Tolerance

A binary relation $R \subseteq A \times A$ on a set $A$ having the properties of reflexivity and symmetry, i.e. such that $aRa$ for all $a \in A$, and $aRb$ implies $b R a$ for all $a,b \in A$. A tolerance $R$ on a universal algebra $A = \{A,\Omega\}$ is said to be compatible if it is a subalgebra of the direct square $A \times A$, that is, if for any $n$-ary operation $\omega$ the conditions $a_i R b_i$, $i = 1,\ldots,n$, imply $(a_1,a_2,\ldots,a_n \omega) R (b_1,\ldots,b_n \omega)$. Thus, a tolerance is a natural generalization of the notion of an equivalence, and a compatible tolerance is a generalization of a congruence. Any compatible tolerance on a relatively complemented lattice is a congruence [1]. The set $\mathrm{LT}(A)$ of all compatible tolerances on a universal algebra $A$, ordered by inclusion, is an algebraic lattice, containing the lattice $\mathrm{Con}(A)$ of all congruences on $A$ as a subset (but not necessarily as a sublattice). For properties of the lattices $\mathrm{LT}(A)$ and $\mathrm{Con}(A)$ see [2], [3].

References

[1]	I. Chajda, J. Niederle, B. Zelinka, "On existence conditions for compatible tolerances" Czechoslovak. Math. J. , 26 : 2 (1976) pp. 304–311
[2]	E.T. Schmidt, "Kongruenzrelationen algebraischer Strukturen" , Deutsch. Verlag Wissenschaft. (1969)
[3]	G. Grätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)

Comments

Let $(M,d)$ be a metric space. Then $R = \{(x,y) : d(x,y) < \epsilon \}$ defines a tolerance on $M$. Tolerances of this type (and generalizations) are used, e.g., in statistics, mechanics, robotics, and dynamical systems. There are, e.g., investigations concerning the structural stability of dynamical systems up to some tolerance, the Zeeman tolerance stability conjecture, [a1], [a2].

References

[a1]	F. Takens, "Tolerance stability" A. Manning (ed.) , Dynamical systems (Warwick, 1974) , Lect. notes in math. , 468 , Springer (1975) pp. 293–304
[a2]	F. Takens, "On Zeeman's tolerance stability conjecture" N.H. Kuiper (ed.) , Manifolds (Amsterdam, 1970) , Springer (1971) pp. 209–219