多种类代数的两个同构定理

Two isomorphism theorems of many-sorted algebras

设A=(Ai,i∈Γ)为Ω -代数 ,φ =( φi,i∈Γ)和θ =(θi,i∈Γ)都是A上同余 ,B =(Bi,i∈Γ)为A的子代数 .类似于一个非空集合上代数的情形 ,定义了 φ/θ和Bθ,证明了 (A/θ) / ( φ/θ) ≌A/ φ ,B/θ B ≌Bθ/θ Bθ.

Let A=(A i,i∈Γ) be an Ω-algebra, bath φ=(φ i,i∈Γ) and θ=(θ i,i∈Γ) are congruence on A, B=(B i,i∈Γ) is a subalgebra of A. Similar to the situation of the algebra on a non-vacuous set, definit φ/θ and B-θ,prove (A/θ)/(φ/θ)≌A/φ,B/θB≌B-θ/θB-θ.