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Given a poset
and a standard closure operator
, we give a necessary and sufficient condition for the lattice of
-closed sets of
to be a frame in terms of the recursive construction of the
-closure of sets. We use this condition to show that, given a set
of distinguished joins from
, the lattice of
fails to be a frame if and only if it fails to be
depending on the cardinalities of sets in
. From this we deduce that if a poset has the property that whenever
$a\wedge (b\vee c)$
is defined for
it is necessarily equal to
$(a\wedge b)\vee (a\wedge c)$
, then it has an
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