Boolean Algebras and Stone Duality
Filters and Ideals in Boolean Algebras
Filters as upward-closed meet-closed subsets, ideals as their duals, and their correspondence with congruences.
Learning objectives
- Define filter and ideal and give the standard examples
- Establish the correspondence between filters and congruences
- Describe generated filters and the finite intersection property
Filters
A non-empty subset F of a Boolean algebra that is upward closed (a ∈ F and a ≤ b imply b ∈ F) and closed under meets (a, b ∈ F implies a ∧ b ∈ F).
The order dual: downward closed and closed under joins.
Every filter contains 1; every ideal contains 0. A filter is proper if it omits 0, equivalently if it is not the whole algebra.
- <em>F</em>(<em>X</em>)
- the filter generated by a set X
- <em>I</em>(<em>X</em>)
- the ideal generated by X
- Principal filter
- {b : b ≥ a} for fixed a
- Fréchet filter
- the cofinite subsets, in a power set algebra
Filters and ideals are interchangeable
Complementation converts one into the other: F is a filter exactly when {a′ : a ∈ F} is an ideal. Results need be proved only once.
Filters are conventional in logic, topology and set theory, where they represent notions of largeness. Ideals are conventional in ring theory, where they match the Boolean-ring presentation. The source uses both.
The congruence correspondence
The map sending a filter F to the relation θF defined by a θF b if and only if (a ∧ b) ∨ (a′ ∧ b′) ∈ F is a lattice isomorphism from the filters of B onto Con B.
The expression (a ∧ b) ∨ (a′ ∧ b′) is the Boolean biconditional — it equals 1 exactly when a = b. So the congruence identifies a and b when their biconditional is “large” in the sense of F.
The correspondence with filters is only possible because congruences are determined by a single class — the class of 1. That in turn requires congruence permutability, which Boolean algebras have. Lattices in general do not, which is why general lattice congruences have no filter description.
Generated filters
For a non-empty X, the filter generated by X consists of all b such that b ≥ x1 ∧ … ∧ xn for some finite subset of X.
A subset X has the finite intersection property if every finite meet of its elements is non-zero.
F(X) is proper if and only if X has the finite intersection property. This is the Boolean-algebraic form of a condition familiar from topology and from the compactness theorem, and it is the hypothesis under which ultrafilters can be produced.
Frequently asked questions
Is every filter principal?
Only in finite Boolean algebras. The Fréchet filter of cofinite subsets of an infinite set is non-principal, and non-principal filters are the interesting case throughout the infinite theory.
Why is the biconditional used in the congruence definition?
Because a θ_F b should mean 'a and b agree, up to something in F'. The biconditional (a ∧ b) ∨ (a′ ∧ b′) is exactly the element measuring their agreement, equal to 1 when they are identical.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section IV.3, book pages 142-146.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.