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On the Structure of Pseudo BL-algebras and Pseudo Hoops in Quantum Logics

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Abstract

The main aim of the paper is to solve a problem posed in Di Nola et al. (Multiple Val. Logic 8:715–750, 2002) whether every pseudo BL-algebra with two negations is good, i.e. whether the two negations commute. This property is intimately connected with possessing a state, which in turn is essential in quantum logical applications. We approach the solution by describing the structure of pseudo BL-algebras and pseudo hoops as important families of quantum structures. We show when a pseudo hoop can be embedded into the negative cone of the reals. We give an equational base characterizing representable pseudo hoops. We also describe some subvarieties: normal-valued, and varieties where each maximal filter is normal. We produce some noncommutative covers and extend the area where each algebra is good. Finally, we show that there are uncountably many subvarieties of pseudo BL-algebras having members that are not good.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    A. Dvurečenskij

  2. Facolta delle Scienze Filosofiche e Pedagogiche, Università degli Studi di Cagliari, Via Is Mirrionis, 09 123, Cagliari, Italy

    R. Giuntini & T. Kowalski

Authors
  1. A. Dvurečenskij
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  2. R. Giuntini
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  3. T. Kowalski
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Correspondence to A. Dvurečenskij.

Additional information

Dedicated to Prof. P. Mittelstaedt on the occasion of his 80th birthday.

AD acknowledges the University of Cagliari for the hospitality during his stay in Autumn 2008.

The paper has been supported by the Center of Excellence SAS - Physics of Information - I/2/2005, the grant VEGA No. 2/0032/09 SAV, by Science and Technology Assistance Agency under the contract APVV-0071-06, Bratislava, Slovakia.

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Dvurečenskij, A., Giuntini, R. & Kowalski, T. On the Structure of Pseudo BL-algebras and Pseudo Hoops in Quantum Logics. Found Phys 40, 1519–1542 (2010). https://doi.org/10.1007/s10701-009-9342-5

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  • DOI: https://doi.org/10.1007/s10701-009-9342-5

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