The University of St Andrews

Research@StAndrews:FullText >
Mathematics & Statistics (School of) >
Pure Mathematics >
Pure Mathematics Theses >

Please use this identifier to cite or link to this item:
This item has been viewed 46 times in the last year. View Statistics

Files in This Item:

File Description SizeFormat
AbdullahiUmarPhDThesis.pdf4.21 MBAdobe PDFView/Open
Title: Semigroups of order-decreasing transformations
Authors: Umar, Abdullahi
Supervisors: Howie, John M.
Issue Date: 1992
Abstract: Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial one-to-one) transformations of X. In this Thesis the study of order-increasing full (partial, partial one-to-one) transformations has been reduced to that of order-decreasing full (partial, partial one-to-one) transformations and the study of order-decreasing partial transformations to that of order-decreasing full transformations for both the finite and infinite cases. For the finite order-decreasing full (partial one-to-one) transformation semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden (1990) concerning products of idempotents (quasi-idempotents), and concerning combinatorial and rank properties. By contrast with the semigroups of order-preserving transformations and the full transformation semigroup, the semigroups of orderdecreasing full (partial one-to-one) transformations and their Rees quotient semigroups are not regular. They are, however, abundant (type A) semigroups in the sense of Fountain (1982,1979). An explicit characterisation of the minimum semilattice congruence on the finite semigroups of order-decreasing transformations and their Rees quotient semigroups is obtained. If X is an infinite chain then the semigroup S of order-decreasing full transformations need not be abundant. A necessary and sufficient condition on X is obtained for S to be abundant. By contrast, for every chain X the semigroup of order-decreasing partial one-to-one transformations is type A. The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing full (partial one-to-one) transformations have been investigated.
Type: Thesis
Publisher: University of St Andrews
Appears in Collections:Pure Mathematics Theses

This item is protected by original copyright

This item is licensed under a Creative Commons License
Creative Commons

Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.


DSpace Software Copyright © 2002-2012  Duraspace - Feedback
For help contact: | Copyright for this page belongs to St Andrews University Library | Terms and Conditions (Cookies)