Carsten Dietzel, Structure groups of orthomodular lattices.

Abstract:

Boolean algebras represent the algebraic structure of classical logic. In the same way, orthomodular lattices (OMLs) represent the algebraic structure of quantum logic. In this talk, we will present a connection between OMLs and right l-groups. In the first part, we will explain how to construct for each OML X a structure group G(X). This assignment gives an equivalence between the category of OMLs and the category of right l-groups with a singular strong order unit. In the second part, we will restrict to the case where X is the OML of subspaces of a finite-dimensional vector space with an anisotropic, symmetric bilinear form. In this case, G(X) admits an easy description in terms of matrix groups over Laurent rings.