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Posets with Projections and their Morphisms
, 1999
"... This paper investigates function spaces of partially ordered sets with some directed family of projections. Given a fixed directed index set (I; ), we consider triples (D; ; (p i ) i2I ) consisting of a poset (D; ) and a monotone net (p i ) i2I of projections of D. We call them (I; )indexed pop' ..."
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This paper investigates function spaces of partially ordered sets with some directed family of projections. Given a fixed directed index set (I; ), we consider triples (D; ; (p i ) i2I ) consisting of a poset (D; ) and a monotone net (p i ) i2I of projections of D. We call them (I; )indexed pop's (posets with projections). Our main purpose is to study structure preserving maps between (I; )indexed pop's. Such a morphism respects both order and projections. In fact, we study weak homomorphisms as well as homomorphisms. In case of (I; ) = (N 0 ; ), weak homomorphisms are precisely all monotone maps that are nonexpansive with regard to some canonical pseudoultrametric induced by the given sequence of projections. Weak homomorphisms become then homomorphisms if they are additionally compatible with a socalled weak weight function. Both weak homomorphisms and homomorphisms between two (I; )indexed pop's turn out to induce (I; )indexed pop's of their own. We prove that pr...
The Classification of Continuous Domains (Extended Abstract)
"... Achim Jung y Technische Hochschule Darmstadt and Imperial College of Science and Technology, London Abstract The longstanding problem of finding the maximal cartesian closed categories of continuous domains is solved. The solution requires the definition of a new class of continuous domains, cal ..."
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Achim Jung y Technische Hochschule Darmstadt and Imperial College of Science and Technology, London Abstract The longstanding problem of finding the maximal cartesian closed categories of continuous domains is solved. The solution requires the definition of a new class of continuous domains, called FSdomains, which contains all retracts of SFPobjects. The properties of FSdomains are discussed in some detail. Keywords: continuous domains, SFPobjects, Lawsontopology, Smyth's Theorem, FSdomains, Ldomains 1 Introduction The first spaces suitable for the interpretation of programming language constructs were continuous lattices discovered by Dana Scott in the late sixties. Continuous lattices turned out to have numerous connections to other fields of mathematics such as algebra, topology, and convex analysis. An indication of this is the voluminous Bibliography of Continuous Lattices contained in [4]. In Computer Science, however, it was soon recognized that the subclass of al...
Topology in Computer Science Problems
, 2000
"... N.Pro ve(o r refute) that A isisoTNj hic to a set o fixed po ints osoj ScoR co tinuo s transfo?RzTR1 o P . (Fo r theco untable case, the pro blem is po sed as an 1 exercise in "Lambda Calculus" textbo o k by Barendregt, butno bo dy seems to kno w the so lutio n, so it is likely that the pro blem s ..."
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N.Pro ve(o r refute) that A isisoTNj hic to a set o fixed po ints osoj ScoR co tinuo s transfo?RzTR1 o P . (Fo r theco untable case, the pro blem is po sed as an 1 exercise in "Lambda Calculus" textbo o k by Barendregt, butno bo dy seems to kno w the so lutio n, so it is likely that the pro blem sho uld be co nsidered o pen even fo r theco untable case.) 3 A problem in doma5 theory Sent in by Ralph Kummetz Let (D,#)beadcpo andletF be a directed familyo ScoR? co ntinuo us maps f : D # D with sup F = id D such that the foj4 wing pro erties are satisfied: 1. #f #F#g #F : f #g #g # id D . (This implies that the co llectio { B f  f<F18.93
An Old Problem From Domain Theory Stated in QuasiUniform Terms
, 2000
"... We pose the problem of whether every FSdomain is a retract of a bifinite domain purely in terms of quasiuniform spaces. ..."
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We pose the problem of whether every FSdomain is a retract of a bifinite domain purely in terms of quasiuniform spaces.
The Largest Cartesian Closed Category of Domains, Considered Constructively
, 2000
"... A conjecture of Smyth [10] is discussed which says that if D and [D # D] are effectively algebraic directedcomplete partial orders with least element (cpo's), then D is an e#ectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an e#ectively ..."
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A conjecture of Smyth [10] is discussed which says that if D and [D # D] are effectively algebraic directedcomplete partial orders with least element (cpo's), then D is an e#ectively strongly algebraic cpo, where it was left open what exactly is meant by an effectively algebraic and an e#ectively strongly algebraic cpo. First, notions of an e#ectively strongly algebraic cpo and an e#ective SFP object are introduced. The effective SFP objects are just the constructive (computable) objects in the effectively given category [9] of indexed # algebraic cpo's. Theorem Every effective SFP object is an effectively strongly algebraic cpo, and vice versa. Moreover, this equivalence holds effectively. This shows that the given notion of an effective SFP object is stable. In e#ectivity considerations of # algebraic cpo's it is usual to require that the partial order be decidable on the compact elements. Here, we use a stronger assumption. Theorem If D is an indexed #algebraic cpo that has a comp...
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"... GRAEME FONBES, D.PHIL. Continuous d,irected complete partial orders (continuous dcpo's) are ordered algebraic structures which serve as mathematical models for the semantics of programming languages.. The class of continuous dcpo's is the closure of the class of algebraic dcpo's under images of Scot ..."
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GRAEME FONBES, D.PHIL. Continuous d,irected complete partial orders (continuous dcpo's) are ordered algebraic structures which serve as mathematical models for the semantics of programming languages.. The class of continuous dcpo's is the closure of the class of algebraic dcpo's under images of Scottcontinuous projections pz D+ D. The paradigm is lhe Cøntor functi,on p:C+ C, which is a Scottcontinuous projection on the Cantor set such that im(p) is isomorphic to the unit interval. A dcpo D is called projectíonsto,ble ifffor ali p € lD\Dl, i*(p) is algebraic. If all orderdense chains in /f(D) are degenerate, then an algebraic dcpo D is projectionstable. The converse is not true. If D has a bottom, then the converse is valid. The class of projectionstable dcpo's is closed under arbitrary products. Let D be a continuous Ldomain (profinite dcpo) with bottom. Then, lDl:Dl is a continuous Ldomain (profinite dcpo) with bottom itr [DSD] is a continuous dcpo iff D is projectionstable. Every dldomain is projectionstable. The dcpo (Proj(D), E) is a dldomain, if