Historical Background
For over 60 years, noncommutative variations of lattices have been studied with differing motivations. For some the motivation has been an interest in the conceptual boundaries of lattice theory; for others it was a search for noncommutative forms of logic and Boolean algebra; and for others it has been the behavior of idempotents in rings. A noncommutative lattice, generally speaking, is an algebra where and are associative, idempotent binary operations connected by absorption identities guaranteeing that in some way dualizes . The precise identities chosen depends upon the underlying motivation, with differing choices producing distinct varieties of algebras. Pascual Jordan, motivated by questions in quantum logic, initiated a study of noncommutative lattices in his 1949 paper, Über Nichtkommutative Verbande, choosing the absorption identities
He referred to those algebras satisfying them as Schrägverbände. By varying or augmenting these identities, Jordan and others obtained a number of varieties of noncommutative lattices. Beginning with Jonathan Leech's 1989 paper, Skew lattices in rings, skew lattices as defined above have been the primary objects of study. This was aided by previous results about bands. This was especially the case for many of the basic properties.
Read more about this topic: Skew Lattice
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