Tolerances as images of congruences in varieties defined by linear identities

TL;DR

This paper demonstrates that tolerances in algebras defined by linear identities can be represented as images of congruences through homomorphisms, linking tolerances and congruences in such varieties.

Contribution

It establishes a new representation theorem connecting tolerances and congruences in varieties with linear identities.

Findings

Tolerances are images of congruences via homomorphisms in these varieties.

Existence of an algebra B with a congruence Theta related to a given tolerance T.

The result applies to algebras in varieties defined by linear identities.

Abstract

An identity s=t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra A in a variety defined by a set of linear identities. We prove that there exist an algebra B in the same variety and a congruence Theta of B such that a homomorphism from B onto A maps Theta onto T.