Канонiчнi форми матричних зображень комутативних моноїдiв четвертого порядку

Abstract

Matrix representations of finite semigroups over fields are studied not so well as for finite groups; in particular, for groups, unlike semi-groups, the criterion of tameness is obtained (V. M. Bondarenko, Yu. A. Drozd). If we talk about the description of indecomposable representations of semigroups, then you should highlight some individual results about quite simple semigroups (I. S. Ponizovsky) and semigroups of all transformations of a finite set (I. S. Poniozovsky and K. Ringel, in the case of finite representation type), and for Riesz semigroups (S. M. Dyachenko) and semigroups generated by idempotents with partial zero multiplication (V. M. Bondarenko, A. M. Tertychna, both in the cases offinite and infinite representation type). Description of the representations of the monomial algebra < a,b|ab = ba = 0 > (I. M. Gel fand, V. A. Ponomarev) and the monomial algebra < a,b|a^2 = b^2 = 0 > (V. M. Bondarenko and C. Ringel) are also naturally regardedas the results on the representations of semigroups.This paper is devoted to the finding of canonical forms of matrix representations over an arbitrary field for semigroups of small order.Semigroups of order n < 4 are studied in sufficient detail. The cases n = 1,2 are trivial. The semigroups of the order n = 3 have been described by T. Tamura, in the form of Kelli’s tables, even in 1953 (under the description traditionally refers to a description up to isomorphism and duality). The semigroups treated with such accuracy are called different.In previous papers, the authors described third-order semigroups and non-commutative fourth-order monoids, which have a finite representation type over a field. For all such semigroups, canonical forms of their matrix representations are indicated. At the same time for each semigroup a minimal system of generators and the corresponding defining relations are indicated.In this paper, similar results are obtained for commutative monoids of the fourth order.