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.
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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.
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.
