Two families of Newman lattices
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.
scientific article; zbMATH DE number 679612
scientific article