**Description: **Define the *Moore
closure* of a generating set, which is the smallest
closed set containing all generating elements. Definition of Moore
closure in [Schechter] p. 79. This
generalizes topological closure
(mrccls 16816) and linear span (mrclsp 15746).
A Moore closure operation is (1) extensive, i.e.,
for all subsets of the base set (mrcssid 13519),
(2) isotone, i.e.,
implies that
for all subsets
and of the base set (mrcss 13518), and (3)
idempotent, i.e., for all subsets
of the base set
(mrcidm 13521.) Operators satisfying these three
properties are in bijective correspondence with Moore collections, so
these properties may be used to give an alternate characterization of a
Moore collection by providing a closure operation on the set of
subsets of a given base set which satisfies (1), (2), and (3); the
closed sets can be recovered as those sets which equal their closures
(Section 4.5 in [Schechter] p. 82.)
(Contributed by Stefan O'Rear,
31-Jan-2015.) (Revised by David Moews,
1-May-2017.) |