1,962 research outputs found
A Universal Characterization of the Double Powerlocale
This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297{321.
The double powerlocale P(X) (found by composing, in either order,the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Locop; Set] to the double exponential SSX where S is the Sierpinski locale. Further PU(X) and PL(X) are shown to be the subobjects P(X) comprising, respectively, the meet semilattice and join
semilattice homomorphisms. A key lemma shows that, for any locales X and Y , natural transformations from SX (the presheaf Loc
Uniqueness of directed complete posets based on Scott closed set lattices
In analogy to a result due to Drake and Thron about topological spaces, this
paper studies the dcpos (directed complete posets) which are fully determined,
among all dcpos, by their lattices of all Scott-closed subsets (such dcpos will
be called -unique).
We introduce the notions of down-linear element and quasicontinuous element
in dcpos, and use them to prove that dcpos of certain classes, including all
quasicontinuous dcpos as well as Johnstone's and Kou's examples, are
-unique. As a consequence, -unique dcpos with their
Scott topologies need not be bounded sober.Comment: 12 pages. arXiv admin note: substantial text overlap with
arXiv:1607.0357
Presenting dcpos and dcpo algebras
Dcpos can be presented by preorders of generators and inequational relations expressed as covers. Algebraic operations on the generators (possibly with their results being ideals of generators) can be extended to the dcpo presented, provided the covers are “stable” for the operations. The resulting dcpo algebra has a natural universal characterization and satisfies all the inequational laws satisfied by the generating algebra. Applications include known “coverage theorems” from locale theory
A Recipe for State-and-Effect Triangles
In the semantics of programming languages one can view programs as state
transformers, or as predicate transformers. Recently the author has introduced
state-and-effect triangles which capture this situation categorically,
involving an adjunction between state- and predicate-transformers. The current
paper exploits a classical result in category theory, part of Jon Beck's
monadicity theorem, to systematically construct such a state-and-effect
triangle from an adjunction. The power of this construction is illustrated in
many examples, covering many monads occurring in program semantics, including
(probabilistic) power domains
Canonical extension and canonicity via DCPO presentations
The canonical extension of a lattice is in an essential way a two-sided
completion. Domain theory, on the contrary, is primarily concerned with
one-sided completeness. In this paper, we show two things. Firstly, that the
canonical extension of a lattice can be given an asymmetric description in two
stages: a free co-directed meet completion, followed by a completion by
\emph{selected} directed joins. Secondly, we show that the general techniques
for dcpo presentations of dcpo algebras used in the second stage of the
construction immediately give us the well-known canonicity result for bounded
lattices with operators.Comment: 17 pages. Definition 5 was revised slightly, without changing any of
the result
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