The structure of \(0\)-bisimple \(\mathcal R\)-unipotent semigroups (Q1849638)

The purpose of this paper is to generalize the theory of Lawson for \(0\)-bisimple inverse monoids to a wider class of \(0\)-bisimple regular semigroups. A function \(\theta\) between regular semigroups \(S\) and \(T\) is a prehomomorphism if \((xy)\theta\leq(x\theta)(y\theta)\) for all \(x,y\) in \(S\) (where \(\leq\) denotes the natural partial order on \(T\)). If \(S\) and \(T\) have zeros then we say that \(\theta\) is \(0\)-restricted if \(0\theta^{-1}=0\). Also, \(\theta\) is said to be idempotent-pure if \(x\) is idempotent whenever \(x\theta\) is idempotent. Necessary and sufficient conditions are given for a \(0\)-bisimple \(\mathcal R\)-unipotent semigroup to admit a \(0\)-restricted idempotent-pure prehomomorphism to a primitive inverse semigroup. There are a series of applications of these results: for example, it is shown that any completely \(0\)-simple \(\mathcal R\)-unipotent semigroup admits a \(0\)-restricted idempotent-pure prehomomorphism to a group with zero.