Про матричнi зображення наднапiвгруп напiвгрупи, породженої двома взаємно анульовними iдемпотентами

Abstract

Matrix representations of finite semigroups over fields are studied not so well as for finite groups. Representations of finite groups over fields are studied sufficiently well; in particular, the criterions of representation type are fully defined for an arbitrary field. If the characteristic $p$ of a field $K$ does not divide the order of a group (classical case), then the group has, up to equivalence, finite number of non-decomposable representations; such group is called a group of finite representation type. If the characteristic $p$ divides the order of a group (modular case), then the group has finite representation type only if its Sylow $p$-subgroup is cyclic. In this case for most finite groups the problems of describing their representations includes the problem on classification, up to similarity, of the pairs of matrices. Such groups are called wild, and groups that allow explicit descriptions of representations are called tame. The tame and wild groups in modular case are fully described by the first author together with Yu. A. Drozd. In the theory of representations of semigroups, the largest number of works are devoted to irreducible representations. Among the old results, there are only some results on semigroups of finite representation type, 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). In cases, when the numbers of non-decomposable representations is infinite, the most famous are the results from the theory of representations of algebras that can be reformulated in terms of representations of semigroups: the description of representations of the algebra $ $ (I. M. Gelfand, V. A. Ponomarev and L. O. Nazarova, A. V. Roiter, V. V. Sergeichuk, V. M. Bondarenko) and the algebra $ $ (V. M. Bondarenko and C. Ringel). If we are not talking about individual semigroups, but about semigroup classes, then it should be noted works about on representations of the semigroups generated by idempotents with partial zero multiplication (V. M. Bondarenko, O. M. Tertychna), representations of the Rees semigroups (S. M. Dyachenko), 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.