Generalizations of Birkhoff's HSP Theorem

Question

Let $\mathbf{C}$ be the class of algebraic structures of some fixed type satisfying some sentence $\phi$. Birkhoff's HSP theorem says that $\mathbf{C}$ is closed under homomorphisms, subalgebras and products if and only if $\phi$ is equivalent to some universally quantified conjuction of equations. In the following article -

Properties preserved under algebraic construction. Author: R. C. Lyndon. Journal: Bull. Amer. Math. Soc. 65 (1959), 287-299. http://www.ams.org/journals/bull/1959-65-05/S0002-9904-1959-10321-9/

Lyndon refers to an "obvious" HD theorem at the top of p293 (as opposed to Birkhoff's UHD(=HSP) theorem) which I think means that the same result holds if $\mathbf{C}$ is only closed under homomorphisms and products, except that we now allow existential as well as universal quantifiers. Let us say we weaken the assumption on $\mathbf{C}$ just a little more, and assume that the homomorphisms considered only come from coordinate functions on products, i.e. what if we only assume that $\mathbf{C}$ satisfies

$X\times Y\in\mathbf{C}\quad\Leftrightarrow\quad X\in\mathbf{C}\textrm{ and }Y\in\mathbf{C}$

(and the same for infinite products, although I think this already follows from the above finite product assumption). What kind of sentence must $\phi$ now be equivalent to?

Answer 1

It is true that a first-order sentence, which is preserved by finite direct products, is also preserved by infinite direct products; see Corollary 6.7 of S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57-103.

In H. J. Keisler's terminology, a first-order sentence is called a product sentence if it holds in a product $X\times Y$ whenever it holds in $X$ and $Y$, a factor sentence if it holds in $X$ and $Y$ whenever it holds in $X\times Y$. If memory serves, rather complicated characterizations of product sentences and factor sentences were found by Keisler in the 1960s; the work of Keisler's student J. M. Weinstein may also be relevant. I'm pretty sure the results or at least the references are in the book Model Theory by C. C. Chang and H. J. Keisler. (I happen to have at hand a reference to Weinstein's dissertation: Joseph M. Weinstein, First order properties preserved by direct product, University of Wisconsin, Madison, 1965.)

Now, if you wanted to know which first-order sentences are preserved by reduced products (direct products reduced modulo a filter on the index set, like ultraproducts but with any old filter instead of an ultrafilter), the answer (also due to Keisler) is very nice: a first-order sentence is preserved by proper reduced products if and only if it's logically equivalent to a Horn sentence. ("Proper" here means that the index set is nonempty and the filter is a filter of nonempty sets; if you want to include the improper reduced product, insert "strict" before "Horn sentence".) I'm sure this is discussed in the Chang-Keisler book.