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TRANSFORMATION SEMIGROUPS
Let Xn={1, 2,& , n}. Then a (partial) transformation a:Doma"XnIma is said to be full or total if Doma=Xn; otherwise it is called strictly partial. The height of a is the size of Ima. Three fundamental semigroups of transformations of Xn under usual composition have been extensively studied:
Tn, the full transformation semigroup (or the symmetric semigroup);
In, the semigroup of partial oneone mappings (or the symmetric inverse semigroup);
and
Pn the semigroup of partial transformations (or the partial symmetric semigroup).
Partial oneone mappings are also known as subpermutations (Cameron and Deza, HYPERLINK "http://taylorandfrancis.metapress.com/media/c28gtnutyn431y32wmf0/contributions/v/y/c/h/vych9atqpg5fkr07_html/bib.htm" \l "CIT0002" 1979).
SOME SUBSEMIGROUPS OF THE TRANSFORMATION SEMIGROUPS
The study of various (finite) semigroups of transformations forms an
integral part of semigroup theory, just as the study of (finite) symmetric
and alternating groups is, to group theory. For one thing, such semigroups
are a rich source of examples. However, it is also clear that they are worth
studying in their own right as 'naturally occuring' objects.
Arguably, amongst the various subsemigroups of transformations the most successful are those of ORDER-PRESERVING and/or ORDER-DECREASING (ORDER-INCREASING, ORDER-REDUCING, REGRESSIVE)
A transformation EMBED Equation.3 in Pn for which EMBED Equation.3 is said to be order-preserving.
A transformation EMBED Equation.3 in Pn for which EMBED Equation.3 is said to be order-decreasing.
GREEN'S RELATIONS AND THEIR STARRED ANALOGUES
In 1951, J. A. Green introduced five equivalences: H, L, R, D and J on a semigroup S which have played a fundamental role in the development of semigroup theory. They are concerned with mutual divisibility of various kinds, and all of them reduce to the universal equivalence relation in a group.
Regular semigroupsthe definition is copied from von Neumann's (1936) definition of a regular ringparticularly amenable to analysis using Green's equivalences.
In 1982, J. B. Fountain introduced one of the most successful generalizations of regular semigroups and called them abundant semigroups 468@BD\^ " T t
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