Definable combinatorics with dense linear orders

Himanshu Shukla; Arihant Jain; Amit Kuber

doi:10.1007/s00153-020-00709-8

Shukla, Himanshu; Jain, Arihant; Kuber, Amit

2020-01-22 00:00:00

We compute the model-theoretic Grothendieck ring, K0(Q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_0({\mathcal {Q}})$$\end{document}, of a dense linear order (DLO) with or without end points, Q=(Q,<)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {Q}}=(Q,<)$$\end{document}, as a structure of the signature {<}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{<\}$$\end{document}, and show that it is a quotient of the polynomial ring over Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {Z}}$$\end{document} generated by N+×(Q⊔{-∞})\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {N}}_+\times (Q\sqcup \{-\infty \})$$\end{document} by an ideal that encodes multiplicative relations of pairs of generators. This ring can be embedded in the polynomial ring over Q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {Q}}$$\end{document} generated by Q⊔{-∞}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q\sqcup \{-\infty \}$$\end{document}. As a corollary we obtain that a DLO satisfies the pigeon hole principle for definable subsets and definable bijections between them—a property that is too strong for many structures.

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Definable combinatorics with dense linear orders

$We compute the model-theoretic Grothendieck ring, K0(Q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_0({\mathcal {Q}})$$\end{document}, of a dense linear order (DLO) with or without end points, Q=(Q,<)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {Q}}=(Q,<)$$\end{document}, as a structure of the signature {<}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{<\}$$\end{document}, and show that it is a quotient of the polynomial ring over Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {Z}}$$\end{document} generated by N+×(Q⊔{-∞})\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {N}}_+\times (Q\sqcup \{-\infty \})$$\end{document} by an ideal that encodes multiplicative relations of pairs of generators. This ring can be embedded in the polynomial ring over Q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {Q}}$$\end{document} generated by Q⊔{-∞}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q\sqcup \{-\infty \}$$\end{document}. As a corollary we obtain that a DLO satisfies the pigeon hole principle for definable subsets and definable bijections between them—a property that is too strong for many structures.$

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Definable combinatorics with dense linear orders

Definable combinatorics with dense linear orders

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Definable combinatorics with dense linear orders

Definable combinatorics with dense linear orders

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