Nearlattices with an overriding operation
type: Item
instance of: scholarly article
title: Nearlattices with an overriding operation
zbMATH Open document ID: 1216.06003
author: Jānis Cīrulis
published in: Order
publication date: 2011-03-31T00:00:00Z
review text: A nearlattice is a meet semilattice in which every initial segment is a lattice. It is said to be Boolean if all of these lattices are Boolean. Overriding of \(a\) and \(b\) is a binary operation defined as the supremum of all elements \(x\) such that \(x\) is below \(a\) and the join of \(x\) and \(b\) exists or \(x\) is below \(b\). In particular, overriding extends the partial join operation. Natural examples of overriding nearlattices are, e.g., trees and appropriate algebras of partial functions. For example, every complete semilattice is an overriding nearlattice. In a particular case, an overriding nearlattice becomes the so-called o-nearlattice where the operation of overriding coincides with join. In the paper, elementary properties of orerriding are studied, an axiomatic description is obtained and several results concerning associativity of this operation are presented. Functional o-semilattices are studied and subdirectly irreducible members of this variety are characterized. The main result is a representation theorem for Boolean nearlattices with associative overriding.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 06A12
Mathematics Subject Classification ID: 08A55
Mathematics Subject Classification ID: 08B26
zbMATH DE Number: 5872635
zbMATH Keywords: nearlattice
zbMATH Keywords: semilattice
zbMATH Keywords: overriding
zbMATH Keywords: tree
zbMATH Keywords: subdirect irreducibility
MaRDI profile type: Publication
full work available at URL: s11083-010-9152-6
OpenAlex ID: W2171528122
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DOI: 10.1007/S11083-010-9152-6
description: scientific article; zbMATH DE number 5872635