Semigroups of inverse quotients (original) (raw)
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Semigroups of left quotients—the uniqueness problem
Proceedings of the Edinburgh Mathematical Society, 1992
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 u...
Semigroups of Left Quotients—The Layered Approach
Communications in Algebra, 2004
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 element of Q can be expressed as a ♯ b where a, b ∈ S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. Here a ♯ denotes the inverse of a in a subgroup of Q. We say that a left order S is straight in Q if in the above definition we can insist that a R b in Q. A complete characterisation of straight left orders in terms of embeddable *-pairs is available. In this paper we adopt a different approach, based on partial order decompositions of semigroups. Such decompositions include semilattice decompositions and decompositions of a semigroup into principal factors or principal *-factors. We determine when a semigroup that can be decomposed into straight left orders is itself a straight left order. This technique gives a unified approach to obtaining many of the early results on characterisations of straight left orders.
We show that the two binary operations in double inverse semigroups, as considered by Kock [2007], necessarily coincide.
The inverse hull of right cancellative semigroups
Journal of Algebra, 1987
The inverse hull of a left reductive right cancellative semigroup S is represented as a quotient semigroup of the free inverse monoid generated by .S. Necessary and sufficient conditions are established on S in order for its inverse hull to be Eunitary, in which case its P-representation is constructed. The inverse hull of a reversible cancellative semigroup is proved to be an F-inverse semigroup, for which an F-representation is constructed. The class of right cancellative monoids is provided with suitable morphisms; the resulting category is proved to be equivalent to a certain category of inverse monoids via the inverse hull.
Some Orthodox Monoids with Associate Inverse Subsemigroups
Communications in Algebra, 2010
By an associate inverse subsemigroup of a regular semigroup S we mean a subsemigroup T of S containing a least associate of each x ∈ S, in relation to the natural partial order ≤ S in S. In this paper we investigate a class of orthodox monoids with an associate inverse subsemigroup and obtain a known description of uniquely unit regular orthodox semigroups as a corollary. Also, by considering a more general situation, we identify the homomorphic image of a kind of semidirect product of a band with identity by an inverse monoid, thus extending a known result for unit regular orthodox semigroups.
O ct 2 01 8 CONJUGACY IN INVERSE SEMIGROUPS
In a group G, elements a and b are conjugate if there exists g ∈ G such that g −1 ag = b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a ∼ i b, if there exists g ∈ S 1 such that g −1 ag = b and gbg −1 = a. The purpose of this paper is to study the conjugacy ∼ i in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.
Clifford semigroups of left quotients
Glasgow Mathematical Journal, 1986
Several definitions of a semigroup of quotients have been proposed and studied by a number of authors. For a survey, the reader may consult Weinert's paper [8]. The motivation for many of these concepts comes from ring theory and the various notions of rings of quotients. We are concerned in this paper with an analogue of the classical ring of quotients, introduced by Fountain and Petrich in [3].
On inverse and right inverse ordered semigroups
arXiv (Cornell University), 2017
A regular ordered semigroup S is called right inverse if every principal left ideal of S is generated by an R-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element a ∈ S are R-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.
Semigroups of left quotients: existence, straightness and locality
Journal of Algebra, 2003
A subsemigroup S of a semigroup Q is a local left order in Q if, for every group H-class H of Q, S ∩ H is a left order in H in the sense of group theory. That is, every q ∈ H can be written as a ♯ b for some a, b ∈ S ∩ H, where a ♯ denotes the group inverse of a in H. On the other hand, S is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as c ♯ d where c, d ∈ S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. If one also insists that c and d can be chosen such that c R d in Q, then S is said to be a straight left order in Q. This paper investigates the close relation between local left orders and straight left orders in a semigroup Q and gives some quite general conditions for a left order S in Q to be straight. In the light of the connection between locality and straightness we give a complete description of straight left orders that improves upon that in our earlier paper.
A note on amalgams of inverse semigroups
Journal of the Australian Mathematical Society, 2001
This note gives a necessary condition, in terms of graded actions, for an inverse semigroup to be a full amalgam. Under a mild additional hypothesis, the condition becomes sufficient.