A note on the convexity of lattices generated by the set of nonnegative integers.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1332.06007
author: Jozef Pócs
published in: Mathematica Slovaca
publication date: 2014-08-21T00:00:00Z
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06B99
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 06A05
zbMATH DE Number: 6333341
zbMATH Keywords: lattice convexities
zbMATH Keywords: ultraproducts
zbMATH Keywords: infinite ordinal numbers
MaRDI profile type: Publication
full work available at URL: s12175-014-0225-7
OpenAlex ID: W2067243980
DOI: 10.2478/S12175-014-0225-7
title: A note on the convexity of lattices generated by the set of nonnegative integers.
review text: The notion of convexity of lattices is due to \textit{E.~Fried} (Krems, 1988); he defined it as a nonempty class of lattices which is closed with respect to homomorphisms, direct products and convex sublattices. \textit{J.~Lihová} [Publ. Math. 72, No. 1-2, 35-43 (2008; Zbl 1164.06001)] investigated relations between convexities generated by chains and particularly, she proved that convexities generated by nonisomorphic finite chains are different.NEWLINENEWLINE The present paper shows that an analogous assertion fails to hold for convexities generated by different infinite ordinals. Namely, it is proved that all infinite ordinals generate the convexity which can be generated by \(\omega_0\); this is a consequence of the fundamental result being as follows: The convexity generated by nonnegative integers contains all ordinal numbers. The used model-theoretic approach throughout the paper appears to be appropriate and effective.
reviewed by: Danica Jakubiková-Studenovská
description: scientific article; zbMATH DE number 6333341