The natural partial order on linear semigroups with nullity and co-rank bounded below

Higgins [‘The Mitsch order on a semigroup’, Semigroup Forum $49$ (1994), 261–266] showed showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are concerned with the study of the natural partial order on non-regular semigroups. Of particular interest are the non-regular semigroups of linear transformations with lower bounds on the nullity or the corank. In this paper, we determine when they exist, characterise the natural partial order on these non-regular semigroups and consider questions of the compatibility, minimality and maximality. In addition, we provide many examples associated with our results. DOI: 10.1017/S000497271400079

The natural partial order on some transformation semigroups

For a semigroup $S$, let $S1$ be the semigroup obtained from $S$ by adding a new symbol 1 as its identity if $S$ has no identity; otherwise let $S1=S$. Mitsch defined the natural partial order ⩽ on a semigroup $S$ as follows: for $a,b∈S$, $a⩽b$ if and only if $a=xb=by$ and $a=ay$ for some $x,y∈S1$. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements. 10.1017/S000497271300058