Lattice Ordered O-Minimal Structures

Abstract

We propose a notion of $o$-minimality for partially ordered structures. Then we study $o$-minimal partially ordered structures $(A, \leq, \ldots)$ such that $(A,\leq)$ is a Boolean algebra. We prove that they admit prime models over arbitrary subsets and we characterize $\omega$-categoricity in their setting. Finally, we classify $o$-minimal Boolean algebras as well as $o$-minimal measure spaces.