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Topology, Domain Theory and Theoretical Computer Science
, 1997
"... In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from ..."
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In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...
Holonomy Embedding of Arbitrary Stable Semigroups
, 1994
"... We show how the Rhodes expansion b S of any stable semigroup S embeds into the cascade integral (a natural generalization of the wreath product) of permutationreset transformation semigroups with zero adjoined. The permutation groups involved are exactly the Schützenberger groups of the Jclasses o ..."
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We show how the Rhodes expansion b S of any stable semigroup S embeds into the cascade integral (a natural generalization of the wreath product) of permutationreset transformation semigroups with zero adjoined. The permutation groups involved are exactly the Schützenberger groups of the Jclasses of S. Since S b S is an aperiodic map via which all subgroups of S lift to b S, this results in a strong KrohnRhodesZeiger decomposition for the entire class of stable semigroups. This class includes all semigroups that are finite, torsion, finite Jabove, compact Hausdorff, or relatively free profinite, as well as many other semigroups. Even if S is not stable, one can expand it using Henckell's expansion and then apply our embedding. This gives a simplified proof of the Holonomy Embedding Theorem for all semigroups.
Profinite semigroups and applications
 IN STRUCTURAL THEORY OF AUTOMATA, SEMIGROUPS, AND UNIVERSAL ALGEBRA
, 2003
"... Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which ..."
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Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which for pseudovarieties play the role of free algebras in the theory of varieties. Combinatorial problems on rational languages translate into algebraictopological problems on profinite semigroups. The aim of these lecture notes is to introduce these topics and to show how they intervene in the most recent developments in the area.
Über das 5. Hilbert'sche Problem
, 1995
"... 69> Crelleschen Journals eine Arbeit von Niels Henrik Abel unter dem Titel ,,Untersuchung der Funktionen zweier unabhangig veranderlicher Groen x und y , wie f(x; y) , welche die Eigenschaft haben, da f(z; f(x; y)) eine symmetrische Funktion von z , x und y ist ". Er beweist den folgenden Satz: Hat ..."
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69> Crelleschen Journals eine Arbeit von Niels Henrik Abel unter dem Titel ,,Untersuchung der Funktionen zweier unabhangig veranderlicher Groen x und y , wie f(x; y) , welche die Eigenschaft haben, da f(z; f(x; y)) eine symmetrische Funktion von z , x und y ist ". Er beweist den folgenden Satz: Hat eine Funktion die im Titel genannte Eigenschaft, so gibt es eine Funktion / derart, da /f(x; y) = /(x) +/(y) gilt. Nehmen wir einmal S = ]4; 1[ ; diese Menge ist zu R homomorph. Schreiben wir ab = minfa; bg und definieren wir f<F14.
QuasiDiscrete Locally Compact Quantum Groups ( ∗)
, 2004
"... Let A be a C ∗algebra. Let A ⊗ A be the minimal C ∗tensor product of A with itself and let M(A ⊗ A) be the multiplier algebra of A ⊗ A. A comultiplication on A is a nondegenerate ∗homomorphism ∆ : A → M(A ⊗ A) satisfying the coassociativity law ( ∆ ⊗ι) ∆ = (ι ⊗∆) ∆ where ι is the identity map a ..."
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Let A be a C ∗algebra. Let A ⊗ A be the minimal C ∗tensor product of A with itself and let M(A ⊗ A) be the multiplier algebra of A ⊗ A. A comultiplication on A is a nondegenerate ∗homomorphism ∆ : A → M(A ⊗ A) satisfying the coassociativity law ( ∆ ⊗ι) ∆ = (ι ⊗∆) ∆ where ι is the identity map and where ∆ ⊗ι and ι ⊗ ∆ are the unique extensions to M(A ⊗ A) of the obvious maps on A ⊗ A. We think of a pair (A, ∆) as a ‘locally compact quantum semigroup’. When these notes where written, in 1993, it was not at all clear what the extra conditions on ∆ should be for (A, ∆) to be a ‘locally compact quantum group’. This only became clear in 1999 thanks to the work of Kustermans and Vaes. In the compact case however, that is when A has an identity, rather natural conditions can be formulated and so there was a good notion of a ’compact quantum group ’ already at the time these notes have been written. These compact quantum groups have been studied by Woronowicz. In these notes, we consider another class of locally compact quantum groups. We assume the existence of a nonzero element h in A such that ∆(a)(1 ⊗ h) = a ⊗ h for all a ∈ A.
Chapter 2 STOCHASTIC COMPUTING SYSTEMS
"... The invention of the steam engine in the late eighteenth century made it possible to replace the musclepower of men and animals by the motive power of machines. The invention of the storedprogram digital computer during the second world war made it possible to replace the lowerlevel ..."
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The invention of the steam engine in the late eighteenth century made it possible to replace the musclepower of men and animals by the motive power of machines. The invention of the storedprogram digital computer during the second world war made it possible to replace the lowerlevel