I want to know if there is a nice characterization of when a formula is preserved under taking reduced factors.


We say that a formula $\phi$ is closed under taking reduced factors if whenever $I$ is a set, $Z$ is a filter on $I$ and $\prod\_{i\in I}A\_{i}/Z\models\phi([(a\_{i,1})\_{i\in I}],...,[(a\_{i,n})\_{i\in I}])$, then $\{i\in I|A\_{i}\models\phi(a\_{i,1},...,a\_{i,n})\}\in Z$. Is there a good characterization of which formulas are preserved under taking reduced factors?


We say that a formula $\phi$ is closed under taking Boolean product stalks if whenever


1. $X$ is a compact zero-dimensional space
2. $A\_{x}$ is a structure for each $x\in X$
3. $A\subseteq\prod\_{x\in X}A\_{x}$ is a subalgebra where $\pi\_{x}[A]=A\_{x}$ whenever $\pi\_{x}:\prod\_{x\in X}A\_{x}\rightarrow A\_{x}$ is the projection.
4. whenever $f\_{1},...,f\_{n}\in A$ and $(C\_{1},...,C\_{n})$ is a partition of $X$ into clopen sets, then $f\_{1}|\_{C\_{1}}\cup...\cup f\_{n}|\_{C\_{n}}\in A$
5. whenever $\phi$ is a formula and $f\_{1},...,f\_{n}\in A$, then
$\{x\in X|A\_{x}\models\phi(f\_{1}(x),...,f\_{n}(x))\}$ is a clopen subset of $X$,


then whenever $A\models\phi(f\_{1},...,f\_{n})$, then $A\_{x}\models\phi(f\_{1}(x),...,f\_{n}(x))$ for each $x\in X$.


Is there a characterization of the formulas closed under taking Boolean product stalks? Are the formulas closed under taking Boolean product stalks precisely the formulas closed under taking reduced factors?


$\textbf{Background}$


A formula of the form $A\_{1}\wedge...\wedge A\_{n}\rightarrow B$ or of the form $(\neg A\_{1})\vee...\vee(\neg A\_{n})$ where $A\_{1},...,A\_{n},B$ are atomic is said to be a basic Horn formula. A formula of the form $Q\_{1}x\_{1}...Q\_{n}x\_{n}(\phi\_{1}\wedge...\wedge\phi\_{n})$ where $Q\_{1},...,Q\_{n}$ are quantifiers and $\phi\_{1},...,\phi\_{n}$ are basic horn formulae is said to be a Horn formula. It is a standard result that a formula is equivalent to a Horn formula if and only if whenever $A\_{i}\models\phi(a\_{1,i},...,a\_{n,i})$ for $i\in I$ and $Z$ is a filter on $I$, then $\prod\_{i\in I}A\_{i}/Z\models\phi([(a\_{1,i})\_{i\in I}],...,[(a\_{n,i})\_{i\in I}])$. 


It is an exercise in Chang and Keisler that every positive sentence is preserved under taking reduced factors, but not every sentence preserved under reduced factors is a positive sentence.



When is a formula preserved under taking factors in a reduced product or the stalk in a Boolean product?
(Only a sketchy partial answer.)


The family of *factorable formulas*, is the smallest set $F$ of formulas
containing every atomic formula that is closed under conjunction,
existential and universal quantification, and the following rule:



> 
> if $\alpha(\vec x),\beta(\vec x, \vec y), \gamma(\vec x, \vec y)\in
> F$ and $\models\forall \vec x \bigl(\alpha(\vec x) \rightarrow \exists \vec y
>  \beta(\vec x, \vec y)\bigr)$ then $\forall \vec y \bigl( \beta(\vec x,
>  \vec y) \rightarrow \gamma(\vec x, \vec y)\bigr) \in F.$
> 
> 
> 


It is easy to see that these formulas are preserved under direct factors **and** direct products. I was told that these formulas are preserved by reduced products and factors, though I didn't checked this. The place I found this is R. Willard, *Varieties Having Boolean Factor Congruences*, J. Algebra **132** (1990) 130--153, where a slight variant (relative to a class of models) is defined.


These are intimately related to *$h$-formulas*, defined by the simpler scheme



> 
> 1. atomic formulas are $h$-formulas;
> 2. if $A$, $B$ are $h$-formulas and $x$ is a variable, then
>  $(A\wedge B)$ and $(\exists xA\wedge\forall x(A\rightarrow B))$
>  are $h$-formulas.
> 
> 
> 


(I was informed about this family of formulas by an anonymous referee, and he cited the paper E. A. Palyutin, *Categorical Horn classes, I.*, Algebra and Logic **19** (1980) 377--400, but I have no access to it.)


Still in the *sketchy & partial* vein, I didn't checked if they are actually the same fam


Finally, although not directly related, the paper by Hugo Volger, *Preservation Theorems for Limits of Structures and Global Sections of Sheaves of Structures*, Math. Z. **166**, 27--53 (1979), might be relevant here. He gives a partial solution to the problem of characterizing sentences preserved by Boolean products.


