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On special elements and pseudocomplementation in lattices with antitone involutions

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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.

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Notes

  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).

References

  • Balbes R, Dwinger P (1975) Distributive lattices. University of Missouri Press, Columbia

    MATH  Google Scholar 

  • Botur M, Halaš R (2008) Finite commutative basic algebras are MV-effect algebras. J Mult Valued Log Soft Comput 14:69–80

    MathSciNet  MATH  Google Scholar 

  • Botur M, Kühr J (2014) On (finite) distributive lattices with antitone involutions. Soft Comput 18:1033–1040

    Article  MATH  Google Scholar 

  • Botur M, Kühr J, Rachůnek J (2014) On states and state operators on certain basic algebras. Int J Theor Phys 53:3512–3530

    Article  MathSciNet  MATH  Google Scholar 

  • Chajda I, Emanovský P (2004) Bounded lattices with antitone involutions and properties of MV-algebras. Discuss Math Gen Algebra Appl 24:31–42

    Article  MathSciNet  MATH  Google Scholar 

  • 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 25:11–19

    MathSciNet  MATH  Google Scholar 

  • Chajda I, Kühr J (2013) Finitely generated varieties of distributive effect algebras. Algebra Univ 69:213–229

    Article  MathSciNet  MATH  Google Scholar 

  • Chajda I, Kühr J (2013) Ideals and congruences of basic algebras. Soft Comput 17:401–410

    Article  MATH  Google Scholar 

  • Cignoli RLO, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer and Ister Science, Dordrecht and Bratislava

    Book  MATH  Google Scholar 

  • Foulis D, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1331–1352

    Article  MathSciNet  MATH  Google Scholar 

  • Grätzer G (2011) Lattice theory: Foundation. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  • Jenča G, Riečanová Z (1999) On sharp elements in lattice-ordered effect algebras. BUSEFAL 80:24–29

    Google Scholar 

  • Kalman JA (1958) Lattices with involution. Trans Am Math Soc 87:485–491

    Article  MathSciNet  MATH  Google Scholar 

  • Kalmbach G (1983) Orthomodular lattices. Academic Press, London

    MATH  Google Scholar 

  • Kôpka F, Chovanec F (1994) D-posets. Math Slov 44:21–34

    MATH  Google Scholar 

  • Krňávek J, Kühr J (2011) Pre-ideals of basic algebras. Int J Theor Phys 50:3828–3843

    Article  MathSciNet  MATH  Google Scholar 

  • Krňávek J, Kühr J (2015) A note on derivations on basic algebras. Soft Comput 19:1765–1771

    Article  MATH  Google Scholar 

  • Krňávek J, Kühr J (2016) On non-associative generalizations of MV-algebras and lattice-ordered commutative loops. Fuzzy Sets Syst 289:122–136

    Article  MathSciNet  MATH  Google Scholar 

  • 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 54:4423–4429

    Article  MathSciNet  MATH  Google Scholar 

  • Riečanová Z (1997) Compatibility and central elements in effect algebras. Tatra Mt Math Publ 10:119–128

    MathSciNet  MATH  Google Scholar 

  • Riečanová Z (1999) Subalgebras, intervals and central elements of generalised effect algebras. Int J Theor Phys 38:3209–3220

    Article  MATH  Google Scholar 

  • Riečanová Z (2009) Pseudocomplemented lattice effect algebras and existence of states. Inf Sci 179:529–534

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous reviewer for his/her comments.

Funding

P. E. was supported by the Palacký University project “Mathematical structures”, No. IGA PrF 2016 006. J. K. was supported by the project “Algebraic, many-valued and quantum structures for uncertainty modelling” of the Czech Science Foundation (GAČR), No. 15-15286S.

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  1. Department of Algebra and Geometry, Faculty of Science, Palacký University in Olomouc, 17. listopadu 12, 771 46, Olomouc, Czechia

    Petr Emanovský & Jan Kühr

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  1. Petr Emanovský
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  2. Jan Kühr
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Correspondence to Petr Emanovský.

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Communicated by A. Di Nola.

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Emanovský, P., Kühr, J. On special elements and pseudocomplementation in lattices with antitone involutions. Soft Comput 22, 4561–4572 (2018). https://doi.org/10.1007/s00500-017-2926-7

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