Local Polarity
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 ![R subset= <span style=]()
{(f,g)|f,g:L->L}^2" src="https://mathworld.wolfram.com/images/equations/LocalPolarity/Inline11.gif" style="height:17px; width:151px" /> which is a one-to-one correspondence between its domain and range.
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