Let 
be a lattice, and let 
.
Then 
is a tolerance if and only if it is a reflexive and symmetric sublattice of 
.
Tolerances of lattices, with their related polarities, play an important role in the study of the structure of finite algebras.
Lattice Tolerance
This entry contributed by Matt Insall (author's link)
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References
Bandelt, H. H. "Tolerance Relations on Lattices." Bull. Austral. Math. Soc. 23, 367-381, 1981.Birkhoff, G. Lattice Theory, 3rd ed. Providence, RI: Amer. Math. Soc., 1967.Chajda, I. and Zelinka, B. "Tolerances and Convexity." Czech. Math. J. 29, 584-587, 1979.Chajda, I. and Zelinka, B. "A Characterization of Tolerance-Distributive Tree Semilattices." Czech. Math. J. 37, 175-180, 1987.Grätzer, G. General Lattice Theory, 2nd ed. Boston, MA: Birkhäuser, 1998.Hobby, D. and McKenzie, R. The Structure of Finite Algebras. Providence, RI: Amer. Math. Soc., 1988.Insall, M. "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math. Soc. 53, 266-280, 1992.Schweigert, D. "Central Relations on Lattices." J. Austral. Math. Soc. 37, 213-219, 1988.Schweigert, D. and Szymanska, M. "On Central Relations of Complete Lattices." Czech. Math. J. 37, 70-74, 1987.Referenced on Wolfram|Alpha
Lattice ToleranceCite this as:
Insall, Matt. "Lattice Tolerance." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LatticeTolerance.html