Orthonormal bases in Inverse semigroups, a categorical approach

Abstract

The role the choices of orthogonal bases play in the structure of the category Hilb, remains a problem in categorical quantum mechanics. In this thesis we take a closer look at the finite dimensional Hilbert spaces. We will show that there is a link between these and certain inverse semigroups. Thus reducing this geometric problem to a problem in the field of combinatorics. Using the work of Samson Abramsky et al., we will show that we have an equivalence between the categories Frob(PInj) and Frob(Hilb) of Frobenius semigroups. Next, we will study symmetric inverse semigroups and construct the category RepInv of representable inverse semigroups. Using the above equivalence and theWagner-Preston representation we prove that we have an adjunction between RepInv and Frob(Hilb). This shows that these inverse semigroups carry a structure similar to that of finite- dimensional Hilbert spaces.