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. 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. Tolerance relations provide a convenient general tool for studying indiscernibility/indistinguishability phenomena. The importance of those for mathematics had been first recognized by Poincaré.
Definitions A tolerance relation on an algebraic structure ( A , F ) {\displaystyle (A,F)} is usually defined to be a reflexive symmetric relation on A {\displaystyle A} that is compatible with every operation in F {\displaystyle F} . A tolerance relation can also be seen as a cover of A {\displaystyle A} 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 ( A , F ) {\displaystyle (A,F)} form an algebraic lattice Tolr ( A ) {\displaystyle \operatorname {Tolr} (A)} under inclusion. Since every congruence relation is a tolerance relation, the congruence lattice Cong ( A ) {\displaystyle \operatorname {Cong} (A)} is a subset of the tolerance lattice Tolr ( A ) {\displaystyle \operatorname {Tolr} (A)} , but Cong ( A ) {\displaystyle \operatorname {Cong} (A)} is not necessarily a sublattice of Tolr ( A ) {\displaystyle \operatorname {Tolr} (A)} .
As binary relations A tolerance relation on an algebraic structure ( A , F ) {\displaystyle (A,F)} is a binary relation ∼ {\displaystyle \sim } on A {\displaystyle A} that satisfies the following conditions.
(Reflexivity) a ∼ a {\displaystyle a\sim a} for all a ∈ A {\displaystyle a\in A}
(Symmetry) if a ∼ b {\displaystyle a\sim b} then b ∼ a {\displaystyle b\sim a} for all a , b ∈ A {\displaystyle a,b\in A}
(Compatibility) for each n {\displaystyle n} -ary operation f ∈ F {\displaystyle f\in F} and a 1 , … , a n , b 1 , … , b n ∈ A {\displaystyle a_{1},\dots ,a_{n},b_{1},\dots ,b_{n}\in A} , if a i ∼ b i {\displaystyle a_{i}\sim b_{i}} for each i = 1 , … , n {\displaystyle i=1,\dots ,n} then f ( a 1 , … , a n ) ∼ f ( b 1 , … , b n ) {\displaystyle f(a_{1},\dots ,a_{n})\sim f(b_{1},\dots ,b_{n})} . That is, the set { ( a , b ) : a ∼ b } {\displaystyle \{(a,b)\colon a\sim b\}} is a subalgebra of the direct product A 2 {\displaystyle A^{2}} of two A {\displaystyle A} . A congruence relation is a tolerance relation that is also transitive.
As covers A tolerance relation on an algebraic structure ( A , F ) {\displaystyle (A,F)} is a cover C {\displaystyle {\mathcal {C}}} of A {\displaystyle A} that satisfies the following three conditions.
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