Cross-connections in Clifford semigroups (Q6191318)

Let \(S\) be a regular semigroup and \(E(S)\) be the set of its idempotents. Let \(\mathbb{L}(S)\) (\(\mathbb{R}(S)\)) denote the so called normal category which objects are principal left (right) ideals \(Se\) (\(eS\)) where \(e\in E(S)\) and morphisms are partial right (left) translations. Following [\textit{K. S. S. Nambooripad}, Theory of cross-connections. Kowdiar, Trivandrum: Centre for Mathematical Sciences (1994; Zbl 0844.20046)], an inter-relationship of these categories ``is abstracted as a pair of functors called cross-connections''. For \(d\in E(S)\), a normal cone with apex \(Sd\) is a function \(\gamma\) which assigns to every \(\mathbb{L}(S)\)-object \(Se\) a morphism \(Se \to Sd\) such that some conditions are satisfied. It is known that the set of all normal cones in a normal category forms a regular semigroup, under a natural binary operation. It is proved that if \(S\) is a Clifford semigroup then the semigroup of all normal cones in \(\mathbb{L}(S)\) is isomorphic to the semigroup \(S\). This fact leads to the complete description of the cross-connection structure of the semigroup \(S\). It is obtained that ``the cross-connection structure degenerates to isomorphisms of the associated normal categories in a Clifford semigroup \(S\)''. For the entire collection see [Zbl 07795952].