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 ${\eufm{L}} $ is the image of some congruence on a subalgebra of ${\eufm{L}} \times{\eufm{L}}$. This is related to recent results by G. Czédli, G. Grätzer and E. W. Kiss.

AMS Subject Classification (1991): 08A30, 06B10; 08B99, 08C15, 06B20, 06B75

Keyword(s): tolerance, representable, image of a congruence, lattice, \hbox{(quasi-)}variety

Received June 12, 2012, and in revised form October 15, 2012. (Registered under 47/2012.)