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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Local Polarity

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Insall, Matt. "Local Polarity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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