  The concept of a tolerance relation, shortly called tolerance, was studied on
various algebras since the seventieth of the twentieth century by B. Zelinka
and the first author. Since tolerances need not be transitive, their blocks may
overlap and hence in general the set of all blocks of a tolerance cannot be
converted into a quotient algebra in the same way as in the case of
congruences. However, G. Cz\'edli showed that lattices can be factorized by
means of tolerances in a natural way, and J. Grygiel and S. Radelecki proved
some variant of an Isomorphism Theorem for tolerances on lattices. The aim of
the present paper is to extend the concept of a tolerance on a lattice to
posets in such a way that results similar to those obtained for tolerances on
lattices can be derived.
