Desiateryk O. Structure of variants of semigroups

Study of the structure of variants of semigroups and their groups of automorphisms is an important task of the abstract theory of semigroups. Lyapin began to study the properties, further such semigroups have been studied by various authors, in particular Chase, Rees, Hickey, Lawson, Gutik, Zhuchok and others. The dissertation is devoted to study of structure of variants of matrix Rees semigroups, commutative bands with zero, semilattices, Brandt semigroups and finite power set automorphism groups, lattices of subspaces, and lattices of finite partitions of a finite set. The main novelty of the dissertation is proving the criterion of isomorphism of variants of a commutative band with zero, establishing the criterion of isomorphic of variants of lattices of partitions of a countable set. For the first time was proved a pairwise isomorphism of all regular variants of the Rice matrix semigroup M0(G; n, m; P), and obtained criterion of regularity of the variant of the Rees semigroup and the isomorphism criterion of variants of the matrix Rees matrix semigroup over a trivial group with zero. It is also important that it is proved that Brandt’s finite semigroup is not a variant of any semigroup. Also, for the first time were investigated the automorphisms groups of the variants of a power set of a finite set, it is proved that this group of automorphisms is isomorphic to wreaths product of two symmetrical groups of pransformation, the first of which acts on subsets. Also groups of automorphisms of variants of lattice subspaces n-dimensional vector space over the field Fq were studied, it is isomorphic to the generalized wreath product, the first multiplier of it is the wreath product of groups of automorphisms of lattices of subspaces and the second is determined by some set of symmetric groups. Also groups of automorphisms of the lattice variant partition of a finite set were studied.