Bicyclic semigroup

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

Biorder-preserving coextensions of fundamental semigroups

Abstract: In any extension theory for semigroups one must determine the basic building blocksand then discover how they fit together to create more complicated semigroups. Forexample, in group theory the basic building blocks are simple groups. In semigrouptheory however there are several natural choices. One that has received considerableattention, particularly since the seminal work on inverse semigroups by Munn ([14,15]), is the notion of a fundamental semigroup. A semigrou fundamentalp i isf calle it dcannot be "shrunk" homomorphically without collapsing some of its idempotents (seebelow for a precise definition).For example, Munn showed how all fundamental inverse semigroups can beconstructed from semilattices, and proved that any inverse semigroup is a coextension ofa fundamental inverse semigroup which possesses the same semilattice of idempotents.(A semigroup S is called a coextension of a semigroup T if T is a homomorphic imageof S.) This work has been generalized by several authors to wider classes of semigroups([12, 1, 11, 16, 4]).The idempotents of an arbitrary semigroup form a biordered set. (Regular biorderedsets form the basis for Nambooripad's generalization of Munn's results to regularsemigroups). One might ask whether an arbitrary semigroup is a coextension of afundamental semigroup possessing the same biordered set of idempotents. The followingexample shows the answer is negative.Let S = (e,f\e

π-Inverse Semigroups whose Lattice of π-Inverse Subsemigroups is 0-Distributive or 0-Modular

Abstract: A semigroup S is called -regular if for every element a of S there exists m 2 Z (the set of positive integers) such that a is regular. Let us denote by r(a) the least positive integer m such that a is regular of S and call it the regular index of an element a: If every regular element of a -regular semigroup S possesses a unique inverse, then S is called -inverse [1]. Let A be a subsemigroup of a -inverse semigroup S: We say that A is a -inverse subsemigroup of S if for any a 2 A; a 2 RegA (the set of all regular elements of A) [2]. Obviously, A is a -inverse subsemigroup of S if and only if for any a 2 A and every m 2 Z; a 2 RegS implies a 2 RegA: For a -inverse semigroup S; a set Sub S of all -inverse subsemigroups (including the empty set) of S forms a lattice with respect to intersection denoted as usual by T and union denoted by h; i , where hA;Bi is the -inverse subsemigroup generated by the union of subsets A;B of S: A lattice L( V ; W ) with zero is 0-distributive [0-modular ] if for any a; b; c 2 L , a V b = a V c = 0[a V b = 0 and a c] implies a V (b W c) = 0[a V (b W c) = c]: In order to avoid misunderstanding, we remark that, in contrast with relation between distributivity and modularity, 0-distributivity is not stronger than 0-modularity. Moreover, for the lattices under examination in this paper, the latter turns out to be stronger than the former (see Corollaries 2.3 and 2.4 below). As to general case, there is no implicative relation between these two conditions. Indeed, it is easy to verify that the 5-element lattice called pentagon is 0-distributive but not 0-modular, and the 5-element lattice called diamond is 0-modular but not 0-distributive. The aim of this paper is to characterize a -inverse semigroup S for which Sub S is 0-distributive or 0-modular. In [2, 3], the author characterized a -inverse semigroup S for which Sub S is modular or complemented, respectively. Suppose S is a semigroup and A is a subset of S: We will denote by ES the set of all idempotents of S; by GrS the set of all group elements of S; by RegS the set of all regular elements of S; and by hAi the subsemigroup of S generated by A: We will denote by Ge the maximal subgroup of S with the idempotent e as identity element. For e 2 ES we put

Bipartite graphs and completely 0-simple semigroups

Abstract: In a manner similar to the construction of the fundamental group of a connected graph, this article introduces the construction of a fundamental semigroup associated with a bipartite graph. This semigroup is a 0-direct union of idempotent generated completely 0-simple semigroups. The maximal nonzero subgroups are the corresponding fundamental groups of the connected components. Adding labelled edges to the graph leads to a more general completely 0-simple semigroup. The basic properties of such semigroups are examined and they are shown to have certain universal properties as illustrated by the fact that the free completely simple semigroup on n generators and its idempotent generated subsemigroup appear as special cases.

On binary quadratic forms with the semigroup property

Abstract: A quadratic form f is said to have the semigroup property if its values at the points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with the semigroup property. If there is an integer bilinear map s such that f(s(x,y)) = f(x)f(y) for all vectors x and y from the integer two-dimensional lattice, then the form f has the semigroup property. We give an explicit integer parameterization of all pairs (f,s) with the property stated above. We do not know any other examples of forms with the semigroup property.