Let
be a lattice, and let .
Then the pair
is a local polarity if and only if for each finite set , there is a finitely generated sublattice of that contains and on which the restriction is a lattice polarity.

Using nonstandard methods, one may show that the following result holds: Let be a locally finite lattice. Then the
set of local polarities of is a relation which is a one-to-one correspondence
between its domain and range.

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.