The lattice theory of \(r\)-ordered partitions
type: Item
instance of: scholarly article
title: The lattice theory of \(r\)-ordered partitions
zbMATH Open document ID: 0932.06006
DOI: 10.1016/S0012-365X(98)00128-9
author: Michel Serfati
published in: Discrete Mathematics
publication date: 2000-03-07T00:00:00Z
review text: The paper deals with ordered partitions of sets considered from the lattice-theoretical viewpoint. Namely, the set \({\mathcal{P}}_r(\Omega)\) is the set of all \(r\)-ordered partitions of some set \(\Omega\), i.e. the sequences \(P=(P^{0},P^{1},\dots,P^{r-1})\) (\(r\in N,r\geq 2\)) of subsets of \(\Omega\) such that \(\bigcup P^{i} = \Omega\) and \(P^{i}\cap P^{j}=\emptyset\), for \(i\neq j\). The author defines an order relation \(\leq\) and operations \(\oplus,\otimes\) on \({\mathcal{P}}_r(\Omega)\) and proves that \({\mathcal{P}}_r(\Omega)\) is a complete Post algebra of order \(r\) (Cor. 2.8.2) and \(({\mathcal{P}}_r(\Omega),\oplus,\otimes)\) is a commutative ring with a unit of characteristic \(r\) (Th. 4.1).
Mathematics Subject Classification ID: 06D25
Mathematics Subject Classification ID: 05A18
zbMATH DE Number: 1321847
zbMATH Keywords: ordered partition
zbMATH Keywords: distributive lattice
zbMATH Keywords: Post algebra
zbMATH Keywords: commutative ring
reviewed by: Leo Esakia
MaRDI profile type: Publication
cites work: Q5331549
cites work: Lukasiewicz-Moisil algebras
cites work: The Lattice Theory of Post Algebras
cites work: Poet algebras and pseudo-Post algebras
cites work: On Postian algebraic equations
cites work: A Note on Postian Matrix Theory
cites work: Aximos and some properties of Post algebras
description: scientific article; zbMATH DE number 1321847