Tensor product of varieties : What's this notation $V_1\otimes V_2$?

Question

I saw this notation $V= V_1\otimes V_2$ in a survey on universal algebra, where $V$ was a variety, but the survey in question didn't define this notation. Could anyone explain what it means ?

Answer 1

I would have to read the survey in question to be sure, but I have most commonly seen this notation used when talking about decidable varieties.

You can find a definition here: https://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/DecidVar.pdf

Answer 2

The tensor product notation, $V_1\otimes V_2$, for some kind of product of varieties is used in (at least) two different ways.

Way 1.

For the Kronecker product, or tensor product. ($V_1\otimes V_2$ is the variety of $V_1$ models in $V_2$, and conversely.) You can find it used this way here

Freyd, P. Algebra valued functors in general and tensor products in particular. Colloq. Math. 14 1966 89-106.

in the language of algebraic theories, and here

Neumann, Walter D. Malʹcev conditions, spectra and Kronecker product. J. Austral. Math. Soc. Ser. A 25 (1978), no. 1, 103-117.

in the language of varieties. The notation $V_1\otimes V_2$ is still used to denote the tensor product of varieties.

Way 2.

$V_1\otimes V_2$ has been used to denote the categorical product in the category of varieties and clone morphisms. This product of the varieties $V_1$ and $V_2$ is the variety whose clone is the product of the clone of $V_1$ with the clone of $V_2$. I have seen it denoted by $V_1\otimes V_2$, or by $V_1\widehat{\times}V_2$, or by $V_1\times V_2$ in, for example,

García, O. C.; Taylor, W. The lattice of interpretability types of varieties. Mem. Amer. Math. Soc. 50 (1984), no. 305, v+125.

McKenzie, Ralph A new product of algebras and a type reduction theorem. Algebra Universalis 18 (1984), no. 1, 29-69.

Grätzer, G.; Lakser, H.; Płonka, J. Joins and direct products of equational classes. Canad. Math. Bull. 12 1969 741-744.

Fortunately, the use of $V_1\otimes V_2$ in the sense of Way 2 seems to be dying out, and instead $V_1\times V_2$ is being used for the categorical product.