Defining filters in closure algebras: reference request - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:07:49Z http://mathoverflow.net/feeds/question/70368 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70368/defining-filters-in-closure-algebras-reference-request Defining filters in closure algebras: reference request Michael Carroll 2011-07-14T20:00:29Z 2011-07-14T20:00:29Z <p>A <em>closure algebra</em> C is a boolean algebra B together with a unary closure operator, and additional axioms, the Kuratowski axioms, that the closure operator must satisfy. (The <a href="http://en.wikipedia.org/wiki/Interior_algebra" rel="nofollow">Wikipedia article</a> prefers the term "interior algebra".)</p> <p>Since C is a boolean algebra, we may define a <em>boolean filter</em> F on C in the usual way, as a non-empty set of elements of C which contains:</p> <ul> <li><p>the meet of x and y whenever x and y are both in F;</p> <p>the join of x and y, for any x in F and y in C.</p></li> </ul> <p>The definition of a boolean filter makes no mention of the closure operator. This raises the question of whether there is another kind of filter, to be called a <em>closure filter</em>, which adds another condition, relating to the closure operator, to the definition of a boolean filter. For example, it might require that:</p> <ul> <li>the interior of x is in F, for any x in F.</li> </ul> <p>where interior is defined in terms of closure as usual. Or perhaps this is not the right choice for an additional condition.</p> <p>If we then define a <em>closure ultrafilter</em> as a maximal closure filter, we might find that these have different properties from maximal boolean filters.</p> <p>I have not been able to locate any research along these lines, and would appreciate hearing of any references.</p>