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Local Polarity


Let L=(L, ^ , v ) be a lattice, and let f,g:L->L. Then the pair (f,g) is a local polarity if and only if for each finite set X subset= L, there is a finitely generated sublattice K of L that contains X and on which the restriction (f|K,g|K) is a lattice polarity.

Using nonstandard methods, one may show that the following result holds: Let L be a locally finite lattice. Then the set of local polarities of L is a relation R subset= {(f,g)|f,g:L->L}^2 which is a one-to-one correspondence between its domain and range.


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.

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Local Polarity

Cite this as:

Insall, Matt. "Local Polarity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LocalPolarity.html

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