On lattices embeddable into lattices of order-convex sets. II: Star-like posets
type: Item
instance of: scholarly article
title: On lattices embeddable into lattices of order-convex sets. II: Star-like posets
zbMATH Open document ID: 1248.06002
author: Anna Zamojska-Dzienio
author: M. V. Schwidefsky
published in: Order
publication date: 2011-12-13T00:00:00Z
review text: The authors continue the study of lattices that embed in lattices of order-convex subsets of posets. This paper is a continuation of Part I [Int. J. Algebra Comput. 17, No. 8, 1667--1712 (2007; Zbl 1152.06002)]. A poset is \textit{tree-like} if (i) whenever \(a, b \in P\) and \(a\) is majorized by \(b\) there is an ascending path from \(a\) to \(b\) in the Hasse diagram, and (ii) the Hasse diagram contains no circuits. The poset \(P\) is \textit{star-like} if it is tree-like and contains no subposet isomorphic to the four-element poset \(N\). Given a class \({\mathcal K}\) of posets, \({\mathbf{SUB}}({\mathcal K})\) denotes the class of sublattices of convexity lattices of posets in \({\mathcal K}\). With the class of star-like posets denoted by \({\mathcal S}\), the main result is that the finite lattices in \({\mathbf{SUB}}({\mathcal S})\) are characterized among all finite lattices by a finite collection of lattice identities. Such identities are given, and these include three identities that were previously shown to characterize the lattices embeddable in the lattice of order-convex sets of some poset, together with four other identities verified in the current paper. A similar characterization is determined for lattices embeddable in lattices of order-convex subsets of star-like lattices of length bounded by a postive integer \(m\).
reviewed by: James F. Lawrence
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 06B20
zbMATH DE Number: 5988078
zbMATH Keywords: embedding
zbMATH Keywords: join-semidistributive
zbMATH Keywords: tree-like
zbMATH Keywords: \(N\)-free
zbMATH Keywords: order-convex
zbMATH Keywords: star-like
MaRDI profile type: Publication
full work available at URL: s11083-010-9189-6
OpenAlex ID: W2107944948
cites work: The convexity lattice of a poset
cites work: Ordres "C.A.C."
cites work: Q5480767
cites work: Sublattices of lattices of order-convex sets. I: The main representation theorem
cites work: SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, II.
cites work: SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III: THE CASE OF TOTALLY ORDERED SETS
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cites work: Lattices of order-convex sets of forests
DOI: 10.1007/S11083-010-9189-6
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description: scientific article; zbMATH DE number 5988078