 3.

Finally, we present the results of computer computations determining the numbers of all pairwise nonisomorphic dimonoids of orders up to 5, as well as all pairwise nonisomorphic commutative, abelian, and rectangular dimonoids of orders up to 6, obtained using GAP, Python, and C++.