Zhuchok A. Structural properties of dimonoids

It is proved that a system of axioms of a dimonoid is independent and Cayley's theorem for semigroups has an analog in the class of dimonoids. The notion of a semiretraction of a dimonoid is defined and studied and some applications of semiretractions are given. Some least congruences on dimonoids are presented. All semilattice congruences on an arbitrary dimonoid are described. In terms of dibands of subdimonoids decompositions of dimonoids are described. The structure of an arbitrary diband of subdimonoids is characterized. Several relatively free dimonoids are constructed. The basic types of decompositions of some relatively free dimonoids are described. The description of seventeen the least congruences on the free dimonoid such that the corresponding quotient dimonoids are relatively free dimonoids is obtained. The new examples of trioids are constructed, in particular, a trioid which is isomorphic to the free trioid of rank 1. A trioid with a commutative periodic semigroup is characterized. Some least congruences on trioids are presented. Properties of the corresponding quotient trioids are described. The translational hulls of an orthogonal sum of arbitrary monoids and a Clifford semigroup are investigated. The automorphism group of an orthogonal sum of orthogonal indecomposable semigroups, the automorphism group of a free product of п-regular semigroups and the automorphism group of an arbitrary semigroup are characterized.