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

Bruno Courcelle

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arXiv · 2111.04083

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

Bruno Courcelle

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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.

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Bruno Courcelle. 2021-11-07. Order-theoretic trees: monadic second-order descriptions and regularity. https://arxiv.org/abs/2111.04083

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