Are Boolean orders axiomatizable?

Question

Consider the class of Boolean algebras $(B, \vee, \land, ', 0,1)$. We define an order relation $\leq$ via $x \leq y$ iff $x \vee y = y$. I define a Boolean order to be any order isomorphic to such an order. Is the class of Boolean orders axiomatizable, and if it is, is it finitely axiomatizable?

Answer 1

The answer to this question is yes, for exactly the same reason as in the case of your earlier question about real closed fields. Basically, there is a definable process for expanding a Boolean order into a Boolean algebra, and so we can "reverse" this to translate the Boolean algebra axioms into a set of Boolean order axioms. Moreover, since the former are finite so are the latter.

This is a special case of the general principle stated in the above-linked answer, which gives a general recipe for arguing that pseudoelementary classes are elementary (of course this doesn't always work but it does frequently work).