Solid Regular Bands, Strong Semilattices Of Completely Simple Semigroups And Their Inverse Semigroups Of Bicongruences

A bicongruence on a semigroup S,like that on any algebra, is exactly a binary relation on S induced by an isomorphism between two factor semigroups of S.The set B M(S)of all bicongruences on S,under an appropriately defined operation,forms an inverse semigroup which contains important information on the structure of S itself.In this paper,a regular band is called solid,if each of its linking morphisms is an isomorphism.The sufficient and necessary conditions that the solid regular bands must satisfy for their inverse semigroups of bicongruences to be fundamental,combinatorial or Clifford ones are given.The similar work is done for strong semilattices of completely simple semigroups.The sufficient and necessary conditions that this kind of strong semilattices must satisfy for their inverse semigroups of bicongruences to be fundamental or combinatorial are also proved.The results obtained here generalize corresponding results on the inverse semigroups of bicongruences on normal bands and some other classes of bands due to Australian scholar D.G.FitzGerald.