Does the category of Lawvere theories have products?

Question

I know Law has a tensor product, is closed with respect to that tensor product, and it has coproducts. Does it have products?

My best guess at the cartesian product of Lawvere theories is the "intersection" of the theories: say $Th_1$ has a sort $X,$ function symbols $f_i\colon X^{n_i} \to X$ and a set of equations $R$; say $Th_2$ has a sort $Y,$ function symbols $g_j\colon Y^{m_j} \to Y$ and a set of equations $S$.

Then the product will have a sort $X\times Y$, function symbols given by all pairs $(f_i, g_j)\colon (X\times Y)^{n_i = m_j} \to (X\times Y)$ with the same signature, and the "intersection" $R \cap S$ of the equations.

For instance, suppose $Th_1$ is the theory of abelian groups with sort $A$ and $Th_2$ is the theory of monoids with sort $M$; then the result would have one sort $A\times M$, function symbols $(+, \cdot)\colon (A\times M) \times (A\times M) \to (A\times M)$ and $(0, e)\colon 1 \to (A\times M)$ subject to associativity and unit laws, but not commutativity. That is, the product is just the theory of monoids.

Answer 1

According to

Fajtlowicz, S. Birkhoff's theorem in the category of non-indexed algebras. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 1969 273-275.

a product of algebras and varieties was introduced by W. Narkiewicz. The terminology ``nonindexed product'' was used. It is stated in the above paper that the nonindexed product is the category-theoretic product in the category of nonindexed algebras.

This is basically the thing that you are describing: the product of the clones of the varieties within the category of clones, or the product of their algebraic theories within the category of algebraic theories. Given varieties $\mathcal U$ and $\mathcal V$, the models of ${\mathcal U}\times {\mathcal V}$ are those algebras isomorphic to a set-theoretical product of some algebra $A\in \mathcal U$ with some algebra $B\in \mathcal V$, whose $n$-ary operations are pairs $(f,g)$ where $f$ is an $n$-ary operation of $A$ and $g$ is an $n$-ary operation of $B$. It is clear how these operations should act on $A\times B$, namely $$ (f,g)((a_1,b_1),\ldots,(a_n,b_n)) = (f(a_1,\ldots,a_n),g(b_1,\ldots,b_n)). $$

The conjecture that the product of the variety $\mathcal A$ of abelian groups with the variety $\mathcal M$ of monoids is the variety of monoids is not correct. There are two isotypes of monoids of size $2$, but there are $3$ isotypes of algebras in $\mathcal A\times \mathcal M$ that have size $2$.

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Let me edit this to respond to comments: "isotype" = "isomorphism type". To expand on the preceding paragraph, let $A_2$ be the $2$-element group. Let $MA_2$ be the $2$-element monoid that is a group, and let $MS_2$ be the $2$-element monoid that is a semilattice. Let $*$ denote a $1$-element algebra of any type. Then the $3$ isotypes of $2$-element algebras in ${\mathcal A}\times {\mathcal M}$ are $A_2\times *$, $*\times MA_2$ and $*\times MS_2$. Observe that $A_2\times *$ and $*\times MA_2$ are not isomorphic, since there is a binary operation of ${\mathcal A}\times {\mathcal M}$ of the form $(f,g)(x,y)=(x+y,x)$. This agrees with the group operation on $A_2\times *$ but not on $*\times MA_2$.

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Let me edit again to respond (slightly) to a question about identities satisfied by a product ${\mathcal U}\times {\mathcal V}$. Given varieties $\mathcal U$ and $\mathcal V$ and their identities, the identities of ${\mathcal U}\times {\mathcal V}$ have been worked out. I learned about a number of sources from Walter Taylor:

Taylor, Walter The fine spectrum of a variety. Algebra Universalis 5 (1975), no. 2, 263-303. [See Proposition 0.9]

McKenzie, Ralph On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model. J. Symbolic Logic 40 (1975), 186–196.

García, O. C.; Taylor, W. The lattice of interpretability types of varieties. Mem. Amer. Math. Soc. 50 (1984), no. 305, v+125 pp. [See Definition preceding Proposition 3.]

I won't list the details here, but will note that this issue (identities defining the product) has a complicated history. Taylor mentioned the names: Newman, Foster, Pixley, Knoebel, Gratzer, Lakser, Plonka, Hu, Kelenson, Bernardi, Draskovicova, Chang, Jonsson, Tarski, Fajtlowicz, Lawvere, McKenzie, Garcia, Taylor.