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Bull. Austral. Math. Soc. 72(1) pp.139--146, 2005.

Generation of diagonal acts of some semigroups of transformations and relations

The authors are grateful to an anonymous referee for his/her suggestions for streamlining the proof of Theorem 4.3.

Abstract

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 = AS ¹ (respectively, S×S = S ¹A, S×S = S ¹AS ¹).

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 semigroups 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.

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ISSN 0004-9727