Tripartite Unions

This note provides conditions under which the union of three well-founded binary relations is also well-founded. This note concerns conditions under which the union of several wellfounded (binary) relations is also well-founded.1 To garner insight, we tackle just three relations, A, B, and C, over some underlying set V . Let {A|B} denote A ∪ B, and so on for other unions of relations. And let juxtaposition indicate composition of relations and superscript ∗ signify transitive closure. We’ll refer to the relations as “colors”. Theorem 1 (Ramsey). The union {A|B|C} is well-founded if {A|B|C}{A|B|C} ⊆ {A|B|C} (1) Proof. The infinite version of Ramsey’s Theorem applies when the union is transitive, so that every two (distinct) nodes within an infinite chain in the union of the colors has a colored (directed) edge. Then, there must lie an infinite monochrome subchain within any infinite chain, contradicting the well-foundedness of each color alone. Only three of the nine cases are actually needed for the limited outcome that we are seeking (an infinite monochromic path, rather than a clique—as in Ramsey’s Theorem), as we observe next. Date: June 6, 2016. By well-founded, we mean the absence of infinite forward-pointing paths. For some of the history of well-foundedness based on Ramsey’s Theorem, see Pierre Lescanne’s Rewriting List, contributions 38–41 at http://www.ens-lyon. fr/LIP/REWRITING/CONTRIBUTIONS and Andreas Blass and Yuri Gurevich, “Program Termination and Well Partial Orderings”, ACM Transactions on Computational Logic 9(3), 2008 (available at http://research.microsoft.com/en-us/um/ people/gurevich/Opera/178.pdf). See Alfons Geser, Relative Termination, Ph.D. dissertation, Fakultät fur̈ Mathematik und Informatik, Universität Passau, Germany, 1990 (Report 91-03, Ulmer Informatik-Berichte, Universität Ulm, 1991; available at http://homepage.cs. uiowa.edu/~astump/papers/geser_dissertation.pdf). 1