Seminar Abstract:

A residuated lattice is an algebra which combines a lattice and a monoid by means of residuation operations (left and right divisions). For example, the set of twosided ideals in any ring forms a residuated lattice (this lattice was studied by Birkhoff in 1934, and I believe it was the very first residuated lattice to be discovered). In 1930s residuated lattices were investigated in a series of papers by Ward and Dilworth, but then not much happened until about 20 years ago, when residuated lattices were rediscovered by two very different groups of researchers:
 algebraists (as a generalisation of latticeordered groups) and
 logicians (as algebraic semantics for proof systems known as Gentzen calculi).
The theory of residuated lattices is to a large extent a result of interactions between these two groups. I will present some of that theory, trying to meet two largely incompatible conditions: (a) to be nontechnical, (b) to focus on open problems.
