摘要
Let S be an ideal nil-extension of a completely regular semigroup K by a nil semigroup Q 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 on K respectively. It is proved that every congruence on S can be uniquely respresented by an admissible congruence pair (δ,ω) of S. Suppose that ρ K denotes the Rees congruence induced by the ideal K of S. Then it is shown that for any congruence σ 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 to K and σ Q=(σ∨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?
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 ?k denotes the Rees congruence induced by the idealK ofS. Then it is shown that for any congruenceσ on S, a mapping Γ: σ |→ (σQ, σK) is an order-preserving bijection from the set of all congruences on S onto the set of all admissible congruence pairs of S, where σK is the restriction of σ toK and σQ = (σ V ?K ) / ?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?
基金
ThisworkwassupportedbytheNationalNaturalScienceFoundationofChina (GrantNo .194710 6 8) ,andbytheAppliedFoundamentalResearchFundationofYunnan (GrantNo .96a0 0 1z)andby 2 0 0 .6 0 0 .380oftheChineseUniversityofHongKong
