Coherence, regularity and permutability of congruences (Q792364)

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 \(t_ i(x,x,z)=z,\quad i=1,...,n,\quad h(y,t_ 1(x,y,z),...,t_ n(x,y,z))=x.\)