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Lattice properties of the symmetric weakly inverse semigroup on a totally ordered set

Part of: Ordered structures Semigroups

Published online by Cambridge University Press:  09 April 2009

C. C. Edwards  and
Marlow Anderson
C. C. Edwards
Affiliation:
Department of Mathematical Sciences, Indiana University-Purdue University at Fort Wayne Fort Wayne, Indiana 46805, U.S.A.
Marlow Anderson
Affiliation:
Department of Mathematical Sciences, Indiana University-Purdue University at Fort Wayne Fort Wayne, Indiana 46805, U.S.A.
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Abstract

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Let T be a totally ordered set, PT the semigroup of partial transformations on T, and A(T) the l-group of order-preserving permutations of T. We show that PT is a regular left l-semigroup. Let be the set of α ∈ PT such that α is order-preserving and the domain of α is a final segment of T. Then is an l-semigroup, and we prove that it is the largest transitive l-subsemigroup of PT which contains A(T). When T is Dedekind complete, we characterize the largest regular l-semigroup of . When A(T) is also 0 − 2 transitive we show that there can be no l-subsemigroup of properly containing A(T) which is either inverse or a union of groups.

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 06F05: Ordered semigroups and monoids
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 4 , December 1981 , pp. 395 - 404
Copyright
Copyright © Australian Mathematical Society 1981

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