14 research outputs found
Quantum Suplattices
Building on the theory of quantum posets, we introduce a non-commutative
version of suplattices, i.e., complete lattices whose morphisms are
supremum-preserving maps, which form a step towards a new notion of quantum
topological spaces. We show that the theory of these quantum suplattices
resembles the classical theory: the opposite quantum poset of a quantum
suplattice is again a quantum suplattice, and quantum suplattices arise as
algebras of a non-commutative version of the monad of downward-closed subsets
of a poset. The existence of this monad is proved by introducing a
non-commutative generalization of monotone relations between quantum posets,
which form a compact closed category. Moreover, we introduce a non-commutative
generalization of Galois connections and we prove that an upper Galois adjoint
of a monotone map between quantum suplattices exists if and only if the map is
a morphism of quantum suplattices. Finally, we prove a quantum version of the
Knaster-Tarski fixpoint theorem: the quantum set of fixpoints of a monotone
endomap on a quantum suplattice form a quantum suplattice.Comment: In Proceedings QPL 2023, arXiv:2308.1548
A characterisation of orthomodular spaces by Sasaki maps
Given a Hilbert space , the set of one-dimensional subspaces of
becomes an orthoset when equipped with the orthogonality relation
induced by the inner product on . Here, an \emph{orthoset} is a pair
of a set and a symmetric, irreflexive binary relation
on . In this contribution, we investigate what conditions on an orthoset
are sufficient to conclude that the orthoset is isomorphic to
for some orthomodular space , where \emph{orthomodular
spaces} are linear spaces that generalize Hilbert spaces. In order to achieve
this goal, we introduce \emph{Sasaki maps} on orthosets, which are strongly
related to Sasaki projections on orthomodular lattices. We show that any
orthoset with sufficiently many Sasaki maps is isomorphic to
for some orthomodular space, and we give more conditions on
to assure that is actually a Hilbert space over ,
or .Comment: 15 pages, 5 gifure
Boolean Subalgebras of Orthoalgebras
We develop a direct method to recover an orthoalgebra from its poset of
Boolean subalgebras. For this a new notion of direction is introduced.
Directions are also used to characterize in purely order-theoretic terms those
posets that are isomorphic to the poset of Boolean subalgebras of an
orthoalgebra. These posets are characterized by simple conditions defining
orthodomains and the additional requirement of having enough directions.
Excepting pathologies involving maximal Boolean subalgebras of four elements,
it is shown that there is an equivalence between the category of orthoalgebras
and the category of orthodomains with enough directions with morphisms suitably
defined. Furthermore, we develop a representation of orthodomains with enough
directions, and hence of orthoalgebras, as certain hypergraphs. This hypergraph
approach extends the technique of Greechie diagrams and resembles projective
geometry. Using such hypergraphs, every orthomodular poset can be represented
by a set of points and lines where each line contains exactly three points.Comment: 43 page
Semantics for a Lambda Calculus for String Diagrams
To appear in a special issue in the series "Outstanding Contributions to Logic" dedicated to Samson Abramsky.Linear/non-linear (LNL) models, as described by Benton, soundly model a LNL term calculus and LNL logic closely related to intuitionistic linear logic. Every such model induces a canonical enrichment that we show soundly models a LNL lambda calculus for string diagrams, introduced by Rios and Selinger (with primary application in quantum computing). Our abstract treatment of this language leads to simpler concrete models compared to those presented so far. We also extend the language with general recursion and prove soundness. Finally, we present an adequacy result for the diagram-free fragment of the language which corresponds to a modified version of Benton and Wadler's adjoint calculus with recursion. In keeping with the purpose of the special issue, we also describe the influence of Samson Abramsky's research on these results, and on the overall project of which this is a part
LNL-FPC: The Linear/Non-linear Fixpoint Calculus
We describe a type system with mixed linear and non-linear recursive types
called LNL-FPC (the linear/non-linear fixpoint calculus). The type system
supports linear typing, which enhances the safety properties of programs, but
also supports non-linear typing as well, which makes the type system more
convenient for programming. Just as in FPC, we show that LNL-FPC supports
type-level recursion, which in turn induces term-level recursion. We also
provide sound and computationally adequate categorical models for LNL-FPC that
describe the categorical structure of the substructural operations of
Intuitionistic Linear Logic at all non-linear types, including the recursive
ones. In order to do so, we describe a new technique for solving recursive
domain equations within cartesian categories by constructing the solutions over
pre-embeddings. The type system also enjoys implicit weakening and contraction
rules that we are able to model by identifying the canonical comonoid structure
of all non-linear types. We also show that the requirements of our abstract
model are reasonable by constructing a large class of concrete models that have
found applications not only in classical functional programming, but also in
emerging programming paradigms that incorporate linear types, such as quantum
programming and circuit description programming languages
Domains of commutative C*-subalgebras
A C*-algebra is determined to a great extent by the partial order of its
commutative C*-algebras. We study order-theoretic properties of this dcpo. Many
properties coincide: the dcpo is, equivalently, algebraic, continuous,
meet-continuous, atomistic, quasi-algebraic, or quasi-continuous, if and only
if the C*-algebra is scattered. For C*-algebras with enough projections, these
properties are equivalent to finite-dimensionality. Approximately
finite-dimensional elements of the dcpo correspond to Boolean subalgebras of
the projections of the C*-algebra, which determine the projections up to
isomorphism. Scattered C*-algebras are finite-dimensional if and only if their
dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and
only if their dcpo is order-scattered.Comment: 42 page
Mixed linear and non-linear recursive types
International audienceWe describe a type system with mixed linear and non-linear recursive types called LNL-FPC (the linear/non-linear fixpoint calculus). The type system supports linear typing which enhances the safety properties of programs, but also supports non-linear typing as well which makes the type system more convenient for programming. Just like in FPC, we show that LNL-FPC supports type-level recursion which in turn induces term-level recursion. We also provide sound and computationally adequate categorical models for LNL-FPC which describe the categorical structure of the substructural operations of Intuitionistic Linear Logic at all non-linear types, including the recursive ones. In order to do so, we describe a new technique for solving recursive domain equations within the category CPO by constructing the solutions over pre-embeddings. The type system also enjoys implicit weakening and contraction rules which we are able to model by identifying the canonical comonoid structure of all non-linear types. We also show that the requirements of our abstract model are reasonable by constructing a large class of concrete models that have found applications not only in classical functional programming, but also in emerging programming paradigms that incorporate linear types, such as quantum programming and circuit description programming languages