Order-theoretic trees: monadic second-order descriptions and regularity

TL;DR

This paper introduces a new algebraic framework for order-theoretic trees and forests, characterizes regular objects via monadic second-order logic, and explores their applications in graph theory.

Contribution

It defines an algebra with four operations generating order-theoretic trees and forests, and proves regular objects correspond exactly to models of monadic second-order sentences.

Findings

Regular objects are exactly the models of monadic second-order sentences.

The algebra generates all order-theoretic trees and forests up to isomorphism.

Applications include defining rank-width and modular decomposition of countable graphs.

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 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, those defined by regular terms, are exactly the ones that are the unique models of monadic second-order sentences.