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Generation of diagonal acts of some semigroups of transformations and relations

Published online by Cambridge University Press:  17 April 2009

Peter Gallagher  and
Nik Ruškuc
Peter Gallagher
Affiliation:
School of Mathematics and Statistics, University of St. Andrews, North Haugh, St. Andrews, Scotland KY16 9SS
Nik Ruškuc
Affiliation:
School of Mathematics and Statistics, University of St. Andrews, North Haugh, St. Andrews, Scotland KY16 9SS
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The diagonal right (respectively, left) act of a semigroup S is the set S × S on which S acts via (x, y) s = (xs, ys) (respectively, s (x, y) = (sx, sy)); the same set with both actions is the diagonal bi-act. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set A ⊆ S × S such that S × S = AS1 (respectively, S × S = S1A, S × S = SlASl).

In this paper we consider the question of finite generation for diagonal acts of certain infinite semigroups of transformations and relations. We show that the semi-groups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite-to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 1 , August 2005 , pp. 139 - 146
Copyright
Copyright © Australian Mathematical Society 2005

References

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