Jonathan Leech - Noncommutative lattices: foundational issues and recent results

Since the 1940s, a number of individuals have considered algebras which, while noncommutative, nonetheless resemble lattices. Typically, such algebras include a pair of binary operations which are assumed to be associative and to satisfy certain absorption identities, but are not necessarily commutative. Because of the different types of absorption possible in a noncommutative context, there has been considerable variation in the classes of algebras introduced. The talk will focus on issues such as: What should one expect a noncommutative lattice to look like? What is the role of lattice theory in framing such an expectation? Likewise, what is the role of semigroup theory? These and other issues will be discussed in the light of recent developments, some as yet unpublished.