Events > Talks by CEMAT Members

Weakly left ample semigroups

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.