Let a and b be distinct positive integers. We show that the equation u + a · p = v + b · p has no solutions with u, v ∈ βN and p ∈ βN\N. More generally, we show that if (S,+) is any commutative cancellative semigroup and S has no nontrivial solutions to n · s = n · t for n ∈ N and s, t ∈ S, then the equation u + a · p = v + b · p has no solutions with u, v ∈ βS and p ∈ βS\S. We characterize com...

If f is an autohomeomorphism of some space X, then βf denotes its Stone-Čech extension to βX. For each n ≤ ω, we give an example of a first countable, strongly zerodimensional, subparacompact X and a map f such that every point of X has an orbit of size n under f and βf has a fixed point. We give an example of a normal, zero-dimensional X such that f is fixed-point-free but βf is not. We note t...

Let fig be the Stone-Cech compactification of a group G, Aa the set of all almost periodic points in G, Ka c[U { supp eLIM(G)}] and Ra the set of all recurrent points in fiG. In this paper we will study the relationships between Ka and Ra, and between Aa and Ra. We will show that for any infinite elementary amenable group G, Aa Ra and RaKa =/= .

Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions onX which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone–Čech compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuo...

Eric van Douwen [5] produced a maximal crowded extremally disconnected regular space and showed that its Čech–Stone compactification is an at most two-to-one image of βN. We construct for any n = 3 an example of a compact crowded space Xn that is an image of βN under a map all of whose fibers have either size n or n− 1. We also show that under CH this is best possible.

Let $S$ be a semitopological semigroup. The $wap-$ compactification of semigroup S, is a compact semitopological semigroup with certain universal properties relative to the original semigroup, and the $Lmc-$ compactification of semigroup $S$ is a universal semigroup compactification of $S$, which are denoted by $S^{wap}$ and $S^{Lmc}$ respectively. In this paper, an internal construction of ...