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Thursday, May 14, 2020 | History

2 edition of **Cartesian closed categories of domains** found in the catalog.

Cartesian closed categories of domains

A. Jung

- 281 Want to read
- 40 Currently reading

Published
**1989**
by Centrum voor Wiskunde en Informatica in Amsterdam, The Netherlands
.

Written in English

- Ordered sets.

**Edition Notes**

Includes bibliographical references (p. [105]-107).

Statement | A. Jung. |

Series | CWI tract -- 66. |

Contributions | Centrum voor Wiskunde en Informatica (Amsterdam, Netherlands) |

The Physical Object | |
---|---|

Pagination | 110 p. : |

Number of Pages | 110 |

ID Numbers | |

Open Library | OL14371741M |

ISBN 10 | 9061963761 |

In this paper we establish a systematic connection between formal concept analysis and domain theory as a categorical equivalence, enriching the link between the two areas as outlined in [25]. Building on a new notion of approximable concept. Passing from pointed domains to general domains which do not necessarily have a least element, it turns out that there are four maximal cartesian closed subcategories of the category CONT of domains: F-L, F-FS, U-L and U-FS, where the notation F-C (U-C) denotes the category whose objects are finite amalgams (disjoint unions) of objects from another category C (see Definition in).Cited by: 2.

Zhang showed that the category of dI-domains is the largest cartesian closed subcategory of ω-SABCω-SABC and ω-SABC˜, with the exponential being the stable function space, where ω-SABCω-SABC and. Usefulness of this categorical idea of cartesian closed categories (CCC): Gérard Béry’s stable functions: continuity: you cannot use an infinite amount of information to compute something (computation in finite time).

Reprints in Theory and Applications of Categories, No. 15, , pp. 1– DIAGONAL ARGUMENTS AND CARTESIAN CLOSED CATEGORIES F. WILLIAM LAWVERE Author Commentary In May I had suggested in my Chicago lectures certain applications of category theory to smooth geometry and dynamics, reviving a direct approach to function spaces. BibTeX @MISC{Jung88cartesianclosed, author = {Achim Jung and Der Technischen Hochschule Darmstadt and Dipl. -math Achim Jung and Referent Prof and Dr. K. Keimel and Koreferent Prof and Dr. K. -h. Hofmann}, title = {Cartesian Closed Categories of Domains}, year = {}}.

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April Tag der mu¨ndlichen Pru¨fung: Juli Darmstadt D Contents. 1 Basic Concepts 11 Ordered sets, directed sets, and directed-complete partial orders 11 Algebraic and continuous posets 15 Scott-topology and continuous functions 20 Biﬁnite domains 32 Directed-complete.

Perhaps the most important and striking fact of domain theory is that important categories of domains are cartesian closed. This means that the category has a terminal object, finite products, and exponents. The only problematic part for domains is the exponent, which in this setting means the space of continuous by: 1.

The cartesian closed category of dI-domains, however, is closed under D → Retr(D) and D → Proj(D), where Retr(D) is the dI-domain of all stable Scott-continuous retractions in the stable order.

The dI-domain Proj(D) consists of all p ∃ Retr(D) which are below the identity in the stable by: 6. Cartesian closed categories of domains. Amsterdam, The Netherlands: Centrum voor Wiskunde en Informatica, © (OCoLC) Material Type: Internet resource: Document Type: Book, Internet Resource: All Authors / Contributors: A Jung; Centrum voor Wiskunde en Informatica (Amsterdam, Netherlands).

Many well-known kinds of domains had been found forming cartesian closed categories together with Scott continuous functions as morphisms, such as continuous (algebraic) lattices, bounded complete domains, Scott domains [15] and SFP domains [16], and so : Zhenchao Lyu, Hui Kou.

Cartesian closed categories of domains (). Pagina-navigatie: Main; Save publication. Save as MODS; Export to Mendeley; Save as EndNote. Cartesian closed categories of domains book The main result of this paper is: the class of ωAML-domains is closed under function spaces and finite cartesian products.

Hence, the category of ωAML -domains together with Scott continuous functions is cartesian : Zhenchao Lyu, Hui Kou. Cartesian closed categories of effective domains Hamrin, Göran Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.

Next, in, Plotkin introduced the cartesian closed category of coherent separable continuous bounded complete pointed domains, and showed that it contains a universal object T ⊥ ω. Then, in, Scott introduced the category of Scott domains, and gave a universal object for the full sub-cartesian closed category of the separable Scott : Andrej Bauer, Gordon D.

Plotkin, Dana S. Scott. The long-standing 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. The Largest Cartesian Closed Category of Domains, Considered Constructively 3 a strongly algebraic domain which has an indexing of its compact elements such that for any ﬁnite set X of compact elements a canonical index of the set of its minimal upper bounds can be.

Corresponding to them, it is proved that, for a suitable subset system Ƶ, the categories F Ƶ CPO of Ƶ-complete posets, FSF Ƶ of finitely separated FƵ-domains and BFF Ƶ of bifinite FƵ-domains are all cartesian closed.

Some examples of these categories are : Min Liu, Bin Zhao. a domain. Using the classi cation, we determine all sub-cartesian closed categories of the category of separable Scott domains that contain a univer-sal object. The separable Scott domain models of the -calculus are then classi ed up to a retraction by their coherence degrees.

Keywords: Scott domain, cartesian closed category, lambda calculus 1. Lemma (5) SAL is cartesian closed in which the one-point domain, the carte- sian product A×B and [ A→ s B ] are the terminal object, the categorical product and the exponential object respectively for algebraic L-domains A,B.

Cartesian Closed Categories of Domains By Achim Jung, Der Technischen Hochschule Darmstadt, Dipl. -math Achim Jung, Referent Prof, Dr. Keimel, Koreferent Prof and Dr. Hofmann. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We classify all sub-cartesian closed categories of the category of separable Scott domains.

The classification employs a notion of coherence degree determined by the possible inconsistency patterns of sets of finite elements of a domain. Using the classification, we determine all sub-cartesian closed categories of the.

Zhang showed that the category of dI-domains is the largest cartesian closed sub- category of Ï‰-SABC and Ï‰-SËœABC, with the exponential being the stable function space, where Ï‰-SABC and Ï‰-SËœABC are full subcategories of SABC and SËœABC respectively which contain countablly based algebraic bounded complete domains as : Xiaoyong Xi, Guohua Wu.

COMP and CON are two Cartesian closed categories. Proof. Here we take the case of COMP as instance. Similarly, we can show that CON is a Cartesian closed category.

(1) Let Φ and D be two single point sets, where the operations combination and focusing are defined naturally. Then (Φ, D) is a compact information algebra and it is a terminal object of COMP. These LUCAs eventually evolved into three different cell types, each representing a domain.

The three domains are the Archaea, the Bacteria, and the Eukarya. Figure \(\PageIndex{1}\): A phylogenetic tree based on rRNA data, showing the separation of bacteria, archaea, and eukaryota domains. Therefore, the category of dI-domains is the largest cartesian closed category within omega-algebraic, bounded complete domains, with the exponential being the stable function space.

Cartesian closed categories ofeffective domains Hamrin, Göran Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of .Part of the Semantic Structures in Computation book series (SECO, volume 1) Abstract A subcategory of the category CONT of continuous dcpos is called topologically cartesian closed(tcc for short) if it is closed with respect to finite topological products and function spaces equipped with the Isbell by: 2.Cartesian Closed Categories Andrea Schalk Novem This is asupplementary part of the notes on category theory as provided by Harold Simmons’ ‘An Introduction to Category Theory’.

It explains the notion of a cartesian closed category, which is merely a special example of an adjunction.