In this article, we establish necessary and sufficient condition on a topological Clifford semigroup to be a semilattice of topological groups. As a consequence, we show that a topological Clifford...

Semilattice of topological groups

Introduction to semigroup theory

background

The theory of topological semigroups

In his article [2] Wallace mentions the following problem: let X be an algebraic group with a locally compact Hausdorff topology such that the map of XXX into X which takes (x, y) into xy for all x, yEX is continuous. Then is X a topological group? The purpose of this note is to answer this, question in the affirmative. Lemma 1 is an immediate consequence of the continuity of multiplication, and the proof of Lemma 2 appears in [1]. The proofs of these lemmas will therefore be omitted.

/pdf/a-note-on-the-continuity-of-the-inverse-tjcpmty25e.pdf

A note on the continuity of the inverse

In ?I, two properties of T are established which hold for arbitrary S; namely, that B is a discrete open subspace of T and T\B is an ideal of T if it is nonvoid. In ?11, we introduce the notion of a topological inverse semigroup and establish several properties of such objects. Some questions are posed. In ?111, it is shown that if S is a topological inverse semigroup, then T\B is a group with a dense cyclic subgroup. ?IV contains a description of three examples of a topological semigroup which contains B as a dense proper subsemigroup. Finally, in ?V, we assume that S is a locally compact topological inverse semigroup and show that either B is closed in S or T is isomorphic with the last of the examples described in ?IV. A corollary about homomorphisms from B into a locally compact topological inverse semigroup is obtained which generalizes a result due to A. Weil [1, p. 96] concerning homomorphisms from the integers into a locally compact group. All spaces are topological Hausdorff in this paper. We state the definitions of Green's equivalence relations in a semigroup and the definition of an inverse semigroup. Green's relations S, 9, a and 9 on a semigroup S are defined by: agtb if and only if a u aS=b u bS, afb if and only if a u Sa=b u Sb, 4=S n 9 and 9=' o M. The notations Ra, La, Ha, and Da stand for the appropriate equivalence class of a in S. A semigroup S is an inverse semigroup provided each element x of S has a unique inverse; that is, an element x - 1 of S such that xx - lx = x and x - lxx - 1 = x - 1. For details about inverse semigroups and Green's relations, see [2]. We assume a certain familiarity with these notions.

/pdf/on-the-closure-of-the-bicyclic-semigroup-1id126ex82.pdf

Semilattice of topological groups

Citations

Introduction to semigroup theory

References

Introduction to semigroup theory

The theory of topological semigroups

A note on the continuity of the inverse

On the closure of the bicyclic semigroup

Related Papers (5)

Embedding topological semigroups into the hyperspaces over topological groups

Embedding topological semigroups in topological groups

A note on the embedding of topological semigroups

On topological Clifford semigroups embeddable into products of cones over topological groups

TheL ∞-representation algebra of a foundation topological semigroup