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Abstrakty
We investigate the structure theory of the variety of PBZ*-lattices and some of its proper subvarieties. These lattices with additional structure originate in the foundations of quantum mechanics and can be viewed as a common generalisation of orthomodular lattices and Kleene algebras expanded by an extra unary operation. We lay down the basics of the theories of ideals and of central elements in PBZ*-lattices, we prove some structure theorems, and we explore some connections with the theories of subtractive and binary discriminator varieties.
Czasopismo
Rocznik
Tom
Strony
3--39
Opis fizyczny
Bibliogr. 26 poz.
Twórcy
autor
- Department of Pedagogy, Psychology, Philosophy University of Cagliari 09123 Cagliari, Italy
autor
- cmuresan@fmi.unibuc.ro
autor
- Department of Pedagogy, Psychology, Philosophy University of Cagliari 09123 Cagliari, Italy
Bibliografia
- [1] P. Aglian`o, A. Ursini, On subtractive varieties II: General properties, Algebra Universalis 36 (1996), 222–259.
- [2] P. Aglian`o, A. Ursini, On subtractive varieties III: From ideals to congruences, Algebra Universalis 37 (1997), 296–333.
- [3] P. Aglian`o, A. Ursini, On subtractive varieties IV: Definability of principal ideals, Algebra Universalis 38 (1997), 355–389.
- [4] R. Bignall, M. Spinks, On binary discriminator varieties, I, II, and III, typescript.
- [5] G. Bruns, J. Harding, Algebraic aspects of orthomodular lattices, In: Current Research in Operational Quantum Logic (Eds. B. Coecke et al.), Springer, Berlin, 2000, pp. 37–65.
- [6] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics 78, Springer–Verlag, New York–Berlin, 1981.
- [7] G. Cattaneo, M.L. Dalla Chiara, R. Giuntini, Some algebraic structures for manyvalued logicst, Tatra Mountains Mathematical Publications 15 (1998), 173–196.
- [8] G. Cattaneo, R. Giuntini, R. Pilla, BZMV and Stonian MV algebras (applications to fuzzy sets and rough approximations), Fuzzy Sets and Systems 108 (1999), 201–222.
- [9] G. Cattaneo, G. Nistic`o, Brouwer-Zadeh posets and three-valued £Lukasiewicz posets, Fuzzy Sets and Systems 33:2 (1989), 165–190.
- [10] I. Chajda, A note on pseudo-Kleene algebras, Acta Univ. Palacky Olomouc 55:1 (2016), 39–45.
- [11] I. Chajda, R. Halaˇs, I.G. Rosenberg, Ideals and the binary discriminator in universal algebra, Algebra Universalis 42 (1999), 239–251.
- [12] M.L. Dalla Chiara, R. Giuntini, R.J. Greechie, Reasoning in Quantum Theory, Kluwer, Dordrecht, 2004.
- [13] J.M. Font, Abstract Algebraic Logic: An Introductory Textbook, College Publications, London, 2016.
- [14] R. Giuntini, A. Ledda, F. Paoli, A new view of effects in a Hilbert space, Studia Logica 104 (2016), 1145–1177.
- [15] R. Giuntini, A. Ledda, F. Paoli, On some properties of PBZ*-lattices, International Journal of Theoretical Physics 56:12 (2017), 3895–3911.
- [16] R. Giuntini, C. Muresan, F. Paoli, On PBZ*-lattices, In: Mathematics, Logic, and Their Philosophies: Essays in Honour of Mohammad Ardeshir (Eds. M. Saleh Zarepour, S. Rahman, M. Mojtahedi), Springer, Berlin, forthcoming.
- [17] G. Gr¨atzer, Universal Algebra, Second Edition, Springer Science+Business Media, LLC, New York, 2008.
- [18] H.F. de Groote, On a canonical lattice structure on the effect algebra of a von Neumann algebra, arXiv:math-ph/0410018v2, 2005.
- [19] E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, New York, 1978.
- [20] C. Mure¸san, A note on direct products, subreducts and subvarieties of PBZ*-lattices, Mathematica Slovaca, forthcoming, ArXiv:1904.10093v2 [math.RA], 2019.
- [21] M.P. Olson, The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice, Proceedings of the American Mathematical Society 28 (1971), 537–544.
- [22] A. Salibra, A. Ledda, F. Paoli, T. Kowalski, Boolean-like algebras, Algebra Universalis 69:2 (2013), 113–138.
- [23] D.W. Stroock, A Concise Introduction to the Theory of Integration (3rd ed.), Birkh¨auser, Basel, 1998.
- [24] A. Ursini, On subtractive varieties, I, Algebra Universalis 31 (1994), 204–222.
- [25] A. Ursini, On subtractive varieties, V: congruence modularity and the commutators, Algebra Universalis 43 (2000), 51–78.
- [26] H. Werner, Discriminator Algebras, Studien zur Algebra und ihre Anwendungen, Band 6, Akademie-Verlag, Berlin, 1978.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-b9970eb8-74fb-4c45-965f-17cb9c93795a