Elegant and general algorithms for handling upwards-closed and downwards-closed subsets of WQOs can be developed using the filter-based and ideal-based representation for these sets. These algorithms can be built in a generic or parameterized way, in parallel with the way complex WQOs are obtained by combining or modifying simpler WQOs.
@article{arxiv.1904.10703,
title = {The Ideal Approach to Computing Closed Subsets in Well-Quasi-Ordering},
author = {Jean Goubault-Larrecq and Simon Halfon and Prateek Karandikar and K. Narayan Kumar and Philippe Schnoebelen},
journal= {arXiv preprint arXiv:1904.10703},
year = {2019}
}