A tolerance on some algebra is a binary, compatible, symmetric and reflexive relation. Thus, a congruence is a transitive tolerance. The following two examples are important in this paper:
1) Suppose \(\mathfrak A\), \(\mathfrak B\) are algebras, and \(\varphi \colon {\mathfrak B}\to {\mathfrak A}\) is a surjective homomorphism. If \(\Psi\) is a tolerance on \(\mathfrak B\), then \[ \varphi(\Psi)=\{(\varphi a, \varphi b)\mid a\Psi b,\;a,b\in B\} \] is a tolerance on \(\mathfrak A\) and \(\varphi(\Psi)\) is called the homomorphic image of the tolerance \(\Psi\). So the homomorphic image of a congruence is a tolerance, but, in general, it is not a congruence. It is known that every tolerance \(\Theta\) on any algebra \(\mathfrak A\) can be represented as a homomorphic image of some congruence on some algebra \(\mathfrak B\).
2) Suppose \(\mathfrak A\) is an algebra, and \(R\) is a compatible reflexive relation on \(\mathfrak A\). If \(R^{-1}\) denotes the converse of \(R\), then \(R\circ R^{-1}\) is a tolerance on \(\mathfrak A\). A tolerance is called representable if it can be expressed in the form \(R\circ R^{-1}\) for some \(R\). In this paper the following results are proved:
a) If \(\mathfrak L\) is a lattice and \(\Theta\) is a tolerance on \(\mathfrak L\), then \(\Theta\) is the homomorphic image of some congruence on some subalgebra of \({\mathfrak L}\times {\mathfrak L}\). b) If a tolerance \(\Theta\) on \(\mathfrak A\) is representable, then \(\Theta\) is the homomorphic image of a congruence on some subalgebra of \({\mathfrak A}\times {\mathfrak A}\). c) Suppose \(\mathfrak A\) is an algebra in a 3-permutable variety \(\mathfrak V\). If \(\Theta\) is a tolerance on \(\mathfrak A\), then \(\Theta\) is representable if and only if \(\Theta\) is a congruence of \(\mathfrak A\). Some generalizations of the mentioned results are considered, too.