Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Soft Computing
Erschienen in: Soft Computing 14/2018

20.11.2017 | Foundations

On special elements and pseudocomplementation in lattices with antitone involutions

Erschienen in: Soft Computing | Ausgabe 14/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The so-called basic algebras correspond in a natural way to lattices with antitone involutions and hence generalize both MV-algebras and orthomodular lattices. The paper deals with several types of special elements of basic algebras and with pseudocomplemented basic algebras.
Vorheriger Artikel Some results on fuzzy -continuous(algebraic) poset
Nächster Artikel Locally adaptive multiple kernel k-means algorithm based on shared nearest neighbors

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren
Fußnoten
1
This identity is one of the aforementioned additional conditions which lead to algebras similar to MV-algebras; see Krňávek and Kühr (2011) and Botur et al. (2014).
 
2
A note on terminology: when speaking of lattices with antitone involution(s), we omit the adjective “bounded”.
 
3
In fact, the quasi-identity \((x\le \lnot y\) & \(x\oplus y\le \lnot z)\) \(\Rightarrow \) \((x\oplus y)\oplus z=x\oplus (z\oplus y)\) was used in Chajda et al. (2009a, b), but it is possible to show that it is equivalent to (2.10).
 
4
In the variety generated by linearly ordered basic algebras, (2.13) is equivalent to the quasi-identity \(x\le y\) \(\Rightarrow \) \(z\oplus x\le z\oplus y\), but we do not know whether this is true in general.
 
5
Given \(B\subseteq A\), this is not equivalent to saying that \((B,\vee ,\wedge ,\lnot ,0,1)\) is a Boolean algebra. It can easily happen that \((B,\vee ,\wedge ,\lnot ,0,1)\) is a Boolean algebra, but \((B,\oplus ,\lnot ,0,1)\) is not a Boolean subalgebra of \((A,\oplus ,\lnot ,0,1)\), because B need not be closed under \(\oplus \).
 
6
As in the case of Boolean subalgebras, this is stronger than saying that \((B,\vee ,\wedge ,\lnot ,0,1)\) is an orthomodular lattice.
 
7
This means that the relative complementation in [a, 1], which is the natural antitone involution in [a, 1], is replaced with another antitone involution. Of course, this is possible, provided that the interval has more than two elements. For a concrete example, see Chajda and Kühr (2013b), Example 3.1 or Krňávek and Kühr (2015), Example 14.
 
8
Namely, the identity \(x\oplus (\lnot x\wedge y)=x\oplus y\) in Krňávek and Kühr (2011), and the identity \(x\le x\oplus y\) in Botur and Kühr (2014).
 
Literatur
Balbes R, Dwinger P (1975) Distributive lattices. University of Missouri Press, ColumbiaMATH
Botur M, Halaš R (2008) Finite commutative basic algebras are MV-effect algebras. J Mult Valued Log Soft Comput 14:69–80MathSciNetMATH
Botur M, Kühr J (2014) On (finite) distributive lattices with antitone involutions. Soft Comput 18:1033–1040CrossRefMATH
Botur M, Kühr J, Rachůnek J (2014) On states and state operators on certain basic algebras. Int J Theor Phys 53:3512–3530MathSciNetCrossRefMATH
Chajda I, Emanovský P (2004) Bounded lattices with antitone involutions and properties of MV-algebras. Discuss Math Gen Algebra Appl 24:31–42MathSciNetCrossRefMATH
Chajda I, Halaš R, Kühr J (2009a) Many-valued quantum algebras. Algebra Univ 60:63–90
Chajda I, Halaš R, Kühr J (2009b) Every effect algebra can be made into a total algebra. Algebra Univ 61:139–150
Chajda I, Kolařík M (2009) Direct decompositions of basic algebras and their idempotent modifications. Acta Univ M Belii Ser Math 25:11–19MathSciNetMATH
Chajda I, Kühr J (2013) Ideals and congruences of basic algebras. Soft Comput 17:401–410CrossRefMATH
Cignoli RLO, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer, DordrechtCrossRefMATH
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer and Ister Science, Dordrecht and BratislavaCrossRefMATH
Jenča G, Riečanová Z (1999) On sharp elements in lattice-ordered effect algebras. BUSEFAL 80:24–29
Kalmbach G (1983) Orthomodular lattices. Academic Press, LondonMATH
Kôpka F, Chovanec F (1994) D-posets. Math Slov 44:21–34MATH
Krňávek J, Kühr J (2015) A note on derivations on basic algebras. Soft Comput 19:1765–1771CrossRefMATH
Krňávek J, Kühr J (2016) On non-associative generalizations of MV-algebras and lattice-ordered commutative loops. Fuzzy Sets Syst 289:122–136MathSciNetCrossRefMATH
Kühr J, Chajda I, Halaš R (2015) The join of the variety of MV-algebras and the variety of orthomodular lattices. Int J Theor Phys 54:4423–4429MathSciNetCrossRefMATH
Riečanová Z (1997) Compatibility and central elements in effect algebras. Tatra Mt Math Publ 10:119–128MathSciNetMATH
Riečanová Z (1999) Subalgebras, intervals and central elements of generalised effect algebras. Int J Theor Phys 38:3209–3220CrossRefMATH
Metadaten
Titel
On special elements and pseudocomplementation in lattices with antitone involutions
Publikationsdatum
20.11.2017
Erschienen in
Soft Computing / Ausgabe 14/2018
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-017-2926-7

Weitere Artikel der Ausgabe 14/2018

Soft Computing 14/2018 Zur Ausgabe

Foundations

Approximate bisimulation relations for fuzzy automata

Methodologies and Application

Qualitative modeling of catastrophe in group opinion

Methodologies and Application

Fuzzy weighted c-harmonic regressions clustering algorithm

Foundations

Epidemic spreading on a complex network with partial immunization

Methodologies and Application

Impact of sensor-based change detection schemes on the performance of evolutionary dynamic optimization techniques

Methodologies and Application

A competitive functional link artificial neural network as a universal approximator