is a sublattice of , and
for any and , .
The first condition can be replaced by a weaker one: for any , .
for any , , and
for any , if , then .
Special Filters. Let be a filter of a lattice . Some of the common types of filters are defined below.
is a proper filter if , and, if contains , .
is a prime filter if it is proper, and implies that either or .
is an ultrafilter (or maximal filter) of if is proper and the only filter properly contains is .
filter generated by a set. Let be a subset of a lattice . Let be the set of all filters of containing . Since (), the intersection of all elements in , is also a filter of that contains . is called the filter generated by , written . If is a singleton , then is said to be a principal filter generated by , written .
Consider the positive integers, with meet and join defined by the greatest common divisor and the least common multiple operations. Then the positive even numbers form a filter, generated by . If we toss in as an additional element, then and consequently any positive integer , since . In general, if are relatively prime, then . In fact, any proper filter in is principal. When the generator is prime, the filter is prime, which is also maximal. So prime filters and ultrafilters coincide in .
Remark. If is both a filter and an ideal of a lattice , then .
|Date of creation||2013-03-22 15:49:01|
|Last modified on||2013-03-22 15:49:01|
|Last modified by||CWoo (3771)|
|Defines||filter generated by|