A band is a semigroup where every element is idempotent: $a a = a$. A collection of bands is a variety if it is closed under taking subobjects (subsemigroups), quotient objects (images of homomorphisms) and products (including infinite products). As usual in universal algebra, any variety of bands is defined by a collection of identities. For example, there's a variety of semilattices, which are the bands obeying the identity
$$ a b = b a $$
for all $a,b$.
You can define a variety of bands using more than one identity. But amazingly --- to me, at least --- every variety of bands can in fact be defined using just one identity!
(That is, one identity in addition to associativity $(a b) c = a (b c)$ and the idempotence law $a a = a$.)
This was shown here:
- Charles Fennemore, All varieties of bands, Semigroup Forum 1 (1970), 172-179.
He even showed that there are exactly $8 + 10(n-2)$ varieties of bands that are determined by an identity involving $n$ variables, for $n \ge 2$... and he has a method for listing these identities explicitly.
The set of varieties of bands is partially ordered, based on whether one identity implies another. It's actually a lattice, as usual in universal algebra, and it seems that Fennemore described the lattice operations explicitly in terms of his chosen identities.
A small portion of this lattice is shown on Wikipedia:
I'm wondering if there's a good explanation of 'why' any variety of bands can be defined using just one identity. Fennemore's proof is not easy for me to follow. Are there are other varieties, besides bands, that have this property?