# Cancellative Orders

## Authors

**David Easdown** and **Victoria Gould**

## Status

Research Report 96-2

To appear in Semigroup Forum

Date: December 1995

## Abstract

A subsemigroup S of a semigroup Q is a *left order* in Q and Q is a
*semigroup of left quotients* of S if every q in Q can be written as
q=a*b for some a,b in S, where a* denotes the inverse of a in a subgroup of
Q, and if, in addition, every square-cancellable element of S lies in a
subgroup of Q.
Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup,
can have non-isomorphic semigroups of left quotients. We show that if S is a
cancellative left order in Q then Q is completely regular and the
*D*-classes of Q are left groups. The semigroup S is right reversible
and its group of left quotients is the minimum semigroup of left quotients
of S.

## Key phrases

cancellative semigroups of left quotients. completely regular semigroups.

## AMS Subject Classification (1991)

Primary: 20M10

Secondary:

## Content

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1996-2.dvi (52kB)

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