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Suppose $K$ is an ordered field, $X$ is any set, and $F$ is a filter over $X$. Let $G$ be the reduced power of $K$ by $F$. That is, we take all functions from $X$ to $K$, then take equivalence classes by $F$. We compute all operations and the ordering of $G$ modulo $F$. (Note $F$ could be the trivial filter $\{ X \}$, in which case we just have a power of $K$.)

It is easy to check that $G$ is a characteristic-zero commutative ring, but we may lose the totality of the ordering and get zero divisors.

Question: Is there a well-known first-order axiomatization for the kind of structure as $G$, which is preserved under further reduced powers?

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  • $\begingroup$ So you're asking for a subtheory of ordered fields that is closed under reduced powers? $\endgroup$
    – Asaf Karagila ♦
    Commented Apr 23, 2021 at 10:26
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    $\begingroup$ The sentences preserved by reduced products are exactly the Horn sentences. Thus, you are asking for an axiomatization of the Horn sentences valid in ordered fields. I’m not sure whether there is a complete characterization, but clearly these include the axioms of von Neumann regular lattice-ordered commutative rings. $\endgroup$
    – Emil Jeřábek
    Commented Apr 23, 2021 at 10:36
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    $\begingroup$ Sorry, I misread power as product. But anyway, the axiomatization has to include what I wrote above. $\endgroup$
    – Emil Jeřábek
    Commented Apr 23, 2021 at 10:48
  • $\begingroup$ @AsafKaragila Yes but a nontrivial one that still does some work. We can abstractly define it as the intersection of all such theories, but I’m wondering if there’s a well-studied axiomatization. $\endgroup$
    – Monroe Eskew
    Commented Apr 23, 2021 at 10:57
  • $\begingroup$ @EmilJeřábek Thanks, this is helpful. $\endgroup$
    – Monroe Eskew
    Commented Apr 23, 2021 at 10:57

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