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Semigroups of left quotients—the uniqueness problem

Published online by Cambridge University Press:  20 January 2009

Victoria Gould
Victoria Gould
Affiliation:
Department of MathematicsUniversity of YorkHeslingtonYork, YO1 5DD
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Abstract

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Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group -class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q.

J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse ω-semigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations ℛ and ℒ in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise.

The above result is then used to show that if R is a subring of rings Q1 and Q2 and the multiplicative subsemigroups of Q1 and Q2 are semigroups of left quotients of the multiplicative semigroup of R, then Ql and Q2 are isomorphic rings.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 2 , June 1992 , pp. 213 - 226
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

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