In universal algebra and lattice theory, a tolerance relation on an algebraic structure is a reflexivesymmetric 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. A set that possesses a tolerance relation can be described as a tolerance space.[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]
Definitions
A tolerance relation on an algebraic structure is usually defined to be a reflexivesymmetric relation on that is compatible with every operation in . A tolerance relation can also be seen as a cover of that satisfies certain conditions. 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 form an algebraic lattice under inclusion. Since every congruence relation is a tolerance relation, the congruence lattice is a subset of the tolerance lattice , but is not necessarily a sublattice of .[4]
A tolerance relation on an algebraic structure is a cover of that satisfies the following three conditions.[5]: 307, Theorem 3
For every and , if , then .
In particular, no two distinct elements of are comparable. (To see this, take .)
For every , if is not contained in any set in , then there is a two-element subset such that is not contained in any set in .
For every -ary and , there is a such that . (Such a need not be unique.)
Every partition of satisfies the first two conditions, but not conversely. A congruence relation is a tolerance relation that also forms a set partition.
Let be an algebraic structure and let be a tolerance relation on . Suppose that, for each -ary operation and , there is a unique such that
Then this provides a natural definition of the quotient algebra
of over . 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. Therefore, for a variety of algebraic structures, we may consider the following two conditions.[4]
(Tolerance factorability) for any and any tolerance relation on , the uniqueness condition is true, so that the quotient algebra is defined.
(Strong tolerance factorability) for any and any tolerance relation on , the uniqueness condition is true, and .
Every strongly tolerance factorable variety is tolerance factorable, but not vice versa.
Examples
Sets
A set is an algebraic structure with no operations at all. In this case, tolerance relations are simply reflexivesymmetric relations and it is trivial that the variety of sets is strongly tolerance factorable.
For a tolerance relation on a lattice, every set in is a convex sublattice of . Thus, for all , we have
In particular, the following results hold.
if and only if .
If and , then .
The variety of lattices is strongly tolerance factorable. That is, given any lattice and any tolerance relation on , for each there exist unique such that
In particular, we can form quotient lattices of distributive lattices and modular lattices over tolerance relations. 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.[7]: 40 [4] Actually, every subvariety of the variety of lattices is tolerance factorable, and the only strongly tolerance factorable subvariety other than itself is the trivial subvariety (consisting of one-element lattices).[7]: 40 This is because every lattice is isomorphic to a sublattice of the quotient lattice over a tolerance relation of a sublattice of a direct product of two-element lattices.[7]: 40, Theorem 3
^Poincare, H. (1905). Science and Hypothesis (with a preface by J.Larmor ed.). New York: 3 East 14th Street: The Walter Scott Publishing Co., Ltd. pp. 22-23.{{cite book}}: CS1 maint: location (link)
^Grätzer, George; Wenzel, G. H. (1990). "Notes on tolerance relations of lattices". Acta Scientiarum Mathematicarum. 54 (3–4): 229–240. ISSN0001-6969. MR1096802. Zbl0727.06011.
Gerasin, S. N., Shlyakhov, V. V., and Yakovlev, S. V. 2008. Set coverings and tolerance relations. Cybernetics and Sys. Anal. 44, 3 (May 2008), 333–340. doi:10.1007/s10559-008-9007-y
Hryniewiecki, K. 1991, Relations of Tolerance, FORMALIZED MATHEMATICS, Vol. 2, No. 1, January–February 1991.