On Definability of Hypergraphs by Their Semigroups of Homomorphisms

Abstract

In studying mathematical objects on the basis of their mapping systems considerable attention is given to the endomorphism semigroups and the partial endomorphisms semigroups of the investigated objects. It is natural to apply the same attention to systems of partial homomorphisms of these objects. However, such systems were rarely considered, since in these situations the usual composition of binary relations cannot be used and it is necessary to consider such mapping systems together with other operations. Experience shows that it is possible to consider the operations of right restrictive multiplication and left restrictive multiplication introduced by Wagner in [14] in the setting of operations for systems of partial homomorphisms. For example, Schein [11] and Molchanov [8], [9] showed that in individual cases systems of partial homomorphisms endowed with the restrictive multiplications have simple structure and contain important information about the investigated objects. This paper continues investigations in this direction. We describe here a wide class of hypergraphs which are determined up to isomorphism by their restrictive semigroups of partial homomorphisms. We also apply this result in analysing some problems of endomorphism semigroups of hypergraphs. The main result of the paper — theorem 3.1 — was announced in [7]. Note also that results of this paper generalize and correct a series of statements formulated without proofs in the end of the survey article [10]. The author wishes to thank Professor Boris M. Schein and the referee of this paper for their numerous valuable suggestions.