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The Topology of Mazurkiewicz Traces
"... The present paper characterizes the topological structure of real traces. This is done in terms of graphtheoretic properties of the underlying dependence alphabet, which may be innite. The topological space of real traces is shown to be homeomorphic to the direct product of (at most) the full b ..."
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The present paper characterizes the topological structure of real traces. This is done in terms of graphtheoretic properties of the underlying dependence alphabet, which may be innite. The topological space of real traces is shown to be homeomorphic to the direct product of (at most) the full binary tree and the full countably branching tree and one higherdimensional grid. The occurrence of each of these factors depends on the existence of nite nontrivial and of innite connected components and on the number of isolated letters of the dependence alphabet. 1 Introduction Trace monoids were introduced by Cartier and Foata [3], who investigated combinatorial problems concerning the rearrangement of words, and by Mazurkiewicz [14], who was motivated to provide a mathematical model for concurrent systems. Since then trace theory has become a very popular topic, see the recent surveys [5, 6]. Corresponding author. This work was written while the second author worked at th...
Domains with Approximating Projections
 Institute of Algebra, Dresden University of Technology
, 1999
"... We investigate approximating posets with projections (approximating pop's). These are triples (D; ; P) consisting of a poset (D; ) and a directed set P of projections with sup P = id D . They carry a canonical uniformity and thus a topology. We relate their properties such as completeness ..."
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We investigate approximating posets with projections (approximating pop's). These are triples (D; ; P) consisting of a poset (D; ) and a directed set P of projections with sup P = id D . They carry a canonical uniformity and thus a topology. We relate their properties such as completeness and compactness to properties of the poset and the projection set. We show that each monotone net in D is convergent if and only if (D; ) is an algebraic domain such that the images of the projections are precisely the compact elements of (D; ). We call these domains Pdomains and characterize them as inverse limits of posets satisfying the ascending chain condition. Moreover, we describe Pdomains by a certain system of socalled "complete" subsets. We prove that if the set of compact elements of an algebraic domain is mubcomplete, then it is a Pdomain if and only if the mubclosure of every finite set of compact elements fulfils the ascending chain condition. Furthermore, we characte...
Uniform Completion versus Ideal Completion of Posets with Projections
 First Irish Conference on the Mathematical Foundations of Computer Science and Information Technology, 2000, volume 40 of Electronic Notes in Theoretical Computer Science
, 2001
"... Posets with (I; )indexed projections are triples D = (D; ; (p i ) i2I ). They consist of a partially ordered set (D; ) and a monotone net (p i ) i2I of projections on D with respect to a xed directed index set (I; ). In the present paper we prove that there are natural \completions" C(D) = (C ..."
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Posets with (I; )indexed projections are triples D = (D; ; (p i ) i2I ). They consist of a partially ordered set (D; ) and a monotone net (p i ) i2I of projections on D with respect to a xed directed index set (I; ). In the present paper we prove that there are natural \completions" C(D) = (C(D); ; ( b p i ) i2I ) and J(D) = (J(D); e ; ( e p i ) i2I ) of D. They are complete with respect to the uniformity induced by the kernels of all b p i and e p i , respectively. Moreover, they satisfy a universal property concerning the extension of special mappings (\homomorphisms"). If sup i2I p i = id D , then D can be viewed as a substructure of C(D) and of J(D). Further, D is dense in C(D); whence C(D) appears as the uniform completion of D. On the other hand, J(D) can be obtained as the ideal completion of a suitable subset of D. A comparison shows that the completion C(D) may be seen as a substructure of the completion J(D). We also investigate under which conditions both completions coincide.
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...
Function Spaces of Posets with Projections
, 2002
"... This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections. ..."
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This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections.
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...
Continuous Domains with Approximating Mappings and their Uniformity
"... 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 g. Fposets give rise to a unif ..."
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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 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 dcpo such that f(d) d for all f 2 F and all d 2 D if and only if each 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 space and closely related to partially ordered sets, Scott [12] presented the rst models of the ty...