Bicyclic semigroup

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

Fundamental Representations for Classes of Semigroups Containing a Band of Idempotents

Abstract: The construction by Hall of a fundamental orthodox semigroup W B from a band B provides an important tool in the study of orthodox semigroups. Hall's semigroup W B has the property that a semigroup is fundamental and orthodox with band of idempotents isomorphic to B if and only if it is embeddable as a full subsemigroup into W B . The aim of this article is to extend Hall's approach to some classes of nonregular semigroups. From a band B, we construct a semigroup U B that plays the role of W B for a class of weakly B-abundant semigroups having a band of idempotents B. The semigroups we consider, in particular U B , must also satisfy a weak idempotent connected condition. We show that U B has subsemigroup V B where V B satisfies a stronger notion of idempotent connectedness, and is again the canonical semigroup of its kind. In turn, V B contains W B as its subsemigroup of regular elements. Thus we have the following inclusions as subsemigroups: either of which may be strict, even in the finite case. The ex...

On the closure of the extended bicyclic semigroup

Abstract: In the paper we study the semigroup CZ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup CZ and prove that every non-trivial congruence C on the semigroup CZ is a group congruence, and moreover the quotient semigroup CZ=C is isomorphic to a cyclic group. Also we show that the semigroup CZ as a Hausdor semitopological semigroup admits only the discrete topology. Next we study the closure clT (CZ) of the semigroup CZ in a topological semigroup T. We show that the non-empty remainder of CZ in a topological inverse semigroup T consists of a group of units H(1T) of T and a two-sided ideal I of T in the case when H(1T) 6 ? and I 6 ?. In the case when T is a locally compact topological inverse semigroup and I 6 ? we prove that an ideal I is topologically isomorphic to the discrete additive group of integers and describe the topology on the subsemigroup CZ [ I. Also we show that if the group of units H(1T) of the semigroup T is non-empty, then H(1T) is either singleton or H(1T) is topologically isomorphic to the discrete additive group of integers.

Regular ordered semigroups and intra-regular ordered semigroups in terms of fuzzy subsets

Abstract: Let $S$ be an ordered semigroup. A fuzzy subset of $S$ is anarbitrary mapping from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper, the concept of fuzzygeneralized bi-ideals of an ordered semigroup $S$ is introduced.Regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.Finally, two main theorems which characterize regular orderedsemigroups and intra-regular ordered semigroups in terms of fuzzyleft ideals, fuzzy right ideals, fuzzy bi-ideals or fuzzyquasi-ideals are given. The paper shows that one can pass fromresults in terms of fuzzy subsets in semigroups to orderedsemigroups. The corresponding results of unordered semigroups arealso obtained.