The natural partial order on linear semigroups with nullity and co-rank bounded below
Keywords:
linear transformation semigroups, natural partial order, left [right] compatible, minimal [maximal] elementsAbstract
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/S0004972714000793Published
2014-11-13
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