DOCSLIB.ORG

Formal Topological Characterizations of Various Continuous Domains$

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Formal Topological Characterizations of Various Continuous Domains$ View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 56 (2008) 444–452 www.elsevier.com/locate/camwa Formal topological characterizations of various continuous domains$ Luoshan Xua,∗, Xuxin Maob a Department of Mathematics, Yangzhou University, Yangzhou 225002, PR China b College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China Received 15 August 2005; received in revised form 2 April 2007; accepted 4 December 2007 Abstract In this paper, some new concepts such as (super-compact) quasi-bases and the consistently (locally) coherent property of super- compact quasi-bases, are introduced. With these concepts, various continuous domains including bc-domains and (s)L-domains, are successfully characterized in formal topological ways. Furthermore, to deal with algebraic domains, the concept of quasi- formal points, a generalization of formal points, is introduced. Formal topological characterizations of various algebraic domains via quasi-formal points are obtained. c 2008 Elsevier Ltd. All rights reserved. Keywords: Continuous domain; Locally super-coherent topology; Quasi-base; (Quasi-)formal point; BC-domain; L-domain 1. Introduction In 1972, Dana Scott introduced a class of lattices called continuous lattices in order to provide models for the type free calculus in logic (see [1]). Later, a more general notion of continuous domains, which are mathematical structures used in semantics as carriers of meaning, was introduced in [2]. Now domain theory has received more and more attention (see [3–8]) of both mathematicians and computer scientists. And the study of domain theory integrates the studies of order structures, topological structures as well as algebraic structures. Meanwhile, domain theory is also closely related to theoretical computer science and yields many applications in various areas. It should be noted that a distinctive feature of the theory of continuous domains is that many of the considerations are closely interlinked with topological ideas (see [3,4,8,9]). The Scott topology, as an order-theoretical topology, is of fundamental importance in domain theory. It lies at the heart of the structure of continuous domains. In [2], Lawson proved that a dcpo is continuous if and only if its Scott topology is completely distributive. However, the definitions or characterizations of the Scott topology always assume the underlying set to be equipped with a partial order. Domain theory can be seen as a branch of formal topology which is the topology as developed in (Martin Lof’s)¨ type theory. Recently, Sambin in [10], introduced the concept of super-coherent topology. A purely topological characterization of the Scott topologies over algebraic dcpos, independently of any orders, was presented. $ Supported by NSFC (10371106, 60774073), the Tianyuan Fund of Mathematics of NSFC (10726028) and the Fund (S0667-082) from Nanjing University of Aeronautics and Astronautics. ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (L. Xu). 0898-1221/$ - see front matter c 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2007.12.010 L. Xu, X. Mao / Computers and Mathematics with Applications 56 (2008) 444–452 445 Since continuous domains are more general and complicated than algebraic ones, it is tempting to find similar topological characterizations of various kinds of continuous domains. For this purpose, in this paper, a series of new concepts such as the locally super-coherent topology, (super-compact) quasi-bases and the consistently (resp., locally) coherent property of super-compact quasi-bases, are introduced. With these concepts, various continuous domains including bc-domains, sL-domains and L-domains, are successfully characterized in formal topological ways. Furthermore, to deal with algebraic domains, the concept of quasi-formal points, a generalization of formal points in some sense, is introduced. And formal topological characterizations of various algebraic domains via quasi-formal points are also thus obtained. 2. Preliminaries Recall that in a poset P, a subset D ⊆ P is directed if each finite subset of D has an upper bound in D. Any directed lower set of P is called an ideal. The set of all ideals of P ordered by set inclusion is called the ideal completion of P, denoted by I dl(P). A poset is called a directed complete poset (briefly dcpo) if each directed subset has a supremum. A poset is said to be bounded complete if every subset with an upper bound has a supremum. This implies that a bounded complete poset has a bottom. A complete lattice is a poset in which every subset has a supremum. Let P be a dcpo and x, y ∈ P. We say that x approximates y, written x y, if whenever D is directed with sup D ≥ y, then x ≤ d for some d ∈ D. An element k ∈ P is compact if k k. The set of all compact elements of P is denoted by K (P). If for every element x ∈ P, the set ↓x := {a ∈ P : a x} is directed and sup ↓x = x, then P is called a continuous domain. A dcpo is called an algebraic domain if every element is the directed supremum of compact elements that approximate it. A bounded complete continuous (algebraic) domain is called a bc-domain (Scott domain). A(n) continuous (algebraic) lattice means a(n) continuous (algebraic) domain which is a complete lattice. An (algebraic) sL-domain (see [6]) is a(n) continuous (algebraic) domain in which every principal ideal is a join-semilattice. An (algebraic) L-domain is a(n) continuous (algebraic) domain in which every principal ideal is a complete lattice. An upper set U of a dcpo P is said to be Scott open if for all directed sets D ⊆ P, sup D ∈ U implies U ∩ D 6= ∅. The set of Scott open sets of P forms a topology, called the Scott topology and denoted by σ (P). Proposition 2.1 (See [4]). If P is a continuous domain, then the approximating relation has the interpolation property (INT): x z ⇒ ∃y ∈ P such that x y z. Proposition 2.2 (See [4]). Let P be a continuous domain. Then for each x ∈ P, the set ↑x = {y ∈ P : x y} is Scott open, and these form a base for the Scott topology. A subset F of a space X is said to be irreducible, if F 6= ∅ and for any pair of closed sets F1 and F2 in X, F ⊆ F1 ∪ F2 implies that F ⊆ F1 or F ⊆ F2.A T0-space is said to be sober if every irreducible closed set is a closure of a unique point. It is known that continuous domains equipped with Scott topologies are sober (see [4, Corollary II-1.12]). A topology O(X) induces a preorder, called the specialization order which is defined by ∀u, v ∈ X, u ≤X v ⇔ u ∈ clX ({v}) ⇔ ∀U ∈ O(X), u ∈ U → v ∈ U, where clX ({v}) is the closure of the set {v} in the space (X, O(X)). The ordered set (X, ≤X ) is denoted by Ω X in short. Note that every open set of X is an upper set of Ω X and the closure of {v} is the lower set ↓ v = {x ∈ X : x ≤X v}. It is clear that (X, O(X)) is a T0-space iff ≤X is a partial order. A T0-space X is called a monotone convergence space (see [4,11]) if every subset D directed relative to the specialization order has a supremum, and the relation sup D ∈ U for any open set U of X implies D ∩ U 6= ∅. Remark 2.3 (See [4]). Each sober space is a monotone convergence space and every monotone convergence space X is a dcpo in its specialization order with O(X) ⊆ σ (Ω X). Lemma 2.4 (See [4]). For a monotone convergence space (X, O(X)) and its order of specialization, the following conditions are equivalent: (1) Ω X is a continuous domain and the topology of X is the Scott topology; (2) O(X) is a completely distributive lattice. 446 L. Xu, X. Mao / Computers and Mathematics with Applications 56 (2008) 444–452 Definition 2.5 (see [12]). Let L be a complete lattice. (1) The complete way-below relation C in L is defined for all a, b ∈ L, a C b ⇔ (∀S ⊆ L, b ≤ sup S ⇒ ∃s ∈ S, a ≤ s); (2) If a ∈ L and B ⊆ {t ∈ L : t C a} with sup B = a, then B is called an approximating complete way-below set of a in L. We will use B(a) to denote any one of the approximating complete way-below sets of a ∈ L. Lemma 2.6 (See [12]). Let L be a complete lattice. Then L is completely distributive if and only if ∀a ∈ L, a has an approximating complete way-below set. 3. Locally super-coherent topologies and characterizations of continuous domains In this section, we introduce the new concept of locally super-coherent topology and present a purely topological characterization of continuous domains. The following definition comes from [10]. Definition 3.1 (See [10]). Let (X, O(X)) be a topological space. Then a subset U is said to be super-compact if for { } ⊆ S ∈ ⊆ any family of open sets Vi i∈I with U i∈I Vi , there is i0 I such that U Vi0 . The topology O(X) is called super-coherent if it is sober and has a base of super-compact open sets. Theorem 1 of [10] says that any super-coherent space (X, O(X)) coincides with the Scott topology of a suitable algebraic domains over X. Since algebraic domains with Scott topologies are all coherent, Theorem 1 of [10], as well as Theorem 6.4 below in this paper, justifies the choice of the terminology of super-coherent.
Load more

Recommended publications
  • Part II Topological Dualities
    Part II Top ological dualities Chapter Top ology and armative predicates In the rst part of this monograph we considered predicates to be subsets of an abstract set of states If we think of the states as the denotations of results of computations of programs then predicates b ecome computationally mean ingful in the sense that we can use partial information ab out a computation to tell whether or not a predicate holds for that computation A predicate for which only nite information ab out a computation is needed to arm whether it holds is called an armative predicate The set of armative predicates is closed under nite intersections and ar bitrary unions Hence armative predicates can be identied with the op en sets of a top ological space The idea that op en sets are observable predi cates was prop osed by Smyth in although it is also brie y mentioned in Smyth interprets op en sets as semidecidable prop erties in some eectively given top ological space More generally op en sets can be inter preted as nitely observable predicates Alp ern and Schneider and Kwiatkowska use op en sets as nite liveness predicates and closed sets as safety predicates to formalize the informal characterization of liveness and safety prop erties of Lamp ort The name armative predicates has b een intro duced by Vickers for denoting the abstract op en sets of a frame Armative predicates are also called veriable predicates by Rewitzky who uses the term observable for predicates which are b oth armative and refutative Bonsangue In this chapter we intro
    [Show full text]
  • Spatio-Temporal Domains: an Overview David Janin
    Spatio-temporal domains: an overview David Janin To cite this version: David Janin. Spatio-temporal domains: an overview. 2018. hal-01634897v2 HAL Id: hal-01634897 https://hal.archives-ouvertes.fr/hal-01634897v2 Preprint submitted on 17 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spatio-temporal domains: an overview David Janin? UMR LaBRI, Bordeaux INP Université de Bordeaux [email protected] Abstract. We consider the possibility of defining a general mathemat- ical framework for the homogeneous modeling and analysis of hetero- geneous spatio-temporal computations as they occur more and more in modern computerized systems of systems. It appears that certain fibra- tions of posets into posets, called here spatio-temporal domains, eventu- ally provide a fully featured category that extends to space and time the category of cpos and continuous functions, aka Scott Domains, used in classical denotational semantics. 1 Introduction Research context. Program semantics is classically divided between two com- plementary approaches : denotational semantics and operational semantics. De- notational semantics generally refers to what the partial functions encoded by programs are : what is the relationship between (models of) their input val- ues (or input memory state) and their output values (or output memory state).
    [Show full text]
  • On the Categorical Semantics of Elementary Linear Logic
    Theory and Applications of Categories, Vol. 22, No. 10, 2009, pp. 269{301. ON THE CATEGORICAL SEMANTICS OF ELEMENTARY LINEAR LOGIC OLIVIER LAURENT Abstract. We introduce the notion of elementary Seely category as a notion of cate- gorical model of Elementary Linear Logic (ELL) inspired from Seely's de¯nition of models of Linear Logic (LL). In order to deal with additive connectives in ELL, we use the ap- proach of Danos and Joinet [DJ03]. From the categorical point of view, this requires us to go outside the usual interpretation of connectives by functors. The ! connective is decomposed into a pre-connective ] which is interpreted by a whole family of functors (generated by id, ­ and &). As an application, we prove the strati¯ed coherent model and the obsessional coherent model to be elementary Seely categories and thus models of ELL. Introduction The goal of implicit computational complexity is to give characterizations of complexity classes which rely neither on a particular computation model nor on explicit bounds. In linear logic (LL) [Gir87], the introduction of the exponential connectives gives a precise status to duplication and erasure of formulas (the qualitative analysis). It has been shown that putting constraints on the use of exponentials permits one to give a quantitative analysis of the cut elimination procedure of LL and to de¯ne light sub-systems of LL characterizing complexity classes (for example BLL [GSS92], LLL [Gir98] or SLL [Laf04] for polynomial time and ELL [Gir98, DJ03] for elementary time). In order to have a better understanding of the mathematical structures underlying these systems, various proposals have been made in the last years with the common goal of de¯ning denotational models of light systems [MO00, Bai04, DLH05, LTdF06, Red07].
    [Show full text]
  • Scott Domain Representability of a Class of Generalized Ordered Spaces
    Scott Domain Representability of a Class of Generalized Ordered Spaces Kevin W. Duke∗ and David Lutzer† Draft of July 25, 2007 Abstract Many important topological examples (the Sorgenfrey line, the Michael line) belong to the class of GO-spaces constructed on the usual set R of real numbers. In this paper we show that every GO- space constructed on the real line, and more generally, any GO-space constructed on a locally compact LOTS, is Scott-domain representable, i.e., is homeomorphic to the space of maximal elements of some Scott domain with the Scott topology. MR Classifications: primary = 54F05; secondary = 54D35,54D45,54D80, 06F30 Key words and phrases: Scott domain, Scott topology, dcpo, domain-representable space, generalized ordered space, GO-space. 1 Introduction A topological space X is a Baire space if every intersection of countably many dense, open sets is dense. The Baire space property does not behave well under topological operations, and in the 1960s several authors (Choquet, deGroot, and Oxtoby, for example) described topological properties now called Choquet completeness, subcompactness, and pseudo-completeness that are stronger than the Baire space property and are well-behaved under the product operation. Such properties were studied in [1] and have come to be thought of as being strong completeness properties. More recently, topologists have borrowed a property called domain representability from theoret- ical computer science and have come to be see it as a kind of completeness property related to the Baire property. A space X is domain representable if X is homeomorphic to the space of maximal elements of some domain, topologized with the relative Scott topology.
    [Show full text]
  • Compact Topologies on Locally Presentable Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 38, No 3 (1997), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES PANAGIS KARAZERIS Compact topologies on locally presentable categories Cahiers de topologie et géométrie différentielle catégoriques, tome 38, no 3 (1997), p. 227-255 <http://www.numdam.org/item?id=CTGDC_1997__38_3_227_0> © Andrée C. Ehresmann et les auteurs, 1997, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIER DE TOPOLOGIE ET J Illume XXXVIII-3 (1997) GEOMETRIE DIFFERENTIELLE CATEGORIQUES COMPACT TOPOLOGIES ON LOCALLY PRESENTABLE CATEGORIES by Panagis KARAZERIS RESUME. Les topologies sur les categories localement pr6sentables g6n6rallsent les notions famill6res suivantes, d’une part les topologies de Grothendieck sur des petites categories, d’autre part les topologies de Gabriel sur des categories abéliennes à generateurs. Dans cet article on introduit une condition, qui peut être v6rifi6e pour les topologies pr6c6dentes, appel6e "compacit6". Dans le cas des topologies de Grothendieck, cette condition signifie qu’un recouvrement quelconque a un sous-recouvrement fini. Les topologies compactes correspondantes ont des localisations ferm6es dans la cat6gorie donnee pour des colimites filtrantes monomorphiques. On examine aussi la fermeture des objets s6par6s et des faisceaux pour les colimites filtrantes. Les topologies compactes sur une cat6gorie localement de presentation finie forment un locale. Si cette cat6gorie est un topos coherent, alors le locale est compact et localement compact.
    [Show full text]
  • Introduction to Coherent Spaces
    Introduction to coherent spaces Arnold Neumaier Fakult¨at f¨ur Mathematik, Universit¨at Wien Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria email: [email protected] WWW: http://www.mat.univie.ac.at/~neum arXiv:1804.01402 September 28, 2018 Abstract. The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept. Coherent spaces provide a setting for the study of geometry in a different direction than traditional metric, topological, and differential geometry. Just as it pays to study the properties of manifolds independently of their embedding into a Euclidean space, so it appears fruitful to study the properties of coherent spaces independent of their embedding into a Hilbert space. Coherent spaces have close relations to reproducing kernel Hilbert spaces, Fock spaces, and unitary group representations, and to many other fields of mathematics, statistics, and physics. This paper is the first of a series of papers and defines concepts and basic theorems about coherent spaces, associated vector spaces, and their topology. Later papers in the series dis- cuss symmetries of coherent spaces, relations to homogeneous spaces, the theory of group representations, C∗-algebras, hypergroups, finite geometry, and applications to quantum physics. While the applications to quantum physics were the main motiviation for develop- ing the theory, many more applications exist in complex analysis, group theory, probability theory, statistics, physics, and engineering. For the discussion of questions concerning coherent spaces, please use the discussion forum https://www.physicsoverflow.org.
    [Show full text]
  • Objects, Interference, and the Yoneda Embedding
    Syracuse University SURFACE College of Engineering and Computer Science - Former Departments, Centers, Institutes and College of Engineering and Computer Science Projects 1995 Objects, Interference, and the Yoneda Embedding Peter W. O'Hearn Syracuse University Uday S. Reddy University of Illinois at Urbana-Champaign Follow this and additional works at: https://surface.syr.edu/lcsmith_other Part of the Programming Languages and Compilers Commons Recommended Citation O'Hearn, Peter W. and Reddy, Uday S., "Objects, Interference, and the Yoneda Embedding" (1995). College of Engineering and Computer Science - Former Departments, Centers, Institutes and Projects. 14. https://surface.syr.edu/lcsmith_other/14 This Article is brought to you for free and open access by the College of Engineering and Computer Science at SURFACE. It has been accepted for inclusion in College of Engineering and Computer Science - Former Departments, Centers, Institutes and Projects by an authorized administrator of SURFACE. For more information, please contact [email protected]. Electronic Notes in Theoretical Computer Science to app ear Ob jects Interference and the Yoneda Emb edding Peter W OHearn Syracuse University Uday S Reddy University of Il linois at UrbanaChampaign Dedicated to John C Reynolds in honor of his th birthday Abstract We present a new semantics for Algollike languages that combines metho ds from two prior lines of development the ob jectbased approach of where the meaning of an imp erative program is describ ed in terms of sequences of observable
    [Show full text]
  • LINEAR LOGIC, -AUTONOMOUS CATEGORIES and COFREE COALGEBRAS1 R.A.G. Seely Girard 1987]
    LINEAR LOGIC AUTONOMOUS CATEGORIES 1 AND COFREE COALGEBRAS RAG Seely ABSTRACT A brief outline of the categorical characterisation of Girards linear logic is given analagous to the relationship b etween cartesian closed cat egories and typed calculus The linear structure amounts to a autonomous category a closed symmetric monoidal category G with nite pro ducts and a closed involution Girards exp onential op erator is a cotriple on G which carries the canonical comonoid structure on A with resp ect to cartesian pro duct to a comonoid structure on A with resp ect to tensor pro duct This makes the Kleisli category for cartesian closed INTRODUCTION In Linear logic JeanYves Girard introduced a logical system he describ ed as a logic b ehind logic Linear logic was a consequence of his analysis of the structure of qualitative domains Girard he noticed that the interpretation of the usual conditional could b e decomp osed into two more primitive notions a linear conditional and a unary op erator called of course which is formally rather like an interior op erator X Y X Y The purp ose of this note is to answer two questions and p erhaps p ose some others First if linear category means the structure making valid the prop ortion linear logic linear category typed calculus cartesian closed category then what is a linear category This question is quite easy and in true categorical spirit one nds that it was answered long b efore b eing put namely by Barr Our intent here is mainly to supply a few details to make the matter more precise
    [Show full text]
  • Variations on the Bagdomain Theme
    Theoretical Computer Science 136 (1994) 3 20 3 Elsevier Variations on the bagdomain theme P.T. Johnstone Department of Pure Mathematics, Cambridqe University, 16 Mill Lane, Cambridge, CB2 I SB, UK Abstract Johnstone, P.T., Variations on the bagdomain theme, Theoretical Computer Science 136 (1994) 3-20. The notion of bagdomain was first introduced by Vickers (1992) and further studied by the present author in (Johnstone, 1992). In these papers, attention was focused on one particular version of the bagdomain construction, the "bag" analogue of the lower (Hoare) powerdomain; but there are many other possibilities. The purpose of the present paper is to introduce some of these possibilities and to describe their basic properties, using the theory of fibrations and partial products developed in (Johnstone, 1993). O. Introduction The notion of bagdomain was first introduced by Vickers [20] and further studied by the present author in [9]. The basic idea is that, for certain potential applications (e.g. in building a mathematical model of databases), the concept of a "powerdomain" PD whose points are sets of points of the original domain D is inadequate: one wants to consider bags (indexed families) of points of D, and in particular the "refinement ordering" on bags of points needs to keep track of which particular points in the first bag are (regarded as being) refined by which points in the second bag. This immediately forces the "refinement ordering" to be a category and not just a preorder, and so the problem cannot be solved within the traditional context of spaces (or loc- ales): one needs instead to think of domains as (Grothendieck) toposes, which gives one precisely the right sort of "generalized spaces of points" for the construction to be possible.
    [Show full text]
  • Introduction to Coherent Spaces
    Introduction to coherent spaces Arnold Neumaier Fakult¨at f¨ur Mathematik, Universit¨at Wien Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria email: [email protected] WWW: http://www.mat.univie.ac.at/~neum arXiv:1804.01402 September 28, 2018 Abstract. The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept. Coherent spaces provide a setting for the study of geometry in a different direction than traditional metric, topological, and differential geometry. Just as it pays to study the properties of manifolds independently of their embedding into a Euclidean space, so it appears fruitful to study the properties of coherent spaces independent of their embedding into a Hilbert space. Coherent spaces have close relations to reproducing kernel Hilbert spaces, Fock spaces, and unitary group representations, and to many other fields of mathematics, statistics, and physics. This paper is the first of a series of papers and defines concepts and basic theorems about coherent spaces, associated vector spaces, and their topology. Later papers in the series dis- cuss symmetries of coherent spaces, relations to homogeneous spaces, the theory of group representations, C∗-algebras, hypergroups, finite geometry, and applications to quantum physics. While the applications to quantum physics were the main motiviation for develop- ing the theory, many more applications exist in complex analysis, group theory, probability theory, statistics, physics, and engineering. arXiv:1804.01402v2 [math-ph] 28 Sep 2018 For the discussion of questions concerning coherent spaces, please use the discussion forum https://www.physicsoverflow.org.
    [Show full text]
  • Introduction to Coherent Quantization
    Introduction to coherent quantization Arnold Neumaier Fakult¨at f¨ur Mathematik, Universit¨at Wien Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria email: [email protected] WWW: http://www.mat.univie.ac.at/~neum Arash Ghaani Farashahi Department of Pure Mathematics, School of Mathematics Faculty of Mathematics and Physical Sciences, University of Leeds Leeds, LS2 9JT, United Kingdom email: [email protected] email: [email protected] WWW: https://sites.google.com/site/ghaanifarashahi/ August 16, 2021 arXiv:1804.01400v2 Abstract. This paper is one of a series of papers on coherent spaces and their appli- cations, defined in the recent book ’Coherent Quantum Mechanics’ by the first author. The paper studies coherent quantization – the way operators in the quantum space of a coherent space can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization arXiv:1804.01400v3 [math-ph] 16 Aug 2021 of coherent maps.
    [Show full text]
  • Decomposition of Domains 1 Introduction
    Decomp osition of Domains y Achim Jung Leonid Libkin Hermann Puhlmann Abstract The problem of decomp osing domains into sensible factors is addressed and solved for the case of dIdomains A decomp osition theorem is proved which allows the represention of a large sub class of dIdomains in a pro duct of at domains Direct pro duct decomp ositions of Scottdomains are studied separately Intro duction This work was initiated by Peter Bunemans interest in generalizing relational databases see He quite radically dismissed the idea that a database should b e forced into the format of an nary relation Instead he allowed it to b e an arbitrary antichain in a Scottdomain The reason for this was that advanced concepts in database theory such as null values nested relations and complex ob jects force one to augment relations and values with a notion of information order Following Bunemans general approach the question arises how to dene basic database theoretic concepts such as functional dep endency for antichains in Scottdomains For this one needs a way to sp eak ab out relational schemes which are nothing but factors of the pro duct of which the relation is a subset Buneman successfully dened a notion of scheme for Scottdomains and it is that denition which at the heart of this work We show that his generalized schemes b ehave almost like factors of a pro duct decomp osition Consequently we cho ose the word semifactor for them In the light of our results Peter Bunemans theory of generalized databases b ecomes less miraculous a large class of
    [Show full text]