Weakly left ample semigroups

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.