Usbrth

Birkhoff's HSP theorem states that if a class of algebras (for a given type) is closed under products, subalgebras and homomorphic images ($iff$ quotient), then it is actually defined by some equations.

I was wondering what could be said about other combinations of these operators $H,S,P$, in syntactic terms. The one I'm most interested about would be a syntactic characterization of classes closed under $HP$, but if there are results for each one of $H, S,P, HS,HP,SP$ I am interested as well. I'm mostly thinking about first order logic, but if there are some results involving various infinitary logics (though reasonable ones, I don't want a formula like $bigvee_Ain C$ "have the same elementary diagram as $A$") I am happy to hear about them too.

A few thoughts :

For $P$ it seems that any formula of the form $forall overlinex, varphi$ with $varphi$ of the form $Aimplies B$, where $A,B$ are conjunctions of equations is preserved under $P$; but this reminds me of Horn formulas, and I seem to recall them characterizing classes that are stable under reduced products, so that's not enough.

For $H$, it seems unreasonable to have a syntactic characterization: assume we have one, and apply it for $mathbbZ$ (as a group, so we have operations $0,+, -$) : $HmathbbZ$ is the class of cyclic groups (including $mathbbZ$), so we would have a theory $T$ whose models are precisely the cyclic groups, but this is impossible by Löwenheim-Skolem since a cyclic group is at most countable.

A similar argument shows that it's unreasonable for $S$ in general.

For $HP$, which is the one I'm most interested in, it seems related to reduce products, so to Horn formulas, but there are more quotients of products than just quotients by filters (or it seems so). I noticed that if $varphi$ is a conjunction of quantifier-free equations, then $forall exists ... forall exists varphi$ seems to be preserved under $HP$. This question already asks for some references about this, but the only comment that gives a reference gives one to a paper I don't have access to, and it doesn't seem to deal with syntactic matters.

For $HS$, we have the same problem as for $H$ and $S$.

For $SP$, a whole bunch of universal formulas seem to work, but I haven't given it much thought since I'm less interested in that one.

(the arguments I gave for $H,S,HS$ of course don't work if I change the context and go to something like infinitary logic, that's why I'm not considering that it completely closes the question; but again, I'm more interested in first order logic and I'm more interested in $HP$ so if you don't want to think about them you can focus on $HP$)

This is not really an answer to the question, just an observation that may help you to refine your question to get better answers.

In fact none of the closure properties you are interested in have syntactic characterizations in first-order logic, because there are classes closed under each of these properties which are not elementary. You've already observed that it's clear from Löwenheim-Skolem that there are classes closed under H, S, and HS that are not elementary (since any homomorphic image and any substructure of $M$ have cardinality $leq |M|$). It also follows from Löwenheim-Skolem that there are non-elementary classes closed under product: e.g. no product of a $2$ element set is countably infinite. So we're left with HP and SP.

For HP, consider the language with a single unary function $S$, and consider the class $K$ of all structures $M$ in this language such that there exists a homomorphism $(mathbbZ,S)to M$, where $S$ is the successor function on $mathbbZ$. $K$ is clearly closed under homomorphic images and products (by composition and the universal property of the product). But it is not closed under elementary equivalence: $(mathbbN,S)$ is not in $K$ (where $S$ is the successor function on $mathbbN$), but any proper elementary extension of this structure is in $K$.

For SP, consider the language with unary functions $(f_n)_nin omega$ and another unary function $g$. Let $K$ be the class of all structures $M$ in this language satisfying the infinitary axiom $$forall x, left(left(bigwedge_nin omega f_n(x) = xright) rightarrow g(x) = xright).$$
It's easy to see that $K$ is closed under substructures and products, but $K$ is not elementary: there is a structure $M$ in $K$ which has an element $x_N$ satisfying $bigwedge_n<Nf_n(x_N) = x_N$ but $g(x_N)neq x_N$ for all $Nin omega$. By compactness, $M$ has an elementary extension $M'$ realizing the type $f_n(x) = xmid nin omegacup g(x)neq x$, and $M'notin K$.

On the other hand, many of the closure properties in your question do have first-order syntactic characterizations if you also assume the class is elementary (or sometimes pseudo-elementary is enough). And as you suggested in your question, if you don't want to do this, you can still sometimes get a characterization in infinitary logic.

For example, let's consider the SP closure condition. An elementary class $K$ of algebras which is closed under substructures and products is called a quasivariety. Now it's a theorem that $K$ is a quasi-variety if and only if $K$ is axiomatized by first-order universal Horn sentences: sentences of the form $$forall overlinex, left(left(bigwedge_i=1^n varphi_i(overlinex) right)rightarrow psi(overlinex)right),$$ where the $varphi_i$ and $psi$ are equations.

It's also a theorem that every pseudo-elementary class closed under substructure and product is actually a quasivariety, and hence elementary.

If you don't even want to assume pseudo-elementary, a general SP-closed class is sometimes called a "prevariety", and these can always be axiomatized by the infinitary logic ($L_kappa,kappa$ for large enough $kappa$) version of universal Horn sentences. The non-elementary example I gave above was axiomatized in this way (in $L_omega_1,omega$).

One of the most interesting things (to me) about Birkhoff's HSP theorem is that it works even without the assumption that $K$ is elementary. This can be explained by the fact that elementary classes can be characterized by preservation under ultraproducts and ultraroots; the former is a homomorphic image of a product, and the latter is a substructure.