Representable tolerances in varieties

TL;DR

This paper explores two methods of representing tolerances in algebraic varieties, analyzing their relationships and applying these concepts to lattices, thereby extending understanding of tolerance representations in algebraic structures.

Contribution

It introduces and compares two tolerance representation methods, clarifies their relationships, and applies these to lattices, connecting to recent research in the field.

Findings

Any lattice tolerance can be represented as a congruence image on a subalgebra of its product

The second representation method is independent of the variety

Relationships between the two representations are clarified

Abstract

We discuss two possible ways of representing tolerances: first, as a homomorphic image of some congruence; second, as the relational composition of some compatible relation with its converse. The second way is independent from the variety under consideration, while the first way is variety-dependent. The relationships between these two kinds of representations are clarified. As an application, we show that any tolerance on some lattice L is the image of some congruence on a subalgebra of L $\times$ L. This is related to recent results by G. Cz\'edli and E. W. Kiss.