Topological semigroup

About: Topological semigroup is a research topic. Over the lifetime, 346 publications have been published within this topic receiving 2877 citations.

Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups

Abstract: Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map ( a , b ) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G ) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC ( G ) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f e C(G ) such that the map a → l a f of G into C(G ) is continuous when C(G ) has a norm topology, where ( l a f )( x ) = f (ax) (a, x e G ). Then LUC ( G ) is a closed subalgebra of C(G ) invariant under translations. Furthermore, if m e LUC ( G )*, f e LUC ( G ), then the function is also in LUC ( G ). Hence we may define a product for n, m e LUC( G )*. LUC ( G )* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l ∞ ( G )* = l 1 ( G )** to LUC ( G )*. We refer the reader to [ 1 ] and [ 10 ] for more details.

On the closure of the bicyclic semigroup

Abstract: 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.

Profinite semigroups and applications

Abstract: Profinite semigroups may be described briefly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which for pseudovarieties play the role of free algebras in the theory of varieties. Combinatorial problems on rational languages translate into algebraic-topological problems on profinite semigroups. The aim of these lecture notes is to introduce these topics and to show how they intervene in the most recent developments in the area.