# Generalizations of Birkhoff's HSP Theorem

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?

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.)