A semigroup of operators in convexity theory

We consider three operators which appear naturally in convexity theory and determine completely the structure of the semigroup generated by them.

RESUMO. Duongrupo de operatoroj en la teorio pri konvekseco. Ni konsideras tri operatorojn kiuj aperas nature en la teorio pri konvekseco kaj plene determinas la strukturon de la duongrupo generita de ili.

     

Every semigroup is isomorphic to a transitive semigroup of binary relations

Every (finite) semigroup is isomorphic to a transitive semigroup of binary relations (on a finite set).

     

Affine semigroup rings that are complete intersections

This paper presents a result concerning the structure of affine semigroup rings that are complete intersections. It generalizes to arbitrary dimensions earlier results for semigroups of dimension less than four. The proof depends on a decomposition theorem for mixed dominating matrices.

     

Radicals of algebras graded by cancellative linear semigroups

We consider algebras over a field of characteristic zero, and prove that the Jacobson radical is homogeneous in every algebra graded by a linear cancellative semigroup. It follows that the semigroup algebra of every linear cancellative semigroup is semisimple.

     

Semigroup properties of factors in the polar decomposition or the operator De-Moivre formula

We give necessary and sufficient conditions under which the factors in the polar decomposition of a semigroup homomorphism are themselves semigroup homomorphisms.

     

The universal nilpotent group compactification of a semigroup

The purpose of this paper is to introduce an algebra of functions on a semitopological semigroup and to study these functions from the point of view of universal semigroup compactification. We show that the corresponding semigroup compactification of this algebra is universal with respect to the property of being a nilpotent group.

     

Automorphisms of the endomorphism semigroup of a free monoid or a free semigroup

We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or ``mirror inner". In particular, we answer a question of B. I. Plotkin.

     

Decidability of the representation extension property for finite semigroups

We prove that the decision problem of whether or not a finite semigroup has the representation extension property is decidable.     

Arbitrary vs. regular semigroups

The notion of regularity for semigroups is studied, and it is shown that an unambiguous semigroup (i.e., whose $\text{L}$ and $\text{R}$ orders are respectively unions of disjoint trees) can be embedded in a regular semigroup with the same subgroups and the same ideal structure (except that a zero is added to the regular semigroup).In a previous paper [1] it was shown that any semigroup is the homomorphic image of an unambiguous semigroup with the same groups and a similar ideal structure.Together these two papers thus prove that an arbitrary semigroup divides a regular semigroup with a similar structure.The resulting regular semigroup is finite (resp. torsion, or bounded torsion) if the given semigroup has that property.     

Existence of ergodic retraction for noncommutative semigroups in Banach spaces

We prove the existence of ergodic retraction for a noncommutative semigroup which is right Eberlein-weakly almost periodic in a uniformly convex Banach space.

     

Supercyclic and chaotic translation semigroups

We give a necessary and sufficient condition for a translation semigroup to be supercyclic and to be chaotic in a weighted function space.

     

An algebraic property of joinings

We show that an ergodic automorphism is semisimple if and only if the set of ergodic self-joinings is a subsemigroup of the semigroup of self-joinings.

     

The quadratic form in the Lévy-Khinchin formula on semigroups

In this paper we obtain the quadratic form in the Lévy-Khinchin formula on a commutative involutive semigroup, with a neutral element, as a sum of two simpler quadratic forms.

     

On the existence of chaotic and hypercyclic semigroups on Banach spaces

We prove that every separable infinite dimensional complex Banach space admits a hypercyclic uniformly continuous semigroup. We also prove that there exist Banach spaces admitting no chaotic strongly continuous semigroups.

     

The sum of two locally nilpotent rings may contain a free non-commutative subring

A family of examples of semigroup algebras is constructed each of which is a sum of two locally nilpotent subalgebras but yet contains a free subalgebra which is freely generated by two elements.

     

Stability and almost periodicity of asymptotically dominated semigroups of positive operators

We discuss conditions such that strong stability and strong asymptotic compactness of a (discrete or continuous) semiflow defined on a subset in the positive cone of an ordered Banach space is preserved under asymptotic domination. This is used to show that on a Banach lattice with order continuous norm strong stability and almost periodicity of a (discrete or strongly continuous) semigroup of positive operators is preserved under asymptotic domination.

     

A construction of a robust family of exponential attractors

Given a dissipative strongly continuous semigroup depending on some parameters, we construct a family of exponential attractors which is robust, in the sense of the symmetric Hausdorff distance, with respect to (even singular) perturbations.

     

Prime and Semiprime Semigroup algebras of cancellative semigroups

Necessary and sufficient conditions are given for a semigroup algebra of a cancellative semigroup to be prime and semiprime. These conditions were proved necessary by Okninski; our contribution is to show that they are also sufficient. The techniques used in the proof are a new variation on the -methods which were developed originally for group algebras.

     

Further results in the theory of generalised inflations of semigroups

We determine the structure of semigroups that satisfy xyzw∈{xy,xw,zy,zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F(X) on a set X. In the latter case the semigroup must be an inflation of F(X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band.     

Archimedean ordered semigroups as ideal extensions

The main result of the paper is a structure theorem concerning the ideal extensions of archimedean ordered semigroups. We prove that an archimedean ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if an ordered semigroup S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean. As a consequence, an ordered semigroup is archimedean and contains an idempotent if and only if it is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup.