Identities in ordered sets
type: Item
instance of: scholarly article
title: Identities in ordered sets
zbMATH Open document ID: 0940.06001
DOI: 10.1023/A:1006214005914
author: Josef Niederle
published in: Order
publication date: 2000-07-24T00:00:00Z
review text: The paper deals with lattice-like features of ordered sets. The author introduces the notions of term and identity for ordered sets in the language using, besides the lattice operations, also the symbols \(L\) and \(U\) which are interpreted as the sets of all lower and upper bounds, respectively, of the unions of arguments. It is shown that the identities of ordered sets are in a one-one correspondence with the lattice identities. In connection with this result a question arises: which lattice identities are inherited from any ordered set to its characteristic lattice (i.e., to the lattice generated by an ordered set in its Dedekind-MacNeille completion). The author shows that this is valid for the distributive identity but it fails in the case of the modular identity.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 1340448
zbMATH Keywords: lattice identity
zbMATH Keywords: identity in ordered sets
zbMATH Keywords: lower bound
zbMATH Keywords: upper bound
zbMATH Keywords: term
zbMATH Keywords: characteristic lattice
zbMATH Keywords: distributive identity
zbMATH Keywords: modular identity
zbMATH Keywords: Dedekind-MacNeille completion
reviewed by: Jiří Rachůnek
MaRDI profile type: Publication
full work available at URL: a:1006214005914
OpenAlex ID: W2256489215
description: scientific article; zbMATH DE number 1340448