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A binary relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929601.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929602.png" /> having the properties of reflexivity and symmetry, i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929603.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929604.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929605.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929606.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929607.png" />. A tolerance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929608.png" /> on a [[Universal algebra|universal algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t0929609.png" /> is said to be compatible if it is a subalgebra of the direct square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296010.png" />, that is, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296011.png" />-ary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296012.png" /> the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296014.png" />, imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296015.png" />. Thus, a tolerance is a natural generalization of the notion of an [[Equivalence|equivalence]], and a compatible tolerance is a generalization of a [[Congruence|congruence]]. Any compatible tolerance on a relatively complemented lattice is a congruence [[#References|[1]]]. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296016.png" /> of all compatible tolerances on a universal algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296017.png" />, ordered by inclusion, is an algebraic lattice, containing the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296018.png" /> of all congruences on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296019.png" /> as a subset (but not necessarily as a sublattice). For properties of the lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296021.png" /> see [[#References|[2]]], [[#References|[3]]].
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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|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|equivalence]], and a compatible tolerance is a generalization of a [[Congruence|congruence]]. Any compatible tolerance on a relatively complemented lattice is a congruence [[#References|[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 [[#References|[2]]], [[#References|[3]]].
  
 
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====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296022.png" /> be a metric space. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296023.png" /> defines a tolerance on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092960/t09296024.png" />. 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, [[#References|[a1]]], [[#References|[a2]]].
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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, [[#References|[a1]]], [[#References|[a2]]].
  
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Takens,  "Tolerance stability"  A. Manning (ed.) , ''Dynamical systems (Warwick, 1974)'' , ''Lect. notes in math.'' , '''468''' , Springer  (1975)  pp. 293–304</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Takens,  "On Zeeman's tolerance stability conjecture"  N.H. Kuiper (ed.) , ''Manifolds (Amsterdam, 1970)'' , Springer  (1971)  pp. 209–219</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Takens,  "Tolerance stability"  A. Manning (ed.) , ''Dynamical systems (Warwick, 1974)'' , ''Lect. notes in math.'' , '''468''' , Springer  (1975)  pp. 293–304</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Takens,  "On Zeeman's tolerance stability conjecture"  N.H. Kuiper (ed.) , ''Manifolds (Amsterdam, 1970)'' , Springer  (1971)  pp. 209–219</TD></TR>
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</table>

Revision as of 20:53, 12 October 2014

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
How to Cite This Entry:
Tolerance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tolerance&oldid=33596
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article