Two families of Newman lattices
type: Item
instance of: scholarly article
title: Two families of Newman lattices
zbMATH Open document ID: 0810.06006
DOI: 10.1007/BF01190819
published in: Algebra Universalis
publication date: 1994-10-30T00:00:00Z
review text: The basic ideas for defining two types of lattices which are studied in the present paper go back to \textit{M. H. A. Newman} [Ann. Math., II. Ser. 43, 223-243 (1942; Zbl 0060.125)]. From the authors' introduction: ``We will here apply Newman's concept to the commutative and associative laws of algebra, considering ``strings'' of letters. In the first context, we will consider a `positive'' move as replacing \(xy\) and \(yx\) in a string of ordered letters (or numbers); and in the second, replacing \dots \(((EF)G)\) \dots by \dots \((E(FG))\) \dots in a ``bracketing'' or parenthesization of a string of letters (i.e. shifting a pair of parentheses from left to right). These moves define covering relations in the usual sense (of lattice theory), and indirectly two very interesting kinds of lattices. Those of the first kind, which may be called Newman ``commutativity lattices'' will be the theme of Part A below, while those of the second kind, which may be referred to as Newman ``associativity lattices'', will be discussed in Part B.'' In both parts, the corresponding historical background in described in detail. The relations to graph theory are emphasized.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 679612
zbMATH Keywords: Newman commutativity lattice
zbMATH Keywords: Newman associativity lattice
zbMATH Keywords: orthocomplemented lattice
zbMATH Keywords: covering relations
zbMATH Keywords: relations to graph theory
author: Mary Katherine Bennett
author: Garrett. Birkhoff
reviewed by: Ján Jakubík
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 679612