Author:
Gould Victoria,Schneider Georgia
Abstract
AbstractLet Q be an inverse semigroup. A subsemigroup S of Q is a left I-order in Q and Q is a semigroup of left I-quotients of S if every element in Q can be written as $$a^{-1}b$$
a
-
1
b
, where $$a, b \in S$$
a
,
b
∈
S
and $$a^{-1}$$
a
-
1
is the inverse of a in the sense of inverse semigroup theory. If we insist on being able to take a and b to be $$\mathscr {R}$$
R
-related in Q we say that S is straight in Q and Q is a semigroup of straight left I-quotients of S. We give a set of necessary and sufficient conditions for a semigroup to be a straight left I-order. The conditions are in terms of two binary relations, corresponding to the potential restrictions of $${\mathscr {R}}$$
R
and $${\mathscr {L}}$$
L
from an oversemigroup, and an associated partial order. Our approach relies on the meet structure of the $$\mathscr {L}$$
L
-classes of inverse semigroups. We prove that every finite left I-order is straight and give an example of a left I-order which is not straight.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
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