Showing papers on "Bicyclic semigroup published in 1988"

The Local Cohomology Groups of an Affine Semigroup Ring

Abstract: Publisher Summary This chapter provides an overview of the local cohomology groups of an affine semigroup ring. A commutative semigroup ring k [ S ] over a field k is said to be an affine semigroup ring if k [ S ] is an integral domain of finite type over k . This is equivalent to the condition that S is finitely generated and is contained in a free Z-module M of finite rank. An affine semigroup ring k [ S ] has a natural structure of an M-graded ring with respect to the free Z-module M. The dualizing complex and the local cohomology groups are also described in the chapter.

Embedding any countable semigroup without idempotents in a 2-generated simple semigroup without idempotents

Abstract: Although the classes of regular simple semigroups and simple semigroups without idempotents are evidently at opposite ends of the spectrum of simple semigroups, their theories involve some interesting connections. Jones [5] has obtained analogues of the bicyclic semigroup for simple semigroups without idempotents. Megyesi and Pollak [7] have classified all combinatorial simple principal ideal semigroups on two generators, showing that all are homomorphic images of one such semigroup Po which has no idempotents.

A nilpotent-generated semigroup associated with a semigroup of full transformations

Abstract: Let X be a set with infinite regular cardinality m and let ℱ( X ) be the semigroup of all self-maps of X . The semigroup Q m of ‘balanced’ elements of ℱ( X ) plays an important role in the study by Howie [ 3,5,6 ] of idempotent-generated subsemigroups of ℱ( X ), as does the subset S m of ‘stable’ elements, which is a subsemigroup of Q m if and only if m is a regular cardinal. The principal factor P m of Q m , corresponding to the maximum ℱ-class J m , contains S m and has been shown in [ 7 ] to have a number of interesting properties. Let N 2 be the set of all nilpotent elements of index 2 in P m . Then the subsemigroup ( N 2 ) of P m generated by N 2 consists exactly of the elements in P m /S m . Moreover P m /S m has 2-nilpotent-depth 3, in the sense that

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

A Note on the PI-Property of Semigroup Algebras

Abstract: Let S be a semigroup. We say that S has the property P n, n ≥ 2, if for any a 1,…, a n Є S there exist σ ≠ 1 in the symmetric group S n such that a 1 a 2… a n = a σ(1) a σ(2)…a σ(n). Further S is said to have the permutational property P if S has P n for some n ≥ 2. Semigroups of this type were first studied in [10], (S-periodic), [2], (groups), implicitly in [4] (S − O-simple) and recently in [5].

G-regular semigroups

Abstract: In this paper, we define a g-regular semigroup which is a generalization of a regular semigroup. And we want to find some properties of g-regular semigroup. G-regular semigroups contains the variety of all regular semigroup and the variety of all periodic semigroup. If a is an element of a semigroup S, the smallest left ideal containing a is Sa.cup.{a}, which we may conveniently write as a, and which we shall call the principal left ideal generated by a. An equivalence relation l on S is then defined by the rule alb if and only if a and b generate the same principal left ideal, i.e. if and only if a= b. Similarly, we can define the relation R. The equivalence relation D is R.L and the principal two sided ideal generated by an element a of S is a . We write aqb if a = b , i.e. if there exist x,y,u,v in for which xay=b, ubv=a. It is immediate that D.contnd.q. A semigroup S is called periodic if all its elements are of finite order. A finite semigroup is necessarily periodic semigroup. It is well known that in a periodic semigroup, D=q. An element a of a semigroup S is called regular if there exists x in S such that axa=a. The semigroup S is called regular if all its elements are regular. The following is the property of D-classes of regular semigroup.group.

Semigroup extensions of partial groupoids

Abstract: For the theory of semigroups as well as for the theory of partial groupoids the situations when a partial operation can be extended to a complete associative operation are of substantial interest (see, for example, [I]). This paper is devoted to an investigation of possibilities of such extensions. In §2 it is shown that this extension problem can be reduced in a sense to the problem of an internal extension. Using results of §2 and the conditions obtained in [2], it is possible to find another form for the criterion from §2 connected with the properties of extensions of partial transformations.