A UNIQUE ATOM IN [C(ω),Cα(ω)]

Abstract

Let E be any nontrivial equivalence on the set X, TE(X) be the α-semigroup. It was shown that the congruence lattice of TE(X) can be decomposed into three disjoint complete sublattices: {Δ(TE(X))}, [C(E),Cα(E)] and [C(ω),Cα(ω)]. In this paper, we investigate the relationship between the two nonsingleton complete sublattices of Con(TE(X)), and we determine a unique atom in the complete sublattice [C(ω),Cα(ω)] of Con(TE(X)).