My journey into noncommutative lattices and their theory

Abstract

This paper describes the motivations leading to a renewed interest in the study of noncommutative lattices, and especially skew lattices, beginning with the initial work of the author. Not only are primary concepts and results recalled, but recognition is given to the individuals involved and their particular contributions. It is the written version of a talk given at the NCS2018 workshop in May,

Regular antilattices

“…For this common congruence, the quotient algebra S/D is the maximal lattice image and each D-class of S is a maximal sub-antilattice of S. In brief, every quasilattice is a lattice of antilattices. (Compare Corollary 3 of [4]; see also [6]. )…”

“…For this common congruence, the quotient algebra S/D is the maximal lattice image and each D-class of S is a maximal sub-antilattice of S. In brief, every quasilattice is a lattice of antilattices. (Compare Corollary 3 of [4]; see also [6].) Antilattices have been studied, not only due to their connection to quasilattices, but also in connection with magic squares and finite plains.…”

“…The reader seeking further information on bands is referred to the presentations given in Clifford and Preston [2] and in Howie [3]. For further background on skew lattices and quasilattices, see [4] and [6]. The basic facts of universal algebras, and in particular varieties, may be found in the second chapter of [1].…”

On elementary, odd, semimagic and other classes of antilattices

“…In [12] it was shown that antilattices play a structural role in the theory of noncommutative lattices, where "noncommutative" is to be understood as "not necessarily commutative". For introductions to the modern theory of noncommutative lattices, see [14,15].…”

Noncommutative frames revisited

“…The definition of noncommutative frame fits in the general theory of skew lattices, a theory that goes back to Pascual Jordan [6] and is an active research topic starting with a series of papers of the third author [8,9,10]. For an overview of the primary results on skew lattices, we refer the reader to [12] or the earlier systematic survey [11].…”