Subsemigroups of Stone-Čech compactifications

Abstract

The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

Separating Milliken-Taylor systems in $\mathbb{Q}$

“…Notice, as was remarked in [1], getting a carry of 1 is relatively rare. For this to happen one has to have v t = w t = 0 and c t = −1.…”

“…Our coloring of Q \ {0} is based on the negative factorial representation of rational numbers introduced by Budak, Işic, and Pym in [1]. Section 2 consists of several results about the arithmetic of this representation and construction of two colorings, one of initial segments of the negative factorial representation and the other of terminal segments.…”

“…Section 2 consists of several results about the arithmetic of this representation and construction of two colorings, one of initial segments of the negative factorial representation and the other of terminal segments. In Section 3 we use those two colorings to obtain a coloring of Q which separates expressions of the form a 1 [5,Theorem 17.32] except that the one occurrence of "x ∈ N" should be replaced by "x ∈ G". …”

The set of idempotents in the weakly almost periodic compactification of the integers is not closed

“…(For more details about arithmetic in negative bases, see [5]. ) We now come to our first inequality.…”

Digital representation of semigroups and groups

“…, t + 1} if t ≥ 0. T. Budak, N. Işik, and J. Pym established this fact in [2]. They used it to show that βQ d , the Stone-Čech compactification of Q with the discrete topology, has 2 c minimal left ideals and 2 c minimal right ideals, and each maximal group in its smallest ideal contains a free group on 2 c generators.…”