Varieties whose tolerances are homomorphic images of their congruences

TL;DR

This paper characterizes varieties where tolerances are homomorphic images of congruences, providing conditions and examples, including semilattices, lattices, and unary algebras, with specific results on permutability.

Contribution

It introduces a Maltsev-like condition to characterize TImC varieties and explores their properties and examples, including a counterexample with a majority term.

Findings

Semilattices, lattices, unary algebras have TImC

A congruence n-permutable variety has TImC iff it is congruence permutable

Constructed an idempotent variety with a majority term that fails TImC

Abstract

The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterize varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.