0. Direct products

Here is an attempt at a decomposition theorem using direct products:

Define an algebra ${\cal A}$ to be directly indecomposable if $\vert A\vert>1$ and there are no ${\cal B},{\cal C}$ with ${\cal A} \equiv {\cal B} \times {\cal C}$ except with $\vert B\vert = 1$ or $\vert C\vert = 1$.

Here is the statement you might hope for: ``Every algebra is the direct product of directly indecomposable algebras (possibly infinitely many).'' This is certainly true for finite algebras, but is false in general. In fact, let ${\cal A}$ be a vector space of countable dimension over the two-element field; observe that any directly indecomposable vector space has dimension 1 by a basis argument, but ${\cal A}$ has the wrong cardinality to be a direct product of either finitely many or infinitely many two-element vector spaces¹.

A modified concept, that of ``subdirect products of subdirectly irreducible algebras'', works much better.

Figure: A subdirect product, heuristically

text/Ddir/sd.eps