34 research outputs found
Classical Structures Based on Unitaries
Starting from the observation that distinct notions of copying have arisen in
different categorical fields (logic and computation, contrasted with quantum
mechanics) this paper addresses the question of when, or whether, they may
coincide. Provided all definitions are strict in the categorical sense, we show
that this can never be the case. However, allowing for the defining axioms to
be taken up to canonical isomorphism, a close connection between the classical
structures of categorical quantum mechanics, and the categorical property of
self-similarity familiar from logical and computational models becomes
apparent.
The required canonical isomorphisms are non-trivial, and mix both typed
(multi-object) and untyped (single-object) tensors and structural isomorphisms;
we give coherence results that justify this approach.
We then give a class of examples where distinct self-similar structures at an
object determine distinct matrix representations of arrows, in the same way as
classical structures determine matrix representations in Hilbert space. We also
give analogues of familiar notions from linear algebra in this setting such as
changes of basis, and diagonalisation.Comment: 24 pages,7 diagram
Automatic structures for semigroup constructions
We survey results concerning automatic structures for semigroup
constructions, providing references and describing the corresponding automatic
structures. The constructions we consider are: free products, direct products,
Rees matrix semigroups, Bruck-Reilly extensions and wreath products.Comment: 22 page