Order-theoretic Trees: Monadic Second-order Descriptions and Regularity

Abstract

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch (a maximal linearly ordered subset) can be any countable linear order. Such generalized infinite trees yield convenient definitions of the rank-width and the modular decomposition of countable graphs. We define an algebra based on only four operations that generate up to isomorphism and via infinite terms these order-theoretic trees and forests.We prove that the associated regular objects, i.e., those defined by regular terms, are exactly the ones that are the unique models of monadic second-order sentences. We adapt some tools that we have used in a previous article for proving a similar result for join-trees, i.e., for order-theoretic trees such that any two nodes have a least upper-bound.