Some Remarkable Congruences on Completely Regular Semigroups
Mario Petrich · Rocky Mountain Journal of Mathematics · 1998
We express a completely regular semigroup S as (Y ; S α ), that is, a semilattice of completely simple semigroups.For each pair α > β, we consider the congruence κ α,β on S generated by the set of pairs (a, b) where a ∈ S α , b ∈ S β and a > b.These congruences play an important role in finding conditions which ensure that the kernel relation K on the congruence lattice of S be a congruence.In particular, the meet and the join of these congruences provide interesting congruences in this context.Another class of congruences, constructed as follows, occurs naturally in this study.Given a congruence ρ on S and ideals I ⊆ J of S, we generalize the Rees congruence relative to I by constructing a congruence which involves ρ, I and J; here ρ must saturate I and I or J may be empty.