Boolean Constructions and Discriminator Varieties
Bounded Boolean Powers and Transfer Results
The bounded variant of the Boolean power construction and the transfer theorems it supports.
Learning objectives
- Distinguish bounded from unbounded Boolean powers
- State the principal transfer results
- Identify where each variant is the right tool
The two variants
Functions from B* to A taking only finitely many values, with clopen preimages — the locally constant functions.
A larger construction permitting functions with infinitely many values, subject to a weaker continuity requirement.
The source distinguishes the two notations. The bounded version is the one used for most transfer results because its elements admit a finite description; the unbounded version is needed when the algebra must be closed under limits.
| Bounded | Unbounded | |
|---|---|---|
| Values taken | Finitely many | Possibly infinitely many |
| Elements | Finite clopen partition plus labels | Requires limiting process |
| Size | Smaller | Larger |
| Equals direct power when | B finite | B complete and atomic |
| Primary use | Transfer of identities and congruence structure | Closure properties, completeness arguments |
Transfer theorems
The forward direction holds because the Boolean power is a subalgebra of a product of copies of A. The converse holds because A embeds in the Boolean power as the constant functions.
The variety generated by an algebra is unchanged by taking Boolean powers. So Boolean powers produce new algebras inside a fixed variety, which is exactly what is needed to study the internal structure of that variety.
For A simple and B arbitrary, Con(A[B]*) is isomorphic to the filter lattice of B.
What transfer buys
Realising congruence lattices
Given a target algebraic lattice realisable as a Boolean filter lattice, a Boolean power of a simple algebra realises it inside the chosen variety.
Constructing large algebras
Boolean powers of a finite algebra by large Boolean algebras give arbitrarily large members of a locally finite variety.
Preserving equational behaviour
Because identities transfer exactly, no equational information is lost or gained.
Boolean powers can only produce algebras whose structure is Boolean-indexed. Varieties whose members are not built from Boolean-like decompositions are not reached by the construction, which is why Boolean products — a genuine generalisation — are introduced later in the chapter.
Historical role
Boolean powers were introduced by Foster in the 1950s and developed extensively for the study of primal and quasiprimal algebras. Their significance in the source is as the first of a graded series of constructions:
Each step generalises the previous while retaining enough structure to prove representation theorems. Discriminator varieties, the culmination of the chapter, are characterised by having Boolean product representations of a particularly strong kind.
Frequently asked questions
When do bounded and unbounded Boolean powers coincide?
When B is finite, both reduce to a finite direct power. For infinite B they differ, and the bounded version is a proper subalgebra of the unbounded one.
Does the Boolean power preserve simplicity?
No. Even for simple A, the Boolean power by a non-trivial B has congruences derived from B's filters. Simplicity is lost precisely because B contributes structure.
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.5, book pages 159-162.
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.