There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick , central , syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in (N,+) any central set may be partitioned into infinitely many...

Introduction. The object of this paper is to characterize the clans (compact connected Hausdorff topological semigroups with an identity element) which are homeomorphic to a unit interval and which have a nondegenerate kernel (minimal two-sided ideal). The corresponding case when the kernel is degenerate has been characterized in a paper by H. Cohen and L. I. Wade [2] together with an earlier p...

The main purpose of this paper is to establish a relation between universality of certain P-compactifications of a semitopological semigroup and their corresponding enveloping semigroups. In particular, we show that if we take P to be the property that the enveloping semigroup of a compactification of a semitopological semigroup s is left simple, a group, or the trivial singleton semigroup, t...

We demonstrate that categories of continuous actions topological monoids on discrete spaces are Grothendieck toposes. exhibit properties these toposes, giving a solution to the corresponding Morita-equivalence problem. characterize toposes in terms their canonical points. identify natural classes representatives with good properties, 'powder monoids' and then 'complete monoids', for monoids. Fi...

Contents Chapter 1. Introduction 5 Chapter 2. Internal measure-theoretic phenomena 13 1. Amenable groups and nuclearity 13 2. Amenable actions, nuclearity, and exactness 16 3. The type semigroup, invariant measures, and pure infiniteness 21 4. The universal minimal system 29 5. Minimal actions, pure infiniteness, and nuclearity 31 Chapter 3. External measure-theoretic phenomena 33 1. Sofic grou...

In this paper, we introduce three notions of topological entropy a free semigroup action generated by proper maps for noncompact subsets, which extends the defined Ju et al. [13] and Ma [17]. By using one-point compactification as bridge, study relations entropies between two dynamical systems. We then skew-product transformations, particular subset, relationship upper capacity maps, transforma...

We show the existence of an infinite monothetic Polish topological group G with the fixed point on compacta property. Such a group provides a positive answer to a question of Mitchell who asked whether such groups exist, and a negative answer to a problem of R. Ellis on the isomorphism of L(G), the universal point transitive G-system (for discrete G this is the same as PG the Stonetech compacti...

Let A" be a topological space. We recall that D, a subset of X is called a C-set if any continuum which meets D and its complement must contain D. Let 5 be a continuum which is a topological semigroup with identity 1, and let H denote the maximal subgroup of 5 containing 1. It is well known that H exists and is compact. The following four conjectures have been raised and shown to be equivalent ...