For a locally compact Hausdorff semigroup S, the L representation algebra R(S) was extensively studied by Dunkl and Ramirez. The FourierStieltjes algebra F (S) of a topological semigroup was studied by Lau. The aim of this paper is to investigate these two algebras and study the amenability of them with respect to the structure of S.

Let $Q$ be an inverse semigroup. A subsemigroup $S$ of $Q$ is a left I-order in $Q$ and $Q$ is a semigroup of left I-quotients of $S$ if every element $qin Q$ can be written as $q=a^{-1}b$ for some $a,bin S$. If we insist on $a$ and $b$ being $er$-related in $Q$, then we say that $S$ is straight in $Q$. We characterize semigroups which are left I-quotients of left regular bands of right cancell...

A theory of semiflows with a discrete acting topological semigroup was developed in the 2000 paper by D. Ellis, R. Ellis and M. Nerurkar ([2]). A theory for the case of an arbitrary acting topological semigroup has still to be developed. This paper can be considered as the beginning of an attempt in that direction. We discuss almost periodicity and G-almost periodicity of points in a semiflow a...

Let X, Y, Z be topological spaces. A function F :X × Y → Z is called jointly continuous if it is continuous from X × Y with the product topology to Z . It is said to be separately continuous if x 7→ F (x, y):X → Z is continuous for each y ∈ Y and y 7→ F (x, y):Y → Z is continuous for each x ∈ X . A semitopological semigroup is a semigroup S endowed with a topology such that the multiplication f...

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame” topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame ...

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame” topological space whose topology is determined by the convergence of sequences. In the latter case we say that the dynamical system is tame. We show that (i) a metric distal minimal system is tame ...

In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bis. Some examples with positive or zero Hausdorff metric entropy are given. Moreov...

Without the restriction of metrizability, topological dynamical systems (X, 〈Ts〉s∈G) are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characteri...

This paper analyses the construction of the kernel graph of a non-synchronizing transformation semigroup and introduces the inverse synchronization problem. Given a transformation semigroup S ≤ Tn, we construct the kernel graph Gr(S) by saying v and w are adjacent, if there is no f ∈ S with vf = wf . The kernel graph is trivial or complete if the semigroup is a synchronizing semigroup or a perm...

We define the notion of a torsor for an inverse semigroup, which is based on semigroup actions, and prove that this is precisely the structure classified by the topos associated with an inverse semigroup. Unlike in the group case, not all set-theoretic torsors are isomorphic: we shall give a complete description of the category of torsors. We explain how a semigroup prehomomorphism gives rise t...