15 research outputs found
Unitless Frobenius quantales
It is often stated that Frobenius quantales are necessarily unital. By taking
negation as a primitive operation, we can define Frobenius quantales that may
not have a unit. We develop the elementary theory of these structures and show,
in particular, how to define nuclei whose quotients are Frobenius quantales.
This yields a phase semantics and a representation theorem via phase quantales.
Important examples of these structures arise from Raney's notion of tight
Galois connection: tight endomaps of a complete lattice always form a Girard
quantale which is unital if and only if the lattice is completely distributive.
We give a characterisation and an enumeration of tight endomaps of the diamond
lattices Mn and exemplify the Frobenius structure on these maps. By means of
phase semantics, we exhibit analogous examples built up from trace class
operators on an infinite dimensional Hilbert space. Finally, we argue that
units cannot be properly added to Frobenius quantales: every possible extention
to a unital quantale fails to preserve negations
Entropic Geometry from Logic
We produce a probabilistic space from logic, both classical and quantum,
which is in addition partially ordered in such a way that entropy is monotone.
In particular do we establish the following equation:
Quantitative Probability = Logic + Partiality of Knowledge + Entropy.
That is: 1. A finitary probability space \Delta^n (=all probability measures
on {1,...,n}) can be fully and faithfully represented by the pair consisting of
the abstraction D^n (=the object up to isomorphism) of a partially ordered set
(\Delta^n,\sqsubseteq), and, Shannon entropy; 2. D^n itself can be obtained via
a systematic purely order-theoretic procedure (which embodies introduction of
partiality of knowledge) on an (algebraic) logic. This procedure applies to any
poset A; D_A\cong(\Delta^n,\sqsubseteq) when A is the n-element powerset and
D_A\cong(\Omega^n,\sqsubseteq), the domain of mixed quantum states, when A is
the lattice of subspaces of a Hilbert space.
(We refer to http://web.comlab.ox.ac.uk/oucl/publications/tr/rr-02-07.html
for a domain-theoretic context providing the notions of approximation and
content.)Comment: 19 pages, 8 figure
Lattice Representations with Set Partitions Induced by Pairings
We call a quadruple , where and are two given non-empty finite sets, is a non-empty set and is a map having domain and codomain , a pairing on . With this structure we associate a set operator by means of which it is possible to define a preorder on the power set preserving set-theoretical union. The main results of our paper are two representation theorems. In the first theorem we show that for any finite lattice there exist a finite set and a pairing on such that the quotient of the preordered set with respect to its symmetrization is a lattice that is order-isomorphic to . In the second result, we prove that when the lattice is endowed with an order-reversing involutory map such that , , and , there exist a finite set and a pairing on it inducing a specific poset which is order-isomorphic to
An Abstract Algebraic Theory of L-Fuzzy Relations for Relational Databases
Classical relational databases lack proper ways to manage certain real-world situations including imprecise or uncertain data. Fuzzy databases overcome this limitation by allowing each entry in the table to be a fuzzy set where each element of the corresponding domain is assigned a membership degree from the real interval [0…1]. But this fuzzy mechanism becomes inappropriate in modelling scenarios where data might be incomparable. Therefore, we become interested in further generalization of fuzzy database into L-fuzzy database. In such a database, the characteristic function for a fuzzy set maps to an arbitrary complete Brouwerian lattice L. From the query language perspectives, the language of fuzzy database, FSQL extends the regular Structured Query Language (SQL) by adding fuzzy specific constructions. In addition to that, L-fuzzy query language LFSQL introduces appropriate linguistic operations to define and manipulate inexact data in an L-fuzzy database. This research mainly focuses on defining the semantics of LFSQL. However, it requires an abstract algebraic theory which can be used to prove all the properties of, and operations on, L-fuzzy relations. In our study, we show that the theory of arrow categories forms a suitable framework for that. Therefore, we define the semantics of LFSQL in the abstract notion of an arrow category. In addition, we implement the operations of L-fuzzy relations in Haskell and develop a parser that translates algebraic expressions into our implementation
Dualities in modal logic
Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics