In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexive symmetric relation that is compatible with all operations of the structure.
Thus a tolerance is like a congruence, except that the assumption of transitivity is dropped.
[1] On a set, an algebraic structure with empty family of operations, tolerance relations are simply reflexive symmetric relations.
[2] Tolerance relations provide a convenient general tool for studying indiscernibility/indistinguishability phenomena.
The importance of those for mathematics had been first recognized by Poincaré.
[3] A tolerance relation on an algebraic structure
is usually defined to be a reflexive symmetric relation on
The two definitions are equivalent, since for a fixed algebraic structure, the tolerance relations in the two definitions are in one-to-one correspondence.
The tolerance relations on an algebraic structure
[4] A tolerance relation on an algebraic structure
A tolerance relation on an algebraic structure
be a tolerance binary relation on an algebraic structure
is the set of all maximal cliques of the graph
is just the quotient set of equivalence classes.
and satisfies all the three conditions in the cover definition.
(The last condition is shown using Zorn's lemma.)
is a one-to-one correspondence between the tolerances as binary relations and as covers whose inverse is
A tolerance is transitive as a binary relation if and only if it is a partition as a cover.
Thus the two characterizations of congruence relations also agree.
such that Then this provides a natural definition of the quotient algebra of
In the case of congruence relations, the uniqueness condition always holds true and the quotient algebra defined here coincides with the usual one.
A main difference from congruence relations is that for a tolerance relation the uniqueness condition may fail, and even if it does not, the quotient algebra may not inherit the identities defining the variety that
belongs to, so that the quotient algebra may fail to be a member of the variety again.
A set is an algebraic structure with no operations at all.
In this case, tolerance relations are simply reflexive symmetric relations and it is trivial that the variety of sets is strongly tolerance factorable.
In particular, this is true for all algebraic structures that are groups when some of their operations are forgot, e.g. rings, vector spaces, modules, Boolean algebras, etc.
[6]: 261–262 Therefore, the varieties of groups, rings, vector spaces, modules and Boolean algebras are also strongly tolerance factorable trivially.
The variety of lattices is strongly tolerance factorable.
However, unlike in the case of congruence relations, the quotient lattices need not be distributive or modular again.
In other words, the varieties of distributive lattices and modular lattices are tolerance factorable, but not strongly tolerance factorable.