generalized Boolean algebra
Clearly, a Boolean algebra is a generalized Boolean algebra. Conversely, a generalized Boolean algebra with a top is a Boolean algebra, since is a bounded distributive complemented lattice, so each element has a unique complement by distributivity. So is a unary operator on which makes into a de Morgan algebra. A complemented de Morgan algebra is, as a result, a Boolean algebra.
As an example of a generalized Boolean algebra that is not Boolean, let be an infinite set and let be the set of all finite subsets of . Then is generalized Boolean: order by inclusion, then is a distributive as the operation is inherited from , the powerset of . It is also relatively complemented: if where , then is the relative complement of in . Finally, is, as usual, the bottom element in . is not a Boolean algebra, because the union of all the singletons (all in ) is , which is infinite, thus not in .
One property of a generalized Boolean algebra is the following: if and are complements of , then ; in other words, relative complements are uniquely determined. This is true because in any distributive lattice, complents are uniquely determined. As is distributive, so is each lattice interval in .
In fact, because of the existence of , we can actually construct the relative complement. Let denote the unique complement of in . Then is the unique complement of : and .
Conversely, if is a distributive lattice with such that any lattice interval is complemented, then is a generalized Boolean algebra. Again, provides the necessary complement of in .
|Title||generalized Boolean algebra|
|Date of creation||2013-03-22 17:08:37|
|Last modified on||2013-03-22 17:08:37|
|Last modified by||CWoo (3771)|
|Synonym||generalized Boolean lattice|