We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product [X × H × Y ]σ of a topological group H over topological spaces X, Y with a continuous sandwich function σ : Y × X → H. We prove that a simple topological semigroup S is a topological paragroup if one of the following conditions is satisfied: (1) S is completely simple and the maximal subgrou...

Chain conditions, finiteness conditions, growth conditions, and other forms of finiteness, Noetherian rings and Artinian rings have been systematically studied for commutative rings and algebras since 1959. In pursuit of the deeper results of ideal theory in ordered groupoids (semigroups), it is necessary to study special classes of ordered groupoids (semigroups). Noetherian ordered groupoids (...

In this paper we give a partial answer to the following question: does a large subsemigroup of a semigroup S with the finite combinatorial property finite derivation type (FDT ) also has the same property? A positive answer is given for large ideals. As a consequence of this statement we prove that, given a finitely presented Rees matrix semigroup M [S; I, J ;P ], the semigroup S has FDT if and...

This paper introduces the topological finiteness condition finite derivation type (FDT) on the class of semigroups. This notion is naturally extended from the monoid case. With this new concept we are able to prove that if a Rees matrix semigroupM[S; I, J ;P ] has FDT then the semigroup S also has FDT. Given a monoid S and a finitely presented Rees matrix semigroup M[S; I, J ;P ] we prove that ...

The aim of this paper is to classify all monogenic ternary semigroups, up to isomorphism. We divide them to two groups: finite and infinite. We show that every infinite monogenic ternary semigroup is isomorphic to the ternary semigroup O, the odd positive integers with ordinary addition. Then we prove that all finite monogenic ternary semigroups with the same index...

Exactly as in semigroups, Green’s relations play an important role in the theory of ordered semigroups—especially for decompositions of such semigroups. In this paper we deal with the -trivial ordered semigroups which are defined via the Green’s relation , and with the nil and Δ-ordered semigroups. We prove that every nil ordered semigroup is -trivial which means that there is no ordered semigr...

An ordered semigroup is a structure S = 〈S, ·,≤〉 with a binary operation · that is associative and a partial ordering ≤ that is compatible with the binary operation. For a given congruence relation θ of the semigroup S = 〈S, ·〉 the quotient structure S/θ = 〈S/θ, , 〉 is not in general an ordered semigroup. In this paper we study quotients of ordered semigroups. We first define a special type of ...

We survey results concerning automatic structures for semigroup constructions, providing references and describing the corresponding automatic structures. The constructions we consider are: free products, direct products, Rees matrix semigroups, Bruck-Reilly extensions and wreath products.

The Abelian Sandpile Model is a diffusion process on (directed) graphs, studied, under various names, in statistical physics, discrete dynamical systems, theoretical computer science, algebraic graph theory, and other fields. The model takes a directed multigraph X with a sink accessible from all nodes; associates a configuration space with X and defines transition rules between the configurati...