We introduce the notion of diagonal ranks of semigroups, which are numerical characteristics of semigroups. Some base properties of diagonal ranks are obtained. A new criterion for a monoid being a group is obtained using diagonal ranks. For some semigroup classes we investigate whether their diagonal acts are finitely generated or not. For the semigroups of full transformations, partial transf...

In “Sverdlovsk notebook” (Sverdlovsk, 1969), I proposed a question: Are any too first-order equivalent finitely generated commutative semigroups isomorphic? In 1970, B.I.Zilber answered the question negatively. A question arises: In what language, any equivalent over the language finitely generated commutative semigroups are isomorphic? In the note, we propose such a language. Moreover, we prov...

We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map f̃ and the Lyapunov exponent of f̃ with respect to the maximal entropy measure for f̃ . Moreover, we show that the equality holds if and only if the generator...

We show that the equational theory of representable lower semilattice-ordered residuated semigroups is finitely based. We survey related results.

We introduce a series of new polynomially computable implicit operations on the class of all finite semigroups. These new operations enable us to construct a finite pro-identity basis for the pseudovariety H of all finite semigroups whose subgroups belong to a given finitely based pseudovariety H of finite groups.

We study first-order definability in the lattice L of equational theories of semigroups. A large collection of individual theories and some interesting sets of theories are definable in L . As examples, if T is either the equational theory of a finite semigroup or a finitely axiomatizable locally finite theory, then the set {T, T } is definable, where T ∂ is the dual theory obtained by invertin...

The concept of an automatic structure has been generalized from groups [ECH+] to semigroups [CRRT]. Several authors [CRR, Hof, Kam] have asked the following question: Let S be a finitely generated semigroup embeddable in a group and let G be its universal group [CP, Chapter ]. If G is automatic, must S be automatic? Examples in favour of this implication include: free groups and s...

We study inverse semigroup amalgams [S1, S2;U ], where S1 and S2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S1 and S2. We use a modified version of Bennett’s work on the structure of Schützenberger graphs of the R-classes of S1 ∗U S2 to state sufficient conditions for the amalgamated free products S1 ∗...

In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra contains a Zorder with hyperbolic unit group. In this paper we complete this classification by handling the case in which the semi...