Weakly left ample semigroups
01/06/2003
Gracinda Maria dos Santos Gomes Moreira da Cunha
(CAUL)
International Meeting on Semigroup Theory and Related Topics, University of Minho, Braga, Portugal
For a set $X$, we define a unary operation $^+$ on the monoid
$PT(X)$ of all partial transformations on X by taking $\alpha^+$ to
be the identity mapping on the domain of $\alpha$. A semigroup $S$
with a unary operation $^+$ such that $e = e^+$, for all $e \in
E(S)$, is said to be {\em weakly left ample} if there is a
$(2,1)$algebra embedding of $S$ into $PT(X)$, for some set $X$.
We present a number of examples of weakly left ample semigroups
that arise naturally, such as the graph and the Szendrei
expansions of unipotent monoids. We also consider the generalized
prefix expansion of a weakly left ample semigroup.
On a weakly left ample semigroup $S$ there is a least unipotent
congruence $\sigma$ and $S$ is said to be {\em proper} if,
for all elements $a, b \in S$,
$$
(a^+ = b^+ \; \rm{and} \; a \, \sigma\, b) \Rightarrow a=b.
$$
Any inverse semigroup $I$ is weakly left ample $$ the unary operation
$^+$ is defined on $I$ by $a^+ = aa^{1}$, for all $a \in I$. In $I$
the congruence $\sigma$ is the least group congruence and it is well
known that proper inverse semigroups are exactly the $E$unitary
inverse semigroups.
A proper weakly left ample semigroup $P$ is said to be a {\em proper
cover} of a weakly left ample semigroup $S$ if there is a surjective
$(2,1)$algebra homomorphism from $P$ onto $S$ that separates idempotents.
We describe the structure of an arbitrary weakly left ample semigroup $S$
and prove that $S$ has a proper cover $T$. We also show that $T$ can be
chosen to be finite whenever $S$ is finite.
