Bicyclic semigroup

About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.

Globals of Idempotent Semigroups

Abstract: If S is a semigroup, the global (or the power semigroup) of S is the set P(S) of all nonempty subsets of S equipped with a naturally defined multiplication. A class K of semigroups is globally determined if any two semigroups of K with isomorphic globals are themselves isomorphic. We study properties of globals of idempotent semigroups and show, in particular, that the class of normal bands is globally determined.

Transformation semigroups and transformed dimensions

Abstract: In the transformation semigroup (X, S) we introduce the height of a closed nonempty invariant subset of X, define the transformed dimension of nonempty subset S of X and obtain some results and relations.

On the topology of a compact inverse Clifford semigroup

Abstract: A description of the topology of a compact inverse Clifford semigroup S is given in terms of the topologies of its subgroups and that of the semilattice X of idempotents. It is further shown that the category of compact inverse Clifford semigroups is equivalent to a full subcategory of the category whose objects are inverse limit preserving functors F:X -* G, where X is a compact semilattice and G is the category of compact groups and continuous homomorphisms, and where a morphism from F:X -* G to G:Y -* G is a pair (e, w) such that e is a continuous homomorphism of X into Y and w is a natural transformation from F to Ge. Simpler descriptions of the topology of S are given in case the topology of X is first countable and in case the bonding maps between the maximal subgroups of S are open mappings. A popular topic of study in compact semigroups has been the question, for a given compact Hausdorff space, how many nonisomorphic continuous, associative multiplications of a given type will it admit? There is an older companion question, and that is, for a given algebraic structure, is there a compact Hausdorff topology which is compatible with all the operations, and if so, how many such topologies exist? It is known that the abelian groups which admit such a compact Hausdorff topology are certain products of copies of the group of rational numbers, p-adic groups, finite groups, and Z(p°°) [4, Theorem 25.25]. Butan abelian group may admit several such topologies. For example, the additive group of real numbers admits a compact /i-dimensional topology for each positive integer n. In the nonabelian case, there is the 1932 result by van der Waerden [9], in which he described a system of \"neighborhoods\" about the identity of any group, which was finer than any compact group topology for which the identity was not isolated in the set of noncentral elements. He further proved that if a group admitted a topology giving it the structure of a compact simple Lie group, then each of these algebraically defined neighborhoods was a neighborhood of the identity relative to the given topology. Thus he gave what amounted to an algebraic description of the topology of a compact simple Lie group, which had an immediate Received by the editors November 1, 1974. AMS (MOS) subject classifications (1970). Primary 22A1S.