Bulletin of the Australian Mathematical
Society
Bull. Austral. Math. Soc.
0004-9727
ALNBAB
2005
72
1
10.wxyz/CV72P1
http://www.austms.org.au/Publ/Bulletin/V72P1/
Generation of diagonal acts of some semigroups of
transformations and relations
Peter
Gallagher
Nik Ruškuc
14 February 2006
2006
2
14
139
146
721-5106-GaRu-2005
10.wxyz/C2005V72P1p139
http://www.austms.org.au/Publ/Bulletin/V72P1/721-5106-GaRu/
The \emph {diagonal right} (respectively, \emph
{left}) \emph {act} of a semigroup $S$ is the set $S \times
S$ on which $S$ acts via $(x,y)s=(xs,ys)$ (respectively,
$s(x,y)=(sx,sy)$); the same set with both actions is the
\emph {diagonal bi-act}. The diagonal right (respectively,
left, bi-) act is said to be finitely generated if there is a
finite set $A \subseteq S \times S$ such that $S \times
S=AS^1$ (respectively, $S \times S=S^1A$, $S \times
S=S^1AS^1$). \par \par 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.
20M20, 20M30
MR2162299
02212191
The authors are grateful to an anonymous
referee for his/her suggestions for streamlining the proof of
Theorem 4.3.
S. Bulman-Fleming and K. McDowell
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