Let
be a lattice, and let . Then the pair is a polarity of if and only if is a decreasing join-endomorphism and is an increasing meet-endomorphism and for each ,.
Polarities, and their related tolerances, play an important role in the study of tame congruences in finite algebras.
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.
be a lattice, and let 
. Then the pair 
is a polarity of 
if and only if 
is a decreasing join-endomorphism and 
is an increasing meet-endomorphism and for each 
, 
.
