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

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