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|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.|
|Publisher: ||University of St Andrews|
|Appears in Collections:||Pure Mathematics Theses|
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