5 research outputs found
The natural partial order on linear semigroups with nullity and co-rank bounded below
Higgins [‘The Mitsch order on a semigroup’, Semigroup Forum (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 , let be the semigroup obtained from by adding a new symbol 1 as its identity if has no identity; otherwise let . Mitsch defined the natural partial order ⩽ on a semigroup as follows: for , if and only if and for some . 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