C-拟正则半群上的可许同余对

令半群S为Clifford半群K的诣零扩张,Q为其Rees商半群S/K。引入S的可许同余对(δ,ω)的概念,其中δ和ω分别为诣零半群Q和Clifford半群K上的同余,证明了S上的任何同余σ都可由S的一个可许同余对唯一表示。另外,关于S上的任何同余σ,用σ_K表示σ在Clifford半群K上的限制,即σ_K=σ|_K,而σ_Q=(σ∨ρ_K)/ρ_K,其中ρ_K为S的理想K诱导的Rees同余,还证明了映射Γ:σ→(σ_Q,σ_k)为从S上的所有同余集合到S的所有可许同余对集合上的保序双射。最后,讨论了S上的同余是正则同余的条件。

Admissible congruence pairs on C-quasiregular semigroups

Let S be a nil-extension of a Clifford semigroup K by a nil semigroup Q=S/K. By introducing a concept of admissible congruence pairs (δ,ω), where δ is a congruence on a nil semigroup Q and ω is a congruence on a Clifford semigroup K respectively, it is proved that every congruence σ on S can be uniquely represented by an admissible congruence pair on S. In addition, for any congruence σ on S, suppose that σ_K is a restriction of σ on a Clifford semigroup K, that is, σ_K=σ|_K and σ_Q=(σ∨ρ_K)/ρ_K, where ρ_K is a Rees congruence on S induced by a ideal K of S, it is proved that there is an order-preserving bijection Γ:σ→(σ_Q,σ_k) from the set of all congruences on S onto the set of all admissible congruence pairs on S. Finally, a condition has been given for a congruence which is a regular congruence on S.