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...

We prove that the semigroup operation of a topological semigroup S extends to a continuous semigroup operation on its the Stone-Čech compactification βS provided S is a pseudocompact openly factorizable space, which means that each map f : S → Y to a second countable space Y can be written as the composition f = g ◦ p of an open map p : X → Z onto a second countable space Z and a map g : Z → Y ...

It is well known that a sum (coproduct) of a family {Xi : i ∈ I} of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces Xu, indexed by the ultrafilters u on the index set I. The nature of those subspaces Xu indexed by the free ultrafilters u is not yet fully u...

Eric van Douwen [vD93] produced a maximal crowded extremally disconnected regular space and showed that its Stone-Čech compactification is an at most two-to-one image of βN. We prove that there are non-homeomorphic such images. We also develop some related properties of spaces which are absolute retracts of βN expanding on earlier of work in [BBS94, Sim87].

We study the range transformations 0p(/fD, Reß) and Op(/fD, B) for Banach function algebras A and B. As a special instance, the harmonicity of functions in Op(/fD, Re A) for a nontrivial function algebra A is established and is compared with previous investigations of Op( An, A) and Op((Re/l);, (Re/1)) for an interval /. In §2 we present some results on Op(AD, B) and use them to show that funct...

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...

Let β(N) denote the Stone–Čech compactification of the set N of natural numbers (with the discrete topology), and let N∗ denote the remainder β(N)−N. We show that, interpreting modal diamond as the closure in a topological space, the modal logic of N∗ is S4 and that the modal logic of β(N) is S4.1.2.