Definability in substructure orderings. II: Finite ordered sets
type: Item
instance of: scholarly article
title: Definability in substructure orderings. II: Finite ordered sets
zbMATH Open document ID: 1209.08003
author: Ralph McKenzie
author: Jaroslav Ježek
published in: Order
publication date: 2010-07-02T00:00:00Z
review text: In this paper the authors study first-order definability in the ordered set \( \mathcal{P}\) of isomorphism types of finite ordered sets (posets), where the ordering is by embeddability. They 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. Also, they prove that the only non-identity automorphism of \(\mathcal{P}\) is the duality map. Then they apply these results to investigate definability in the closely related lattice of universal classes of posets. Also, they prove that this lattice has only one non-identity automorphisms, the duality map; that the set of finitely generated universal classes as well as 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, a definable subset \(\{K,K^{\partial }\}\) is a definable subset of the lattice. For Part I see Algebra Univers. 61, No.~1, 59--75 (2009; Zbl 1181.08004).
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 03C40
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 5729753
zbMATH Keywords: category
zbMATH Keywords: first-order definability
zbMATH Keywords: isomorphism types of finite ordered sets
zbMATH Keywords: dual poset
zbMATH Keywords: lattice of universal classes of posets
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
full work available at URL: s11083-010-9141-9
OpenAlex ID: W1992111324
cites work: Avoidable structures. I: Finite ordered sets, semilattices and lattices
cites work: Avoidable structures. II: Finite distributive lattices and nicely structured ordered sets
cites work: Definability in substructure orderings. I: Finite semilattices
cites work: Definability in substructure orderings. III: Finite distributive lattices
cites work: Definability in substructure orderings. IV: Finite lattices
cites work: What is reconstruction for ordered sets?
DOI: 10.1007/S11083-010-9141-9
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description: scientific article; zbMATH DE number 5729753