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.
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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 
, 
.
