The lattice of R-unipotent congruences on a regular semigroup (Q1068952)

In this paper the lattice RC(S) of all R-unipotent congruences on a regular semigroup S is studied. A relation \(\equiv\) defined by \(\sigma\) \(\equiv \theta\) if and only if \(\sigma |_{E(S)}=\theta |_{E(S)}\) is considered in RC(S). It is proved that on a regular semigroup S, the relation \(\equiv\) in RC(S) is a congruence and that each \(\equiv\)-class is a complete modular sublattice of RC(S). A characterization of the least and the greatest elements of each \(\equiv\)- class is presented.