José São João: Representations of Finite Lattices

Abstract

Lattices are everywhere: from logic to collections of substructures to geometry. An important result in lattice theory is Birkhoff’s duality between finite distributive lattices and posets from the 1930’s. Later Urquhart gave a representation for arbitrary finite lattice, generalising Brikhoff’s result. Recently, it has been found that there is a one-to-one correspondence between finite lattices and a special class of digraphs called TiRS graphs. In this talk, I will present Birkhoff’s representation and this correspondence between lattice and TiRS. I will also include some results characterizing the TiRS graphs of semidistributive lattices which is joint work with Andrew Craig (University of Johannesburg, South Africa) and Miroslav Haviar (Matej Bel University, Slovakia).