Higher-arity distality and forking triviality

Mathematics > Logic [Submitted on 21 May 2026 (v1), last revised 18 Jun 2026 (this version, v2)] Title:Higher-arity distality and forking triviality View PDF HTML (experimental)Abstract:Answering a question of Goode, we show that $k$-triviality collapses to (1-)triviality among simple theories. In particular, every stable theory with quantifier elimination in a relational language of bounded arity is trivial. We use our collapse result, along with other facts about $k$-triviality and $k$-total triviality, to generate examples of (strongly) $k$-distal theories. The collapse result immediately implies that no stable theory can be strictly $k$-distal for some $k\geq 3$, partially answering a question of Walker. Moreover, all known examples of non-distal (strongly) $k$-distal theories are $k$-ary, rendering (strong) $k$-distality moot as a $(k+1)$-ary dividing line; we give four classes of examples that are not $k$-ary. We also show that just as distality is not preserved under taking reducts, neither is (strong) $k$-distality. Submission history From: Mervyn Tong [view email][v1] Thu, 21 May 2026 10:59:58 UTC (47 KB) [v2] Thu, 18 Jun 2026 09:30:34 UTC (47 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.