Some algebraic universal semigroup compactifications

Universal compactifications of semitopological semigroups with respect to the properties satisfying the varieties of semigroups and groups are studied through two function algebras.

Separating Points of βℕ By Minimal Flows

We consider minimal left ideals L of the universal semigroup compactification of a topological semigroup S. We show that the enveloping semigroup of L is homeomorphically isomorphic to if and only if given q ≠ r in , there is some p in the smallest ideal of with qp ≠ rp. We derive several conditions, some involving minimal flows, which are equivalent to the ability to separate q and r in this fashion, and then specialize to the case that S = , and the compactification is . Included is the statement that some set A whose characteristic function is uniformly recurrent has .

Universal varieties of semigroups

A category V is called universal (or binding) if every category of algebras is isomorphic to a full subcategory of V. The main result states that a semigroup variety V is universal if and only if it contains all commutative semigroups and fails the identity xnyn = (xy)n for every n ≥ 1. Further-more, the universality of a semigroup variety V is equivalent to the existence in V of a nontrivial semigroup whose endomorphism monoid is trivial, and also to the representability of every monoid as the monoid of all endomorphisms of some semigroup in V. Every universal semigroup variety contains a minimal one with this property while there is no smallest universal semigroup variety.

A universal semigroup

J.H. Michael recently proved that there exists a metric semigroup U such that every compact metric semigroup with property P is topologically isomorphic to a subsemigroup of U; where a semigroup S has property P if and only if for each x, y in S, x ≠ y, there is a z in S such that xs ≠ yz or zx ≠ zyA stronger result is proved here more simply. It is shown that for any set A of metric semigroups there exists a metric semigroup U such that each S in A is topologically isomorphic to a subsemigroup of U. In particular this is the case when A is the class of all separable metric semigroups, or more particularly the class of all compact metric semigroups.

A universal semigroup

In [4] S. Ulam asks the following question. ‘Does there exist a universal compact semigroup; i.e., a semigroup U such that every compact topological semigroup is continuously isomorphic to a subsemigroup of it?’ The author has not been able to answer this question. However, in this paper, a proof is given for the following related result.Let Q denote the Hilbert cube of countably infinite dimension and C(Q) the Banach space of continuous real-valued functions on Q with the usual norm. Let U denote the semigroup consisting of all bounded linear operators T: C(Q)→ C(Q) with ∥T∥ ≦ 1 and let U be endowed with the strong topology. Then, for every compact metric semigroup S with the property: (1.1) for all x, y ∈ S, with x ≠ y, there exists a z ∈ S, such that xz ≠ yz or zx ≠ zy;there exists a 1 − 1 mapping φ of S into U such that φ is both a semigroup isomorphism and a homeomorphism.U is metrizable, but is not compact; hence it does not provide an answer to the question of Ulam. The proof of the above statement leans heavily on a result of S. Kakutani [1].