Unit regularity of semigroup of normal cones

Abstract

Let \(S\) be a unit regular semigroup and \(\mathscr{L}(S)\) be the normal category of principal left ideals of \(S\). Then the semigroup \(T \mathscr{L}(S)\) of normal cones in \(\mathscr{L}(S)\) is a unit regular semigroup. We characterize the category of principal left ideals of a unit regular semigroup as a \(\mathscr{U} \mathscr{R}\)-category. This is a normal category \(\mathscr{C}\) with the additional property that there is a maximum object \(m\) in \(v \mathscr{C}\) and that every isomorphism in \(\mathscr{C}\) is the restriction of an isomorphism of \(m\). We prove that in this case the semigroup \(T \mathscr{C}\) is a unit regular semigroup.