Zatsikha Y. Representations of semigroups of small orders

The thesis is devoted to the study of semigroups of the order of less than five and their matrix representations. The first chapter is written out Kelly’s tables of all (up to isomorphism and duality) semigroups of order less than four, basic initial information is given on the theory of representations of semigroups and partially ordered sets, and formulate a basic classification problem for bundles of chains. In the second chapter, in terms of generators and determining relations, one describes all third order semigroups and the set of seven properties that is characteristic for the class of all third order semigroups. It is proved that the set of common properties P3(7) consisting of properties P(C): commutative; P(1): existence of an identity element; P(0): existence of a null element; P+(0): existence of an attached null element; Pid(1): the number of idempotents is 1; Pid(2): the number of idempotents is 2; Pgen(2): the smallest number of generators is 2, is the minimal characteristic complete set of properties for the class of all semigroups of order 3. In the third chapter one studies matrix representations of the third order semigroups. The criteria of representation type of such semigroups are proved. In particular, it is proved that all third order semigroups have tame type. In the case of the finite type, the canonical form of the matrix representations are specified. All (up to equivalence) indecomposable representations of the third order semigroups are described. In the fourth chapter one studies the fourth order monoids and their matrix representations. In terms of the generators and determining relations all monoids of the fourth order are described. The criteria of representation type of such monoids are proved. In particular, it is proved that all fourth order monoids have the same type. In the case of the finite type, the canonical form of the matrix representations are specified. All (up to equivalence) indecomposable representations of the fourth order monoids are described.