generalized Boolean algebra

A latticeMathworldPlanetmath L is called a generalized Boolean algebra if

Clearly, a Boolean algebraMathworldPlanetmath is a generalized Boolean algebra. Conversely, a generalized Boolean algebra L with a top 1 is a Boolean algebra, since L=[0,1] is a bounded distributive complemented lattice, so each element aL has a unique complement a by distributivity. So is a unary operator on L which makes L 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 A be an infinite setMathworldPlanetmath and let B be the set of all finite subsets of A. Then B is generalized Boolean: order B by inclusion, then B is a distributive as the operationMathworldPlanetmath is inherited from P(A), the powerset of A. It is also relatively complemented: if C[X,Y] where C,X,YB, then (Y-C)X is the relative complement of C in [X,Y]. Finally, is, as usual, the bottom element in B. B is not a Boolean algebra, because the union of all the singletons (all in B) is A, which is infinite, thus not in B.

One property of a generalized Boolean algebra L is the following: if y and z are complements of x[a,b], then y=z; in other words, relative complements are uniquely determined. This is true because in any distributive latticeMathworldPlanetmath, complents are uniquely determined. As L is distributive, so is each lattice interval [a,b] in L.

In fact, because of the existence of 0, we can actually construct the relative complement. Let b-x denote the unique complement of x in [0,b]. Then (b-x)a is the unique complement of x[a,b]: x((b-x)a)=(x(b-x))vee(xa)=0a=a and x((b-x)a)=(x(b-x))a=ba=b.

Conversely, if L is a distributive lattice with 0 such that any lattice interval [0,a] is complemented, then L is a generalized Boolean algebra. Again, (b-x)a provides the necessary complement of x in [a,b].

Title generalized Boolean algebra
Canonical name GeneralizedBooleanAlgebra
Date of creation 2013-03-22 17:08:37
Last modified on 2013-03-22 17:08:37
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 06E99
Classification msc 06D99
Synonym generalized Boolean lattice