On the tolerance lattice of tolerance factors

Abstract

Although the notion of a tolerance is a natural generalization of the notion of a congruence, many properties of factor lattices modulo congruences are not, in general, valid for factor lattices modulo tolerances. In this paper, for a lattice L of a finite length, we define a new partial orderon Tol (L )s uch that for every S ∈ Tol (L )w ithTS, a tolerance S/T is induced on the factor lattice L/T. This partial order is a particular restriction of and thus we can prove for tolerances some analogous results to the homomorphism theorem and the second isomorphism theorem for congruences. The poset (Tol (L), � ) is not always a lattice, but it can be converted into a specific commutative join-directoid. Then, for every T ∈ Tol (L), (Tol (L/T ), � ) constitutes a subdirectoid of the directoid based on the poset (Tol (L), � ) and this specific directoid structure is preserved by the direct product of lattices.