Boolean ideal

Let A be a Boolean algebraMathworldPlanetmath and B a subset of A. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    If A is interpreted as a Boolean ringMathworldPlanetmath, B is a ring ideal.

  2. 2.

    If A is interpreted as a Boolean lattice, B is a lattice ideal.

Before proving this equivalence, we want to mention that a Boolean ring is equivalent to a Boolean lattice, and a more commonly used terminology is a Boolean algebra, which is also valid, as it is an algebra over the ring of integersMathworldPlanetmath. The standard way of characterizing the ring structureMathworldPlanetmath from the latticeMathworldPlanetmathPlanetmath structure is by defining a+b:=(ab)(ab) (called the symmetric differenceMathworldPlanetmathPlanetmath) and ab=ab. From this, we can “solve” for in terms of + and : ab=a+b+ab.


First, suppose B is an ideal of the “ring” A. If a,bB, then ab=a+b+abB. Suppose now that aB and cA with ca. Then c=ca=caB as well. So B is a lattice ideal of A.

Next, suppose B is an ideal of the “lattice” A. If a,bB, then both ab and ab are in B since the first one is less than or equal to b and the second less than or equal to a, so their join is in B as well, this means that a-b=a+b=(ab)(ab)B. Furthermore, if aB and cA, then ca=caaB as well. As a result, B is a ring ideal of A. ∎

A subset of a Boolean algebra satisfying the two equivalent conditions above is called a Boolean ideal, or an ideal for short. A prime Boolean ideal is a prime lattice ideal, and a maximal Boolean ideal is a maximal lattice ideal. Again, these notions and their ring theoretic counterparts match exactly. In fact, one can say more about these ideals in the case of a Boolean algebra: prime idealsMathworldPlanetmathPlanetmath are precisely the maximal idealsMathworldPlanetmath. If A is a Boolean ring, and M is a maximal ideal of A, then A/M is isomorphic to {0,1}.

Remark. The dual notion of a Boolean ideal is a Boolean filter, or a filter for short. A Boolean filter is just a lattice filter of the Boolean algebra when considered as a lattice. To see the connection between a Boolean ideal and a Boolean lattice, let us define, for any subset S of a Boolean algebra A, the set S:={aaS}. It is easy to see that S′′=S. Now, if I is an ideal, then I is a filter. Conversely, if F is a filter, F is a ideal. In fact, given any Boolean algebra, there is a Galois connection


between the set of Boolean ideals and the set of Boolean filters in A. In additionPlanetmathPlanetmath, I is prime iff I is. As a result, a filter is prime iff it is an ultrafilter (maximal filter).

Title Boolean ideal
Canonical name BooleanIdeal
Date of creation 2013-03-22 17:01:59
Last modified on 2013-03-22 17:01:59
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 03G05
Classification msc 03G10
Related topic BooleanRing
Defines Boolean filter
Defines prime Boolean ideal
Defines maximal Boolean ideal