Varieties, Free Algebras and Equational Logic
Simple Algebras and Congruence Simplicity
Algebras whose only congruences are the two trivial ones. Simplicity is the strongest indecomposability condition and appears throughout the structure theory of varieties.
Learning objectives
- Define simple algebra and relate it to simple groups and rings
- Show that simple implies subdirectly irreducible and directly indecomposable
- Identify the simple members of standard varieties
Definition
An algebra A with more than one element is simple if Con A = {Δ, ∇} — the only congruences are the identity and the all-pairs relation.
Equivalently, every surjective homomorphism from A is either an isomorphism or maps onto a one-element algebra.
| Variety | Simple means |
|---|---|
| Groups | No proper non-trivial normal subgroup — the classical notion |
| Rings | No proper non-trivial two-sided ideal |
| Modules | No proper non-trivial submodule — an irreducible module |
| Lattices | No proper non-trivial congruence |
| Boolean algebras | Only 2 is simple |
The implication chain
A simple algebra has ∇ as its unique atom, hence is subdirectly irreducible. A subdirectly irreducible algebra has no non-trivial factor congruences, since a complemented pair would meet to Δ without either being Δ, hence is directly indecomposable.
Counterexamples in both directions are easy: C4 is subdirectly irreducible and not simple; C4 is also directly indecomposable and not simple.
Simple algebras in the variety generated
The simple members of a variety matter because they are the extreme case of subdirect irreducibility, and because several structural conditions are stated in terms of them.
A variety in which every subdirectly irreducible member is simple.
Semisimple varieties are treated in Chapter IV §12. Boolean algebras, distributive lattices and vector spaces over a fixed field are all semisimple; abelian groups are not, since the Prüfer groups are subdirectly irreducible and far from simple.
In a semisimple variety, Birkhoff's theorem represents every algebra as a subdirect product of simple algebras. Combined with a classification of the simple members, this can amount to a complete structure theory — which is exactly what happens for Boolean algebras.
Existence of simple algebras
Chapter II §10 derives the existence of simple algebras in any non-trivial variety from the theory of free algebras. The argument takes a free algebra and quotients by a congruence maximal among those excluding a fixed pair; when the variety is well behaved the quotient is simple.
Every non-trivial variety contains simple algebras, but they need not be few or classifiable. The variety of all groups contains every finite simple group and all infinite simple groups — a class whose classification in the finite case took decades and whose infinite case is wide open.
Frequently asked questions
Is a one-element algebra simple?
No, by convention. It has only one congruence, since Δ and ∇ coincide, and excluding it keeps statements like 'every algebra is a subdirect product of subdirectly irreducibles' clean.
Is every simple algebra finite?
No. Infinite simple groups exist, as do infinite simple rings. Finiteness is unrelated to simplicity.
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 65-66.
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.