Congruences on ideal nil-extension of completely regular semigroups

Abstract

Let S be an ideal nil-extension of a completely regular semigroupK by a nil semigroupQ with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a congruence onK respectively. It is proved that every congruence on S can be uniquely respresented by an admissible congruence pair (δ,ω) ofS. Suppose that ϱ_k denotes the Rees congruence induced by the idealK ofS. Then it is shown that for any congruenceσ on S, a mapping Γ: σ ¦→ (σ_Q, σ_K) is an order-preserving bijection from the set of all congruences on S onto the set of all admissible congruence pairs of S, where σ_K is the restriction of σ toK and σ_Q = (σ V ϱ_K ) / ϱ_K. Moreover, the lattice of congruences of S is also discussed. As a special case, every congruence on completely Archimedean semigroups S is described by an admissible quarterple of S. The following question is asked: Is the lattice of congruences of the completely Archimedean semigroup a semimodular lattice?