Quantale

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (May 2024) (Learn how and when to remove this message)

In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras).^[1] Quantales are sometimes referred to as complete residuated semigroups.

Overview

A quantale is a complete lattice $Q$ with an associative binary operation $\ast \colon Q\times Q\to Q$ , called its multiplication, satisfying a distributive property such that

x*\left(\bigvee _{i\in I}{y_{i}}\right)=\bigvee _{i\in I}(x*y_{i})

and

\left(\bigvee _{i\in I}{y_{i}}\right)*{x}=\bigvee _{i\in I}(y_{i}*x)

for all $x,y_{i}\in Q$ and $i\in I$ (here $I$ is any index set). The quantale is unital if it has an identity element $e$ for its multiplication:

x*e=x=e*x

for all $x\in Q$ . In this case, the quantale is naturally a monoid with respect to its multiplication $\ast$ .

A unital quantale may be defined equivalently as a monoid in the category Sup of complete join-semilattices.

A unital quantale is an idempotent semiring under join and multiplication.

A unital quantale in which the identity is the top element of the underlying lattice is said to be strictly two-sided (or simply integral).

A commutative quantale is a quantale whose multiplication is commutative. A frame, with its multiplication given by the meet operation, is a typical example of a strictly two-sided commutative quantale. Another simple example is provided by the unit interval together with its usual multiplication.

An idempotent quantale is a quantale whose multiplication is idempotent. A frame is the same as an idempotent strictly two-sided quantale.

An involutive quantale is a quantale with an involution

(xy)^{\circ }=y^{\circ }x^{\circ }

that preserves joins:

{\biggl (}\bigvee _{i\in I}{x_{i}}{\biggr )}^{\circ }=\bigvee _{i\in I}(x_{i}^{\circ }).

A quantale homomorphism is a map $f\colon Q_{1}\to Q_{2}$ that preserves joins and multiplication for all $x,y,x_{i}\in Q_{1}$ and $i\in I$ :

f(xy)=f(x)f(y),

f\left(\bigvee _{i\in I}{x_{i}}\right)=\bigvee _{i\in I}f(x_{i}).

References

^ Paeska, Jan; Slesinger, Radek (2018). "A representation theorem for quantale valued sup-algebras". IEEE 48th International Symposium on Multiple-Valued Logic: 1 – via IEEE Xplore.

C.J. Mulvey (2001) [1994], "Quantale", Encyclopedia of Mathematics, EMS Press [1]
J. Paseka, J. Rosicky, Quantales, in: B. Coecke, D. Moore, A. Wilce, (Eds.), Current Research in Operational Quantum Logic: Algebras, Categories and Languages, Fund. Theories Phys., vol. 111, Kluwer Academic Publishers, 2000, pp. 245–262.
M. Piazza, M. Castellan, Quantales and structural rules. Journal of Logic and Computation, 6 (1996), 709–724.
K. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990.

This mathematics-related article is a stub. You can help Wikipedia by expanding it.

Read other articles:

Nuke

NukeNama lahirNuke LuciyantiNama lainNukeLahir29 September 1980 (umur 43)AsalSurabayaGenrePopTahun aktif2010-sekarang Nuke (lahir 29 September 1980) adalah penyanyi Indonesia. Diskografi Cerita Cinta (2010) Artikel bertopik biografi Indonesia ini adalah sebuah rintisan. Anda dapat membantu Wikipedia dengan mengembangkannya.lbs

Kerajaan Ahom

Kerajaan Ahom Kerajaan Ahom (1228–1826, dijuluki Kerajaan Assam pada abad pertengahan) adalah kerajaan di lembah Brahmaputra di Assam yang berhasil mempertahankan kemerdekaannya selama hampir 600 tahun dan berhasil melawan ekspansi Mughal di India Timur Laut. Kerajaan ini didirikan oleh Sukaphaa. Kerajaan ini melemah dengan terjadinya pemberontakan Moamoria, dan runtuh akibat invasi Burma. Dengan kekalahan Burma setelah Perang Inggris-Burma Pertama dan Traktat Yandaboo tahun 1826, kekuasan ...

Carnival Corporation & plc

Carnival Corporation & plcCarnival Place, Kantor pusat Carnival Cruise Line di Doral, FloridaJenisPublikKode emiten NYSE: CCL NYSE: CUK LSE: CCL Komponen S&P 500 (NYSE: CCL) Komponen FTSE 250 (LSE: CCL) ISINPA1436583006/GB0031215220IndustriPenyantunan, pariwisataPendahuluCarnival Cruise Line[1]P&O Princess Cruises[2]Didirikan1972; 52 tahun lalu (1972)[3] (sebagai Carnival Cruise Line, yang kini menjadi anak usaha)1993; 31 tahun lalu (1993) (sebaga...

العلاقات الأوكرانية الروسية

العلاقات الأوكرانية الروسية روسيا أوكرانيا تعديل مصدري - تعديل بدأت رسميًا العلاقات الثنائية الحديثة بين روسيا وأوكرانيا خلال الحرب العالمية الأولى في الحين الذي كانت به الإمبراطورية الروسية السابقة تخوض فترة إصلاحها السياسي. في العام 1920، تغيرت العلاقا...

Guido Zappa

Italian mathematician (1915–2015) Guido ZappaZappa in 1961Born(1915-12-07)7 December 1915Naples, ItalyDied17 March 2015(2015-03-17) (aged 99)Florence, ItalyAlma materScuola Normale SuperioreKnown forAlgebraic geometryGroup theoryScientific careerFieldsMathematicsInstitutionsUniversity of FlorenceAcademic advisorsFrancesco Severi Guido Zappa (7 December 1915 – 17 March 2015) was an Italian mathematician and a noted group theorist: his other main research interests were geome...

محمد مجتهد شبستري

محمد مجتهد شبستري معلومات شخصية الميلاد سنة 1936 (العمر 87–88 سنة) تبريز، وشبستر[1] مواطنة قصر بهلوي (1936–1979) إيران (1979–) إخوة وأخوات محسن مجتهد شبستري[2] مناصب عضو مجلس الشورى الإسلامي في المنصب1980 – 1981 الحياة العملية المدرسة الأم حوزة قم[3&...

List of awards and nominations received by Cardi B

Cardi B awards and nominationsCardi B in 2018Awards and nominationsAward Wins Nominations American Music Awards 6 10BET Awards 6 20BET Hip Hop Awards 14 37Billboard Latin Music Awards 1 4Billboard Music Awards 8 34Billboard Women in Music 1 1Brit Awards 0 2Global Awards 1 2Grammy Awards 1 9iHeartRadio Music Awards 4 25Juno Awards 0 1Latin American Music Awards 3 4Lo Nuestro Awards 1 3MOBO Awards 0 1MTV Europe Music Awards 2 10MTV Millennial Awards 2 5MTV Movie & TV Awards 0 1MTV Video Mu...

Otoritas Transportasi Darat

Otoritas Transportasi Darat (bahasa Inggris: Land Transport Authority, LTA) adalah badan hukum di bawah Kementerian Transportasi Pemerintah Singapura. Artikel bertopik Singapura ini adalah sebuah rintisan. Anda dapat membantu Wikipedia dengan mengembangkannya.lbs Artikel rintisan ini tidak memiliki kategori.Tolong bantu Wikipedia untuk menambahkan kategori. Tag ini diberikan pada Desember 2022.

Léon Bertrand (homme politique)

Pour les articles homonymes, voir Bertrand et Léon Bertrand. Léon Bertrand Fonctions Ministre délégué au Tourisme (secrétaire d'État jusqu'au 30 mars 2004) 17 juin 2002 – 15 mai 2007(4 ans, 10 mois et 28 jours) Président Jacques Chirac Gouvernement Raffarin II et IIIVillepin Prédécesseur Gilles de Robien (ministre) Successeur Luc Chatel (secrétaire d'État) Député français 23 juin 1988 – 19 juillet 2002(14 ans et 26 jours) Élection 12 juin 1988 Ré...

Marvel Feature

Questa voce sull'argomento Marvel Comics è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Segui i suggerimenti del progetto di riferimento. Marvel Feature (vol. 1)fumetto Copertina di Albi dei Super-Eroi n. 1 (edizione italiana, ristampa dalla Marvel Italia), che riproduce quella di Marvel Feature (vol. 1[1]) n. 1, disegni di Neal Adams Lingua orig.inglese PaeseStati Uniti TestiRoy Thomas, Mike Friedrich, Len Wein DisegniRoss Andru...

Oğuz Savaş

Oğuz Savaş Savaş con la maglia del Bahçeşehir Nazionalità Turchia Altezza 211 cm Peso 132 kg Pallacanestro Ruolo Centro Squadra Çayırova Belediyesi CarrieraSquadre di club 2004-2006 Ülkerspor41 (253)2006-2015 Fenerbahçe Ülker243 (2.054)2015-2017 Darüşşafaka20 (86)2018-2019 Darüşşafaka24 (126)2019-2020 Beşiktaş19 (141)2020-2021 Bursaspor28 (303)2021-2023 Bahçeşehir Koleji45 (331)2023- Çayırova Belediyesi15 (143)Naziona...

哈比卜·布尔吉巴

哈比卜·布尔吉巴الحبيب بورقيبة‎第1任突尼斯总统任期1957年7月25日—1987年11月7日（30年105天）总理巴希·拉德加姆（英语：Bahi Ladgham）（1969年－1970年）赫迪·努伊拉（英语：Hedi Nouira）（1970年－1980年）穆罕默德·姆扎利（英语：Mohammed Mzali）（1980年－1986年）拉希德·斯法尔（英语：Rachid Sfar）（1986年－1987年）宰因·阿比丁·本·阿里（1987年）继任宰因·阿比丁·本·...

2010–11 Nottingham Forest F.C. season

Nottingham Forest F.C. 2010–11 football seasonNottingham Forest F.C.2010–11 seasonChairmanNigel DoughtyManagerBilly DaviesStadiumCity GroundFootball League Championship6thFootball League Championship play-offsSemi-finalsFA CupFourth roundLeague CupFirst roundTop goalscorerLeague: Lewis McGugan (13)All: Lewis McGugan (13)Highest home attendance29 490 vs. Derby County (Championship, 29 December 2010)Lowest home attendance19 411 vs. Burnley (Championship, 12 April 2011) Home colours Away co...

Centre Party (Turkey)

Turkish political party Centre Party Merkez PartiLeaderAbdurrahim KarslıFoundedJune 2, 2014 (2014-06-02)Preceded byPeople's Voice MovementHeadquartersAnkaraMembership (2024) 3,114[1]IdeologyLiberalismEuroscepticismAnti-establishmentPolitical positionCentre to centre-right[2]Colours Turquoise, blue and orangeSloganÖnce insan (People first)Websitewww.merkezparti.org.trPolitics of TurkeyPolitical partiesElections The Centre Party (Turk...

Cinema of Denmark

Filmmaking industry in Denmark This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Cinema of Denmark – news · newspapers · books · scholar · JSTOR (August 2023) (Learn how and when to remove this message) Cinema of DenmarkNo. of screens458 (2017)[1] • Per capita7.9 per 100,000 (2011)[1 ...

Bhagat Singh

Indian revolutionary (1907–1931) This article is about the Indian socialist revolutionary. For the Indian-American civil rights activist, see Bhagat Singh Thind. Bhagat SinghSingh in 1929Born(1907-09-28)28 September 1907[1]Banga, Punjab, British IndiaDied23 March 1931(1931-03-23) (aged 23)Lahore Central Jail, Punjab, British IndiaCause of deathExecution by hangingMonumentsHussainiwala National Martyrs MemorialOther namesShaheed-e-AzamOrganization(s)Naujawan Bharat Sab...

Needtobreathe discography

Needtobreathe discographyNeedtobreathe performing at the Fox Theatre in Atlanta, Georgia, December 2015Studio albums10Live albums4Compilation albums1Music videos10EPs12Singles49 The discography of American alternative rock band Needtobreathe consists of ten studio albums, four live albums, twelve extended plays and more than forty singles. Four of the albums charted in the Billboard 200 with Rivers in the Wasteland reaching No. 3.[1] The band was signed to Atlantic Records until 2020...

Warang Citi

Writing script for Ho language of India Warang Citi𑢹𑣗𑣁𑣜𑣊 𑢯𑣂𑣕𑣂UppercaseLowercaseThe word 'Warang Citi (Hbårŋ citi-Undying Writing)' in Warang Citi uppercase and lowercase script[1]Script type Alphabet CreatorLako BodraTime period1950s — presentDirectionLeft-to-right LanguagesHoRelated scriptsParent systemsoriginal inventionWarang CitiISO 15924ISO 15924Wara (262), Warang Citi (Varang Kshiti)UnicodeUnicode aliasWarang CitiUnicode rangeU+11...

Primitive abundant number

Abundant number whose proper divisors are all deficient numbers Euler diagram of numbers under 100: Abundant Primitive abundant Highly abundant Superabundant and highly composite Colossally abundant and superior highly composite Weird Perfect Composite Deficient In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers.[...

Spellow railway station

Former railway station in England SpellowGeneral informationLocationWalton, LiverpoolEnglandCoordinates53°26′34″N 2°58′14″W / 53.4428°N 2.9705°W / 53.4428; -2.9705Grid referenceSJ356944Platforms2[1]Other informationStatusDisusedHistoryOriginal companyLondon and North Western RailwayPre-groupingLondon and North Western RailwayPost-groupingLondon Midland and Scottish RailwayKey datesSeptember 1882Opened31 May 1948Closed[2] Spellow railway stat...

PROFILPELAJAR.COM