On Idempotent Ranks of Semigroups of Partial Transformations |
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Authors: | George Barnes Inessa Levi |
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Institution: | (1) Department of Mathematics, University of Louisville, Louisville, KY 40292, USA;(2) Department of Mathematics, Western Illinois University, Macomb, IL 61455, USA |
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Abstract: | A subset U of a semigroup S is a generating set for S
if every element of S may be written as a finite product of
elements of U. The rank of S is the size of a minimal
generating set of S, and the idempotent rank of S is
the size of a minimal generating set of S consisting of
idempotents in S. A partition of a q-element subset of the set Xn={1,2,...,
n} is said to be of type if the sizes of its classes form
the partition of q n. A non-trivial partition
of a positive integer q consists of k < q elements. For a
non-trivial partition of q n, the semigroup
S(), generated by all the transformations with kernels of
type , is idempotent-generated. It is known that if is a non-trivial partition of n, that
is, S() consists of total many-to-one transformations, then
the rank and the idempotent rank of S() are both equal to
max{nd, N()}, where
N() is the number of partitions of Xn of type
. We extend this result to semigroups of partial
transformations, and prove that if is a non-trivial
partition of q < n, then the rank and the idempotent rank of
S() are both equal to N(). |
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Keywords: | |
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