Abstract. It is well known that for every congruence $\Theta\in\rm Con(\mathcal A)$ and each surjective homomorphism $h\colon\mathcal A\to\mathcal B$, the image $h(\Theta )=\{(h(a),h(b)) \mid(a,b)\in\Theta \}$ 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 $\mathcal K$ be a class of algebras such that every tolerance on $\mathcal B\in\mathcal K$ is a homomorphic image of a congruence of an algebra that belongs to $\mathcal K$. Then $\mathcal K$ is tolerance factorable if and only if each $\mathcal B\in\mathcal K$ is factorable by the tolerance $\Theta\circ \Phi\circ \Theta $ for all $\Theta,\Phi\in \Con(\mathcal B)$. This result is extended for a strongly tolerance factorable variety.

DOI: 10.14232/actasm-012-861-x

AMS Subject Classification (1991): 08A30, 08B99

Keyword(s): tolerance, congruence, tolerance factorable algebra, strongly tolerance factorable variety, TImC-property, free algebra, $3$-permutability

Received December 19, 2012, and in revised form June 10, 2013. (Registered under 111/2012.)