NDLI: Definability in Substructure Orderings, II: Finite Ordered Sets

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Definability in Substructure Orderings, II: Finite Ordered Sets

Content Provider	Springer Nature Link
Author	Ježek, Jaroslav McKenzie, Ralph
Copyright Year	2010
Abstract	Let ${{\uppercase {\mathcal{p}}}} $ be the ordered set of isomorphism types of finite ordered sets (posets), where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite poset P, the set $\{p,p^{\partial}\}$ is definable, where p and $p^{\partial}$ are the isomorphism types of P and its dual poset. We prove that the only non-identity automorphism of ${{\uppercase {\mathcal{p}}}}$ is the duality map. Then we apply these results to investigate definability in the closely related lattice of universal classes of posets. We prove that this lattice has only one non-identity automorphism, the duality map; that the set of finitely generated and also the set of finitely axiomatizable universal classes are definable subsets of the lattice; and that for each member K of either of these two definable subsets, $\{K,K^{\partial}\}$ is a definable subset of the lattice. Next, making fuller use of the techniques developed to establish these results, we go on to show that every isomorphism-invariant relation between finite posets that is definable in a certain strongly enriched second-order language $\textup{\emph L}_2$ is, after factoring by isomorphism, first-order definable up to duality in the ordered set ${{\uppercase {\mathcal{p}}}}$ . The language $\textup{\emph L}_2$ has different types of quantifiable variables that range, respectively, over finite posets, their elements and order-relation, and over arbitrary subsets of posets, functions between two posets, subsets of products of finitely many posets (heteregenous relations), and can make reference to order relations between elements, the application of a function to an element, and the membership of a tuple of elements in a relation.
Starting Page	115
Ending Page	145
Page Count	31
File Format	PDF
ISSN	01678094
Journal	Order
Volume Number	27
Issue Number	2
e-ISSN	15729273
Language	English
Publisher	Springer Netherlands
Publisher Date	2010-01-23
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Definability Ordered set Lattice Universal class Category Convex and Discrete Geometry Theory of Computation Geometry
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Computational Theory and Mathematics Geometry and Topology

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