## Coherence, regularity and permutability of congruences.(English)Zbl 0537.08006

A variety $${\mathcal V}$$ of algebras is regular, if any two congruences on each algebra $${\mathfrak A}\in {\mathcal V}$$ coincide whenever they have a congruence class in common. It is coherent, if for any two algebras $${\mathfrak A,B}$$ of $${\mathcal V}$$ the condition that $${\mathfrak A}\subseteq {\mathfrak B}$$ and $${\mathfrak A}$$ contains a class of a congruence $$\theta$$ on $${\mathfrak B}$$ implies that $${\mathfrak A}$$ is a union of classes of $$\theta$$. A variety $${\mathcal V}$$ is permutable, if $$\theta_ 1\cdot \theta_ 2=\theta_ 2\cdot \theta_ 1$$ for any two congruences on every algebra $${\mathfrak A}\in {\mathcal V}.$$
An algebra $${\mathfrak A}$$ has subalgebras closed under principal congruence classes (briefly $${\mathfrak A}$$ is CUT), if for every subalgebra $${\mathfrak B}$$ of $${\mathfrak A}$$, all $$x\in {\mathfrak A},\quad y\in {\mathfrak B},\quad z\in {\mathfrak B}$$ and any algebraic functions $$\phi$$ over $${\mathfrak A}$$ the condition $$[z]_{\theta}\subseteq {\mathfrak B}$$ and $$\phi(z,...,z)=y$$ implies $$\phi([z]_{\theta})\in {\mathfrak B},$$ where $$\theta =\theta(x,y)$$ and $$[z]_{\theta}$$ denotes the class of $$\theta$$ containing z.
The main result of the paper is the following theorem. For a variety $${\mathcal V}$$ the following conditions are equivalent: (1) $${\mathcal V}$$ is coherent, (2) $${\mathcal V}$$ is CUT, regular and permutable, (3) there exist an $$(n+1)$$-ary polynomial h and ternary polynomials $$t_ i$$ over $${\mathcal V}$$ such that $$t_ i(x,x,z)=z,\quad i=1,...,n,\quad h(y,t_ 1(x,y,z),...,t_ n(x,y,z))=x.$$