On matrix representations of oversemigroups of semigroups generated by mutually annihilating 2-potent and 2-nilpotent elements

Among the old results, there are only some results on the representation type of semigroups, namely, for a finite quite simple semigroup (I. S. Ponizovsky) and some semigroups of all transformations of a finite set (I. S. Ponizovsky, C. Ringel); these papers were discussed on finite representation type. If we talk about new results, and even for semigroup classes, then it should be noted works on representations of the semigroups generated by idempotents with partial zero multiplication (V. M. Bondarenko, O. M. Tertychna), semigroups generated by the potential elements (V. M. Bondarenko, O. V. Zubaruk) and representations of direct products of the symmetric second-order semigroup (V. M. Bondarenko, E. M. Kostyshyn). Such semigroups can have both a finite and infinite representation type. V. M. Bondarenko and Ja. V. Zatsikha described representation types of the third-order semigroups over a field, and indicate the canonical form of the matrix representations for any semigroup of finite representation type. This article is devoted to the study of similar problems for oversemigroups of commutative semigroups.