Working Seminars on Noncommutative Lattices - Talk 2: Distributivity and cancellation. Varieties of skew lattices. Skew lattices in rings.

Pascual Jordan was the first to study noncommutative lattices in 1949. Skew lattices have been the most successful variation of noncommutative lattices. Jonathan Leech studied a more general version of these algebras and was later interested in their Boolean version termed skew Boolean algebras. The left-handed version of that case includes the class of Boolean skew algebras earlier studied by W.D. Cornish. R.J. Bignall, following ideas of Keimal and Werner, observed a subclass of skew Boolean algebras constitutes a decidable discriminator variety. In collaboration with J. Leech, R. Veroff, R.J. Bignall and M. Spinks have studied general properties of these algebras and used them in the study of multiple valued logic. A special attention has been always devoted to skew lattices in rings, that constitute a large class of examples, where Karin Cvetko-Vah and JPC answered several open questions. Today the classical dualities as Stone’s and Priestley’s are a focus of research in this context, where several relevant results have been achieved. http://www.joaopitacosta.info/home/news/skewlatticeworkshopatcaullisbon/