An equational theory for \(\sigma \)-complete orthomodular lattices


The condition of \(\sigma \)-completeness related to orthomodular lattices places an important role in the study of quantum probability theory. In the framework of algebras with infinitary operations, an equational theory for the category of \(\sigma \)-complete orthomodular lattices is given. In this structure, we study the congruences theory and directly irreducible algebras establishing an equational completeness theorem. Finally, a Hilbert style calculus related to \(\sigma \)-complete orthomodular lattices is introduced and a completeness theorem is obtained.

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  1. Beran L (1985) Orthomodular lattices. Algebraic approach. Kluwer, Dordrecht

    Book  Google Scholar 

  2. Birkhoff G, von Neumann J (1936) The logic of quantum mechanics. Ann Math 27:823–843

    MathSciNet  Article  Google Scholar 

  3. Burris S, Sankappanavar HP (1981) A course in universal algebra, graduate text in mathematics, vol 78. Springer, New York

    MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  5. Kalmbach G (1974) Orthomodular logic. Z Math Logik Grundl Math 20:395–406

    MathSciNet  Article  Google Scholar 

  6. Kalmbach G (1983) Ortomodular lattices. Academic Press, London

    MATH  Google Scholar 

  7. Maeda F, Maeda S (1970) Theory of symmetric lattices. Springer, Berlin

    Book  Google Scholar 

  8. Słomiński J (1959) The theory of abstract algebras with infinitary operations. Rozprawy Matematyczne 18:1–67

    MathSciNet  MATH  Google Scholar 

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Correspondence to Hector Freytes.

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Communicated by F. Holik.

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Freytes, H. An equational theory for \(\sigma \)-complete orthomodular lattices. Soft Comput 24, 10257–10264 (2020).

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  • \(\sigma \)-Complete orthomodular lattices
  • Infinitary operations
  • Hilbert style calculus