Zhuchok Y. Quasi-orderings and structure properties of semigroups

Some types of relative representations and monotonous homomorphisms of quasiordered semigroups are described. The generalization of results of Zarethskiy about exact representations of ordered semigroups by binary relations for quasiordered semigroups is obtained. It is proved that each maximal ideal of a quasi-ordered commutative monoid is a prime ideal. The converse statement does not hold, in general. Regular representations of locally associative semigrupoids in symmetric categories are investigated. The idempotent congruences of symmetric 0-category and symmetric inverse 0-category on the arbitrary set are described. The greatest decomposition of symmetric 0-category in a left (right, orthogonal) sum of semigroups is found. The characteristics of the group of automorphisms of a symmetric inverse 0-category on a finite set is obtained.