On the product of two finitely generated $\Omega$-algebras.

Question

Let $\Omega$ be a signature without predicate symbols and let $\mathcal{A}$ and $\mathcal{B}$ be two finitely generated $\Omega$-algebras. Consider their product in the category of all $\Omega$-algebras, let it be $\mathcal{A} \times \mathcal{B}$.

Is it true that $\mathcal{A} \times \mathcal{B}$ is also finitely generated?

I know that it is true for monoids, for example. But maybe there exists a simple example why it is not true for an arbitrary two finitely generated $\Omega$-algebras? Or is it always true?

Answer 1

Here is an explicit counterexample (covered by the theorem cited in Martin's answer). Let $N$ be the semigroup of natural numbers (excluding $0$) under addition. This is finitely generated by $\{ 1 \}$. On the other hand, $N \times N$ is the semigroup of pairs of natural numbers (excluding $0$) under addition. This semigroup is not finitely generated, because there are infinitely many pairs of the form $(1, n)$ or $(n, 1)$ and they are indecomposable; they can't be written as a nontrivial sum of other pairs at all.

Answer 2

When $A,B$ are two finitely generated monoids, then $A \times B$ is finitely generated because it is generated by $\{1\} \times F \,\cup\, E \times \{1\}$, where $E \subseteq A$, $F \subseteq B$ are finite generating sets. This hints that the claim might be false for semigroups, which lack a neutral element.

Theorem. If $A,B$ are two infinite semigroups, then $A \times B$ is finitely generated iff $A$ and $B$ are finitely generated and $A^2 = A$, $B^2 = B$ (i.e., every element of $A$ is decomposable, the same for $B$).

This is Theorem 2.1 in Generators and relations of direct products of semigroups by E. F. Robertson, N. Ruskuc, J. Wiegold (pdf). In particular, there are lots of finitely generated semigroups whose product is not finitely generated (see Qiaochu's answer for an explicit one).

The proof of the Theorem also gives the estimate $\mathrm{rank}(A \times B) \leq 4 \cdot \mathrm{rank}(A) \cdot \mathrm{rank}(B)$ if $A \times B$ is finitely generated (which is much "worse" than the case of monoids).

More general questions of this type (When is a direct product finitely generated / presented?) are covered in Finiteness properties of direct products of algebraic structures by P. Mayr and N. Ruskuc (pdf). In particular, Theorem 2.2 in that paper gives a generalization of the result for monoids, groups, rings, modules, Lie algebras.

It is worthwhile to mention that the following is always true: if $A,B$ are finitely generated, then their coproduct $A \sqcup B$ is finitely generated. Actually, any finite colimit of finitely generated objects is finitely generated. You can either prove this directly with generating sets, or prove it in a much more categorical setting where $A$ is called finitely generated when $\hom(A,-)$ preserves filtered colimits of monomorphisms. Then the result follows from the well-known fact that filtered colimits commute with finite limits (in $\mathbf{Set}$). From this perspective, it is also not a coincidence that finitely generated objects are not closed under finite limits, even under binary products: there is nothing we can say about $\hom(A \times B,-)$ in general, the arrow direction is "wrong". Of course, the product of two finitely cogenerated objects is again finitely cogenerated. If $A,B$ are monoids, however, we can describe $\hom(A \times B,C)$ as the subset of $\hom(A,C) \times \hom(B,C)$ consisting of element-wise commuting homomorphisms, and this is how you get a categorical proof that $A \times B$ is finitely generated when $A,B$ are.

Answer 3

The example that jumps out at me is the set of natural numbers $\Bbb N$ equipped with the successor operation (so $\Omega$ just has a unary function symbol). The finitely generated $\Omega$-algebras are the ones that decompose into a union of finitely many orbits of this function (by orbit I mean a set which is obtained by repeatedly applying this function to some value). In this case $\Bbb N$ is generated by $1$, but its direct product with itself is $\Bbb N^2$ equipped with the operation $(a, b) \mapsto (a + 1, b + 1)$, which is very far from being finitely generated - the subalgebra generated by a point is the infinite diagonal starting from that point.

I know more model theory than universal algebra, which is perhaps why this example appeals to me (it's a pretty radical departure from the "usual" algebraic structures like groups and monoids). I think it's quite nice, and I find it easy to visualise. It is somewhat similar to the other answers (which are both excellent!) but I post it in the hope that someone else will find it useful.