Decidability of the restricted theories of a class of partial orders

Classical algebraic geometry studies the solution sets of algebraic equations over the fields of real and complex numbers. In the past 20 years, the so-called universal algebraic geometry, which studies systems of equations over arbitrary algebraic structures, has been actively developed. In this frameworks, universal and existential theories are very important, the prospect for constructing good algebraic geometry over algebraic systems depends on their complexity. In this paper, we prove that the existential and universal theories of the class of all finite orders are decidable.