A CONSTRUCTION OF A SEMITOPOLOGICAL SEMIGROUP OF SOFT ULTRAFILTERS

In this paper, we construct a semitopological semigroup consisting entirely of soft ultrafilters.

When a locally compact monothetic semigroup is compact

A semigroup endowed with a topology is monothetic if it contains a dense monogenic subsemigroup. A semigroup (group) endowed with a topology is semitopological (quasitopological) if the translations (the translations and the inversion) are continuous. If S is a nondiscrete monothetic semitopological semigroup, then the set {S^{\prime}} of all limit points of S is a closed ideal of S. Let S be a locally compact nondiscrete monothetic semitopological semigroup. We show that (1) if the translations of {S^{\prime}} are open, then {S^{\prime}} is compact, and (2) if {S^{\prime}} can be topologically and algebraically embedded in a quasitopological group, then {S^{\prime}} is a compact topological group.

On locally compact semitopological O-bisimple inverse ω-semigroups

We describe the structure of Hausdorff locally compact semitopological O-bisimple inverse ω- semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological O-bisimple inverse ω-semigroup with a compact maximal subgroup is either compact or it is a topological sum of its H-classes. We describe the structure of Hausdorff locally compact semitopological O-bisimple inverse ω-semigroups with a monothetic maximal subgroups. We show the following dichotomy: a T1 locally compact semitopological Reilly semigroup (B(Z+, θ)0, τ) over the additive group of integers Z+, with adjoined zero and with a non-annihilating homomorphism is either compact or discrete. At the end we establish some properties of the remainder of the closure of the discrete Reilly semigroup B(Z+, θ) in a semitopological semigroup.

Reductive compactifications of semitopological semigroups

We consider the enveloping semigroup of a flow generated by the action of a semitopological semigroup on any of its semigroup compactifications and explore the possibility of its being one of the known semigroup compactifications again. In this way, we introduce the notion ofE-algebra, and show that this notion is closely related to the reductivity of the semigroup compactification involved. Moreover, the structure of the universalEℱ-compactification is also given.

Asymptotic behaviour for an almost-orbit of nonexpansive semigroups in Banach spaces

In this paper, by using the technique of product nets, we are able to prove a weak convergence theorem for an almost-orbit of right reversible semigroups of nonexpansine mappings in a general Banach space X with Opial's condition. This includes many well known results as special cases. Let C be a weakly compact subset of a Banach space X with Opial's condition. Let G be a right reversible semitopological semigroup,  = {T (t): t ∈ G} a nonexpansive semigroup on C, and u (·) an almost-orbit of . Then {u (t): t ∈ G} is weakly convergent (to a common fixed point of ) if and only if it is weakly asymptotically regular (that is, {u (ht) − u (t)} converges to 0 weakly for every h ∈ G).

Recapturing semigroup compactifications of a group from those of its closed normal subgroups

We know that ifSis a subsemigroup of a semitopological semigroupT, and𝔉stands for one of the spaces𝒜𝒫,𝒲𝒜𝒫,𝒮𝒜𝒫,𝒟orℒ𝒞, and(ϵ,T𝔉)denotes the canonical𝔉-compactification ofT, whereThas the property that𝔉(S)=𝔉(T)|s, then(ϵ|s,ϵ(S)¯)is an𝔉-compactification ofS. In this paper, we try to show the converse of this problem whenTis a locally compact group andSis a closed normal subgroup ofT. In this way we construct various semigroup compactifications ofTfrom the same type compactifications ofS.