If a class of binary relations is first-order axiomatizable, are their reflexive reductions axiomatizable w/o equality?

Question

Let $C$ be an arbitrary first-order-with-equality axiomatizable class of binary relations, and let $C'$ be the class of the reflexive reductions of those relations, meaning those relations minus the identity relation. Is $C'$ a first-order-without-equality axiomatizable class of relations? The second question I want to mention is that I have become very interested in the model theory of first order logic w/o equality, as some of my questions show. Is there some book or paper where I may find further information?

Answer 1

No. For example, the class of (strict) linear orders in the language $\{<\}$ is equal to its own reflexive reduction. But this class is not axiomatizable in first-order logic without equality. The problem is with expressing the trichotomy condition: $\forall x\, \forall y\, (x<y \lor y<x \lor x = y$).

As a silly way of seeing that this class is not axiomatizable, note that any two nonempty sets in which $<$ is interpreted as $\emptyset$ are indistinguishable in first-order logic without equality. But such a set is a strict linear order if and only if it has size $1$.

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More generally, elementary-without-equality classes are closed under the following "blow-up" operation. Let $M$ be any first-order structure in a relational language, let $M'$ be any set, and let $f\colon M'\to M$ be any surjective map of sets. We turn $M'$ into a structure by setting $(a_1,\dots,a_n)\in R^{M'}$ if and only if $(f(a_1),\dots,f(a_n))\in R^M$. The idea is that we've replaced every element $b\in M$ with a nonempty set of elements $f^{-1}(\{b\})$.

Now you can prove by induction that for any formula $\varphi$ without equality and any tuple $\overline{a}$ from $M'$, $M'\models \varphi(\overline{a})$ if and only if $M\models \varphi(f(\overline{a}))$. In particular, $M$ and $M'$ satisfy the same sentences. So if you have a class of structures which is not closed under "blow-up", there's no hope for it to be axiomatizable in first-order logic without equality.