Tolerance relation
In mathematics, a tolerance relation is a relation that is reflexive and symmetric, but not necessarily transitive; a set X that possesses a tolerance relation can be described as a tolerance space.[1] Tolerance relations provide a convenient general tool for studying indiscernibility/indistinguishability phenomena. The importance of those for mathematics had been first recognized by Poincaré.[2]
Tolerances in lattice theory and universal algebra
Tolerances are frequently used in lattice theory[3] and universal algebra. When dealing with algebraic structures, tolerances are assumed to be compatible. Thus a tolerance is like a congruence, except that the assumption of transitivity is dropped.[4]
See also
- Dependency relation
- Quasitransitive relation—a generalization to formalize indifference in social choice theory
- Rough set
References
- Sossinsky, Alexey (1986-02-01). "Tolerance space theory and some applications". Acta Applicandae Mathematicae. 5 (2): 137–167. doi:10.1007/BF00046585.
- 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 (2011). Lattice Theory: Foundations. Basel: Birkhäuser. p. 43.
- Kearnes, Keith; Kiss, Emil W. (2013). The Shape of Congruence Lattices. American Mathematical Soc. p. 20. ISBN 978-0-8218-8323-5.
Further reading
- 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.
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