The thesis is devoted to studying abstract properties of the endomorphism semigroups of equivalence relations, symmetric binary relations, effective and connected binary relations. Endotopism semigroups and correspondences of endomorphism semigroups of these relational systems are studied.
The thesis consists of an annotation, an introduction, three sections, conclusions, a list of publications on the topic of the thesis and additions. In the introduction the urgency of the topic of the dissertation is substantiated, the goal, objectives, object and subject of the study are determined, research methods are indicated. The scientific novelty and the theoretical and practical significance of the results are formulated as well as the author personal contribution is given. The list of seminars and conferences on which the results of the thesis have been represented is indicated, and the structure of the work is characterized.
The first section contains the necessary theoretical information from the theory of semigroups and known results that are used below. Six types of endotopisms of binary relation are defined. For an arbitrary equivalence relation an endotopism semigroup a set of semi-strong endotopisms, a set of locally strong endotopisms, a set of quasi-strong endotopisms, a strong endotopism monoid and an autotopism group are described. It is established that the endotopism semigroup, the strong endotopism monoid and the autotopism group of an equivalence relation are correspondences of the monoid of all endomorphisms of the equivalence relation. Necessary and sufficient conditions under which sets , and are correspondences of are found.
In the second section, correspondences of the endomorphism semigroup of an equivalence relation are studied. For an equivalence we define a small category such that and the morphism set consist of all mappings between any two classes of . Using the construction of the wreath product of a monoid with a small category , faithful representations of three correspondences of the semigroup of all endomorphisms of an arbitrary equivalence relation are described, namely, of the semigroup of all endotopisms, of the monoid of all strong endotopisms, and, correspondingly, of the group of all autotopisms of the given equivalence. Using properties of the minimal ideal of the endotopism semigroup it is proved that arbitrary equivalence relations are determined up to an isomorphism by their endotopism semigroups. All isomorphisms are constructed between endotopism semigroups of arbitrary equivalences. The regularity and coregularity conditions for endotopism semigroups of the given type are established.
The concept of an endotype of the binary relation with respect to its endotopisms is defined. We put where , , take a value 0 or 1. In addition, if at the -th position in the chain of inclusions corresponding sets are coincide and otherwise. A classification of all equivalences according to a value of their endotypes relative to endotopisms is obtained. All possible endotype values of an arbitrary strict partial equivalence with respect to its endomorphisms are found.
The concept of an endospectrum of the binary relation with respect to its endotypes is defined as a sequence of powers . For an arbitrary equivalence relation on a finite set the endospectrum is studied.
In the third section, we study the semigroup of endotopisms of effective and connected binary relations and the monoid of strong endotopisms of a symmetric binary relation. For binary relations that satisfy the condition of efficiency and connectivity, it is proved that the endotopism semigroup of any such relation characterizes this binary relation up to an isotopism or an antiisotopism. For symmetric binary relations of a certain class, the faithful representation of the monoid of strong endotopisms in the form of a wreath product of the strong injective endomorphism monoid and some small category is obtained.
