Notes on tolerance factorable classes of algebras

Abstract

It is well known that for every congruence Θ ∈ Con($\mathcal{A}$) and each surjective homomorphism h: $\mathcal{A}$ → $\mathcal{B}$, the image h(Θ) = {(h(a), h(b)) (a, b) ∈ Θ }is a tolerance on $\mathcal{B}$. We study algebras and classes of algebras whose every tolerance is a homomorphic image of a congruence. In particular, we prove that every homomorphic image of a congruence on $\mathcal{A}$ is a congruence on $\mathcal{B}$ if and only if $\mathcal{A}$ is 3-permutable. Let $\mathscr{K}$ be a class of algebras such that every tolerance on $\mathcal{B}$ ∈ $\mathscr{K}$ is a homomorphic image of a congruence of an algebra that belongs to $\mathscr{K}$. Then $\mathscr{K}$ is tolerance factorable if and only if each $\mathcal{B}$ ∈ $\mathscr{K}$ is factorable by the tolerance Θ o Φ o Θ for all Θ, Φ ∈ Con($\mathcal{B}$). This result is extended for a strongly tolerance factorable variety.