Absolutely flat completely 0-simple semigroups of left quotients
type: Item
instance of: scholarly article
title: Absolutely flat completely 0-simple semigroups of left quotients
zbMATH Open document ID: 0658.20034
DOI: 10.1016/0022-4049(88)90119-3
author: Victoria Gould
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: It is proposed a semigroup analogue of Goldie's theorem describing rings which have a simple Artinian ring of left quotients. In this paper the author is concerned with a semigroup analogue of the classical ring of quotients, first proposed by \textit{J. B. Fountain} and \textit{M. Petrich} [J. Algebra 101, 365-402 (1986; Zbl 0589.20041)]. If Q is a semigroup of left quotients of S then S is called a left order in Q. If Q is both a semigroup of left quotients of S and a semigroup of right quotients of S then Q is called a semigroup of quotients and S is called an order in Q. Fountain and Petrich described all orders in completely 0-simple semigroups. If one wishes to classify the left orders in a given class of semigroups, then one requires the semigroups of left quotients from that class to be unique up to semigroups. Unfortunately this is not true for completely 0-simple semigroups. The notion of absolutely flat semigroups one can find in [\textit{S. Bulman-Fleming} and \textit{K. McDowell}, Pac. Math. 107, 319-333 (1983; Zbl 0488.20054)]. The author proves the uniqueness of absolutely flat completely 0-simple semigroups of left (right) quotients. Further it is given a set of necessary and sufficient conditions for a semigroup S to be a left order in an absolutely flat completely 0-simple semigroup. In conclusion these results are applied to Brandt semigroups and abundant semigroups.
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 20M20
zbMATH DE Number: 4075400
zbMATH Keywords: semigroup of left quotients
zbMATH Keywords: semigroup of right quotients
zbMATH Keywords: orders in completely 0-simple semigroups
zbMATH Keywords: left orders
zbMATH Keywords: absolutely flat completely 0-simple semigroups
zbMATH Keywords: Brandt semigroups
zbMATH Keywords: abundant semigroups
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90119-3
OpenAlex ID: W2050838416
cites work: Absolutely flat semigroups
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description: scientific article; zbMATH DE number 4075400