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Bicyclic semigroup

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


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01 Jun 1957
TL;DR: In this article, the authors give a representation of any inverse semigroup S as a semigroup of isomorphisms between the semigroups Se, the representation is faithful if (a more general condition is given below) the center of each maximal subgroup of S is trivial.
Abstract: It is known [1; 2] that every inverse semigroup S has a faithful representation as a semigroup of (1, 1)-mappings of subsets of a set A into A. The set A may be taken as the set of elements of S and the (1, 1)-mappings as mappings of principal left ideals of S onto principal left ideals of S. If E is the set of idempotents of S then there is also a representation of S, not necessarily faithful, as a semigroup of (1, 1)-mappings of subsets of E into E [2]. If eEE denote by S. the subsemigroup eSe of S. In this note we give a representation of any inverse semigroup S as a semigroup of isomorphisms between the semigroups Se, The representation is faithful if (a more general condition is given below) the center of each maximal subgroup of S is trivial. We recall that an inverse semigroup [3] is a semigroup S in which for any a E S the equations xax = x and axa = a have a unique common solution xES called the inverse of a and denoted by a-' [5; 6]. This implies that the idempotents of S commute and that to each aGS there corresponds a pair of idempotents e, f such that aa= e, a-la =f, ea = a, af = a. The idempotents e, f are called respectively the left and right units of a. For any two elements a, bES, (ab)-'=b-1a(see [3]). Throughout what follows S will denote an inverse semigroup and E will denote its set of idempotents. If eEE then Se will denote the subsemigroup eSe of S.

5 citations

Journal ArticleDOI
TL;DR: In this paper, the character space of the Rees matrix semigroups is studied and its character amenability is investigated as an application, and the results of these results are applied to certain matrix classes.
Abstract: In this paper, for an arbitrary . Especially, we study the character space of this algebra. Then, as an application, its character amenability is investigated. Finally, we apply these results to certain semigroups, which are called Rees matrix semigroups.

5 citations

01 Jan 2012
TL;DR: In this article, the authors give some characterizations of the intra-regular ordered ternary semigroups in terms of bi-ideals and quasi-ideal, left ideals and right ideals.
Abstract: In this paper we give some characterizations of the intra-regular or-dered ternary semigroups in terms of bi-ideals and quasi-ideals, bi-idealsand left ideals, bi-ideals and right ideals of ordered ternary semigroups. Mathematics Subject Classification: 06F05, 06D72, 08A72, 20N99,06F99Keywords: Ordered ternary semigroup, right (left, bi-, quasi- ) ideal,ideal, intra-regular ordered ternary semigroup 1 Introduction In 1932, Lehmer gave the definition of ternary semigroups. A nonempty set T is calleda ternary semigroup ifthereexistsa ternaryoperation T×T×T → T ,written as ( a,b,c ) −→ abc satisfying the following identity ( ∀a,b,c,d,e ∈T )((( abc ) de )=( a ( bcd ) e )=( ab ( cde )).Any semigroup can be reduced to a ternary semigroup but a ternary semi-group does not necessarily reduce to a semigroup. However, Banach showsthat a ternary semigroup does not necessarily reduce to a semigroup by thisexample.Example 1.1 ([4]) T = {−i, 0 ,i} is a ternary semigroup while T is not asemigroup under the multiplication over complex numbers.

5 citations

Journal ArticleDOI
TL;DR: In this paper, isometric representations of the semigroup ℤ+\{1] have been studied and the notion of an inverse representation is introduced and a complete description of such representations is given up to unitary equivalence.
Abstract: We study isometric representations of the semigroup ℤ+\{1}. Notion of an inverse representation is introduced and a complete description (up to unitary equivalence) of such representations is given. Also, we study a class of non-inverse irreducible representations — β-representations of the semigroup ℤ+\{1}.

5 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202312
202229
20217
20203
20194
201810