Algebraic and logical descriptions of generalized trees

TL;DR

This paper introduces algebraic and logical frameworks for generalized trees, including quasi-trees and join-trees, and characterizes regular objects as models of monadic second-order sentences, extending prior work on linear orders.

Contribution

It develops algebraic and logical descriptions for generalized trees like quasi-trees and join-trees, and characterizes regular objects through monadic second-order logic.

Findings

Regular objects correspond to models of MSO sentences.

Algebras with finitely many operations generate these generalized trees.

Results extend W. Thomas's work on countable linear orders.

Abstract

Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound of the rank-widths of its finite induced subgraphs. Join-trees are the corresponding directed trees. They are useful to define the modular decomposition of a countable graph. We also consider ordered join-trees, that generalize rooted trees equipped with a linear order on the set of sons of each node. We define algebras with finitely many operations that generate (via infinite terms) these generalized trees. 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. These results use and generalize a similar result by W. Thomas for countable linear orders.