Semigroups of left quotients—the uniqueness problem

Abstract

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 J^-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 wsemigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations 3? and if 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 Q, and Q 2 and the multiplicative subsemigroups of Q, and Q 2 are semigroups of left quotients of the multiplicative semigroup of R, then Q, and Q 2 are isomorphic rings.

Orders and Straight Left Orders in Completely Regular Semigroups

“…This led to the question, answered in [7], of under what circumstances are semigroups of left quotients of a given semigroup isomorphic. More generally, if S is a left order in a semigroup Q and 0 is a morphism from S to a semigroup P, when can 0 be extended to a morphism from Q to P?…”

“…To answer the question of the existence of such an extension, we consider a ternary relation on S, introduced in [7].…”

“…Then 0 can be extended to a morphism from Q to P if and only if aY bY c P TQY S implies that a0Y b0Y c0 P TPY S0 ÃX Proof. The proof may be lifted from part of the proof of Theorem 3.1 of [7], which considers the case where 0 is an embedding. &…”

“…These results appear in [7]. For the purposes of this paper we need a different application of Corollary 3.11.…”

Algebras of quotients of path algebras

“…This localization was carried out by considering generalized inverses like group inverses, and agrees with the classical one (where usual inverses are taken) when the ring is semiprime and coincides with its socle [14,Corollary 3.4]. If a ring R has a Fountain-Gould right quotient ring Q, then Q is unique up to isomorphisms (see [16,Theorem 5.9] or [4,Corollary 3]). …”