Does every commutative variety of algebras have a cogenerator?

Jacob: 2 days ago

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.

Equivalently (i) every interpretation of every operation defines an algebra homomorphism, or (ii) the hom-sets have pointwise algebraic structure, or (iii) $\mathcal{V}$ forms a symmetric closed monoidal category with its tensor product (the latter being definable as a bi-functor in any variety).

By a cogenerator I mean some $K \in \mathcal{V}$ such that any two distinct morphisms $\alpha,\beta : A \to B$ have a respective `predicate' $h : B \to K$ satisfying $h \circ \alpha \neq h \circ \beta$.

I have two questions:

  1. Do commutative varieties necessarily have a cogenerator?
  2. Do locally finite commutative varieties necessarily have a finite cogenerator?

I include some positive examples.

  1. If $(\Sigma,E)$ at least contains a binary operation and a respective unit, then $\mathcal{V}$ is essentially the modules for some commutative semiring. Examples are abelian groups with cogenerator $\mathbb{Q}/\mathbb{Z}$, vector spaces with cogenerator $k$, join-semilattices with zero with cogenerator $2$.

  2. Other examples with a two element cogenerator include sets, pointed sets, semilattices without zero, and the variety defined by a single operation and equation $u(x) = u(y)$ (pointed sets with an additional initial object).

  3. Actions of a commutative monoid have a cogenerator, since they form a topos $[M,\mathsf{Set}]$. If the monoid is finite, there is a finite cogenerator.


Let me add that:

  1. I do not know if commutative monoids have a cogenerator, or whether there is one which works for the finitely-generated = finitely-presentable $\mathbb{N}$-modules.

  2. The commutative inverse monoids define a commutative variety. They extend commutative monoids with a single involutive unary operation, which (i) preserves the monoid structure, (ii) satisfies $x = x \cdot u(x) \cdot x$. They have a cogenerator $2 \times \mathbb{Q}/\mathbb{Z}$ where $2$ is the two-chain.

Sadie: 2 days ago

The answer is no.

Let $A$ be the algebra with universe $\{0,1\}$ and fundamental operations $f(x,y,z)=x+y+z \pmod{2}$ and $g(x)=x+1\pmod{2}$. Then $f$ and $g$ commute with each other and with themselves, so the variety generated by $A$ is commutative. This variety has a weird property: on every $B\in \mathcal V(A)$ the operation $g$ interprets either as a fixed point free involution (type 1) or as the identity function (type 2). Moreover, every group of exponent 2 can be modified slightly to make it an algebra of type i in this variety for i = 1 OR 2.

Now, suppose that $K$ is a cogenerator for $\mathcal V(A)$. Necessarily $|K|>1$.

$K$ is not of type 1.

Assume otherwise. Let $B\in\mathcal V(A)$ be the 2-element type 2 algebra. There are maps $\alpha,\beta\colon B\to B$ where $\alpha = id$ and $\beta$ is a constant function. These maps cannot be separated by a map $h\colon B\to K$, since there is no homomorphism from a type 2 algebra to a type 1 algebra.

$K$ is not of type 2.

Assume otherwise. If $B\in\mathcal V(A)$ is of type 1, then $id, g\colon B\to B$ cannot be separated by any map $h\colon B\to K$ since you cannot separate elements of the same $g$-orbit of a type 1 algebra by a homomorphism into a type 2 algebra.