Varieties, Free Algebras and Equational Logic
Subdirectly Irreducible Algebras
The algebras that admit no non-trivial subdirect decomposition. They are characterised by a single lattice-theoretic condition and serve as the atoms of the structure theory.
Learning objectives
- Characterise subdirect irreducibility by the monolith condition
- Identify subdirectly irreducible algebras in standard varieties
- Explain their role as building blocks
The characterisation
An algebra A with more than one element is subdirectly irreducible if for every subdirect embedding into ∏i Ai, at least one projection is an isomorphism.
The unique minimal non-trivial congruence of a subdirectly irreducible algebra, usually written μ.
A subdirect representation corresponds to a family of congruences meeting to Δ. If every non-trivial congruence lies above a fixed μ > Δ, then any family meeting to Δ must contain Δ itself — so some projection is an isomorphism and the representation is trivial.
Congruence-lattice picture
The condition is easiest to hold as a picture: the congruence lattice has a bottom element Δ, immediately above it a single element μ, and above μ anything at all.
- ∇ — the top
- … arbitrary structure …
- μ — the monolith, unique atom
- Δ — the bottom
- μ — the monolith, unique atom
- … arbitrary structure …
A simple algebra has congruence lattice exactly {Δ, ∇}, so ∇ is the monolith. Every simple algebra is subdirectly irreducible, but not conversely — the cyclic group of order 4 has congruence lattice a three-element chain, is subdirectly irreducible with monolith the middle element, and is not simple.
Examples across varieties
| Variety | Subdirectly irreducibles | How many |
|---|---|---|
| Boolean algebras | Only 2 | One, up to isomorphism |
| Distributive lattices | Only the two-element chain | One |
| Abelian groups | Cpn and the Prüfer groups Cp∞ | Countably many |
| Vector spaces over a field K | The one-dimensional space | One |
| Groups | All simple groups, and many non-simple ones | A proper class |
| Lattices | Very many | A proper class |
| Semigroups | Very many | A proper class |
When a variety has only one subdirectly irreducible algebra, every member is a subdirect power of it — an extremely strong structural statement. This is why Boolean algebras and distributive lattices are so completely understood, and why the search for varieties with few irreducibles is a recurring theme.
Bounding the irreducibles
Two of the subject's major results are precisely bounds on the subdirectly irreducible members of a variety.
If a variety's subdirectly irreducible members are bounded in size, then only finitely many identities in a bounded number of variables are needed to exclude all the non-members. Baker's theorem makes this precise, and it is the reason the two topics sit together in Chapter V.
Frequently asked questions
Can an infinite algebra be subdirectly irreducible?
Yes. The Prüfer p-group is infinite, subdirectly irreducible as an abelian group, and its congruence lattice is an infinite chain with a unique atom.
Does every variety have subdirectly irreducible members?
Every non-trivial variety does. Birkhoff's theorem guarantees that every algebra is a subdirect product of them, so if there were none the variety would contain only trivial algebras.
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 II.8, book pages 63-65.
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.