Completely 0-simple semigroups of quotients

Fountain, John; Petrich, Mario

doi:10.1016/0021-8693(86)90200-0

Cited by 49 publications

(88 citation statements)

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“…Now if 5 is a weak order in Q and q e Q, then q = a*b for some a,b e 5. But then q = (a 2 )*ab and ab^^a 2 Consider the three element semilattice Q = {a, /3, y} where a < /3 and a ^ y. Clearly Q is an order in Q\ if &{Q) were a weak order in 8P(Q) then {/3, -y} could be written as {0, y} = U*V for some U,V e 3>(Q) with V< X U.…”

Section: P(g) Is Idempotent If and Only If E Is A Finite Subgroup Momentioning

confidence: 99%

Orders in power semigroups

Easdown

Gould

1996

Glasgow Math. J.

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1. Introduction. In this paper we consider examples of orders in restricted power semigroups, where for any semigroup S the restricted power semigroup $P(S) is given by = {X c 5:1 < \X\ < X o } with multiplication XY = {xy :x e X, y e Y} for all X,Y e ). We use the notion of order introduced by Fountain and Petrich in [2] which first appears in the form used here in [3]. If 5 is a subsemigroup of Q then S is an order in Q and Q is a semigroup of quotients of S if any q e Q can be written as q = a*b = cd* where a,b,c,d e S and a*(d*) is the inverse of a{d) in a subgroup of Q, and in addition, all elements of 5 satisfying a weak cancellability condition called square-cancellability lie in a subgroup of (?.It is clear that the concept of a semigroup of quotients extends that of the group of quotients G of a commutative cancellative semigroup 5. Our first result shows that for such an S and G, the restricted power semigroup SP(S) is an order in £?(G).In the latter part of the paper we turn our attention to orders in a semigroup Q which is a semilattice V of commutative groups G a , a"e V. To handle the idempotents of 2P(Q) we make the further assumption that the groups G a , a e Y, are torsion-free. We find a necessary and sufficient condition for an order 5 in such a semigroup Q to have the property that 3P(S) is an order in 2P(Q). In fact we prove a slightly stronger result. We say that a subsemigroup 5 of Q is a weak order in Q if any q e Q can be written as q = a*b = cd* where a,b,c,d e S and a*(d*) is the inverse of a(d) in a subgroup of Q. Proposition 4.1 gives a necessary and sufficient condition on 5 such that SP(S) is a weak order in ^(Q), where 5 is an order in Q and Q is a semilattice of torsion-free commutative groups. We then show that for such an 5 and Q, if 2P(S) is a weak order in ®(Q) then necessarily 9>{S) is an order in 0>(Q).Section 2 consists of some preliminary definitions and results concerning orders and Green's relations in certain restricted power semigroups. Section 3 considers restricted power semigroups of orders in commutative groups. In our last section we turn our attention to restricted power semigroups of orders in semilattices of commutative torsion-free groups, and prove the results mentioned in the previous paragraph.We comment that if 5 is an order in Q then 5° (the semigroup S with a zero adjoined) is clearly an order in Q°. Moreover it is easy to see that if T is an order in Q° then T = S°w here S is an order in Q. Including the empty set in 2P(S) would correspond to considering ^(5)°. Thus excluding the empty set from SP(S) does not affect our results in any essential way, it is merely convenient.

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Section: P(g) Is Idempotent If and Only If E Is A Finite Subgroup Momentioning

confidence: 99%

Orders in power semigroups

Easdown

Gould

1996

Glasgow Math. J.

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“…Inspired by the ring theory concepts of orders and classical rings of quotients, Fountain and Petrich introduced the notion of a completely 0-simple semigroup of quotients in [19]. This was generalised to a much wider class of semigroups by Gould in [20].…”

Section: Introductionmentioning

confidence: 99%

Maximal orders in semigroups¹

Fountain

Gould

Smith

1999

Communications in Algebra

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“…We call such an isomorphism an isomorphism over S. Our notion of a semigroup of left quotients is that introduced by Fountain and Petrich in [6] and developed in [7]; in Section 2 we give a precise definition, restricting ourselves here to remarking that the approach is via consideration of group inverses of elements. If Q is a semigroup of left quotients of S then we also say that S is a left order in Q.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, the corresponding result is not true for semigroups. In the first paper of this area, Fountain and Petrich [6] give an example of a semigroup having two nonisomorphic semigroups of left quotients. However, by restricting the class of semigroups of left quotients under consideration, one can obtain some positive results.…”

Section: Introductionmentioning

confidence: 99%

Semigroups of left quotients—the uniqueness problem

Gould

1992

Proceedings of the Edinburgh Mathematical Society

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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 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.

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Completely 0-simple semigroups of quotients

Cited by 49 publications

References 3 publications

Orders in power semigroups

Orders in power semigroups

Maximal orders in semigroups¹

Semigroups of left quotients—the uniqueness problem

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Completely 0-simple semigroups of quotients

Cited by 49 publications

References 3 publications

Orders in power semigroups

Orders in power semigroups

Maximal orders in semigroups1

Semigroups of left quotients—the uniqueness problem

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Maximal orders in semigroups¹