69,154 research outputs found
Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality
We study representations of MV-algebras -- equivalently, unital
lattice-ordered abelian groups -- through the lens of Stone-Priestley duality,
using canonical extensions as an essential tool. Specifically, the theory of
canonical extensions implies that the (Stone-Priestley) dual spaces of
MV-algebras carry the structure of topological partial commutative ordered
semigroups. We use this structure to obtain two different decompositions of
such spaces, one indexed over the prime MV-spectrum, the other over the maximal
MV-spectrum. These decompositions yield sheaf representations of MV-algebras,
using a new and purely duality-theoretic result that relates certain sheaf
representations of distributive lattices to decompositions of their dual
spaces. Importantly, the proofs of the MV-algebraic representation theorems
that we obtain in this way are distinguished from the existing work on this
topic by the following features: (1) we use only basic algebraic facts about
MV-algebras; (2) we show that the two aforementioned sheaf representations are
special cases of a common result, with potential for generalizations; and (3)
we show that these results are strongly related to the structure of the
Stone-Priestley duals of MV-algebras. In addition, using our analysis of these
decompositions, we prove that MV-algebras with isomorphic underlying lattices
have homeomorphic maximal MV-spectra. This result is an MV-algebraic
generalization of a classical theorem by Kaplansky stating that two compact
Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous
[0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl
A Categorical View on Algebraic Lattices in Formal Concept Analysis
Formal concept analysis has grown from a new branch of the mathematical field
of lattice theory to a widely recognized tool in Computer Science and
elsewhere. In order to fully benefit from this theory, we believe that it can
be enriched with notions such as approximation by computation or
representability. The latter are commonly studied in denotational semantics and
domain theory and captured most prominently by the notion of algebraicity, e.g.
of lattices. In this paper, we explore the notion of algebraicity in formal
concept analysis from a category-theoretical perspective. To this end, we build
on the the notion of approximable concept with a suitable category and show
that the latter is equivalent to the category of algebraic lattices. At the
same time, the paper provides a relatively comprehensive account of the
representation theory of algebraic lattices in the framework of Stone duality,
relating well-known structures such as Scott information systems with further
formalisms from logic, topology, domains and lattice theory.Comment: 36 page
Graphical Reasoning in Compact Closed Categories for Quantum Computation
Compact closed categories provide a foundational formalism for a variety of
important domains, including quantum computation. These categories have a
natural visualisation as a form of graphs. We present a formalism for
equational reasoning about such graphs and develop this into a generic proof
system with a fixed logical kernel for equational reasoning about compact
closed categories. Automating this reasoning process is motivated by the slow
and error prone nature of manual graph manipulation. A salient feature of our
system is that it provides a formal and declarative account of derived results
that can include `ellipses'-style notation. We illustrate the framework by
instantiating it for a graphical language of quantum computation and show how
this can be used to perform symbolic computation.Comment: 21 pages, 9 figures. This is the journal version of the paper
published at AIS
Hilbert C*-modules over a commutative C*-algebra
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