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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...
Completion of Posets with Projections
"... Posets with (I; )indexed projections are triples (D; ; (p i ) i2I ). They consist of a partially ordered set (D; ) and a monotone net (p i ) i2I of projections on D with a fixed directed index set (I; ). In this paper we prove that there is a unique "completion" ( b D; b ; ( b p i ..."
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Posets with (I; )indexed projections are triples (D; ; (p i ) i2I ). They consist of a partially ordered set (D; ) and a monotone net (p i ) i2I of projections on D with a fixed directed index set (I; ). In this paper we prove that there is a unique "completion" ( b D; b ; ( b p i ) i2I ) of (D; ; (p i ) i2I ). It is complete with respect to the uniformity induced by the kernels of all b p i . Moreover, it satisfies a universal property concerning the extension of special mappings ("homomorphisms"). If sup i2I p i = id D , then (D; ; (p i ) i2I ) can be viewed as a dense substructure of ( b D; b ; ( b p i ) i2I ). In the second part of this paper we obtain the ideal completion of (D; ) to be a poset with (I; )indexed projections by extending the projections p i in a natural way. A comparison shows that the completion ( b D; b ; ( b p i ) i2I ) may be seen as a substructure of the ideal completion. We also investigate under which conditions these structures co...
Continuous Domains via Approximating Mappings
"... We study the interplay of order and topology in the context of approximating Fposets (D; ; F ). These consist of a poset (D; ) and a directed family F of monotone mappings below the identity with sup F = id D such that for all f 2 F there is some g 2 F with f g ffi g. Fposets give rise to a u ..."
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We study the interplay of order and topology in the context of approximating Fposets (D; ; F ). These consist of a poset (D; ) and a directed family F of monotone mappings below the identity with sup F = id D such that for all f 2 F there is some g 2 F with f g ffi g. Fposets give rise to a uniformity whose properties are closely related to properties of (D; ) and F . We show that (D; ) is a continuous (local) dcpo such that f(d) d for all f 2 F and all d 2 D if and only if each (bounded) monotone net in D converges with respect to the uniform topology. Moreover, we prove that a pointed poset is an FSdomain if and only if it arises as an approximating Fposet whose uniform topology is compact. In this case, we also obtain that the uniform topology coincides with the Lawson topology of the domain. 1 Introduction When discovering T 0 topological spaces homeomorphic to their own function spaces and closely related to partially ordered sets, Scott [14] presented the firs...