Academic Journal
Relatively additive states on quantum logics
| العنوان: | Relatively additive states on quantum logics |
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| المؤلفون: | Pták, Pavel, Weber, Hans |
| بيانات النشر: | Charles University in Prague, Faculty of Mathematics and Physics |
| سنة النشر: | 2005 |
| المجموعة: | DML-CZ (Czech Digital Mathematics Library) |
| مصطلحات موضوعية: | keyword:(weak) state on quantum logic, keyword:Greechie paste job, keyword:Boolean algebra, msc:03G12, msc:46C05, msc:81P10 |
| الوصف: | summary:In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too. |
| نوع الوثيقة: | text |
| وصف الملف: | application/pdf |
| اللغة: | English |
| تدمد: | 0010-2628 1213-7243 |
| Relation: | mr:MR2176895; zbl:Zbl 1121.03085; reference:[1] Beran L.: Orthomodular Lattices. Algebraic Approach.Academia, Praha, 1984. Zbl 0558.06008, MR 0785005; reference:[2] Binder J., Pták P.: A representation of orthomodular lattices.Acta Univ. Carolin. - Math. Phys. 31 (1990), 21-26. MR 1098124; reference:[3] Dvurečenskij A., Pulmannová S.: New Trends in Quantum Structures.Kluwer/Dordrecht & Ister/Bratislava, 2000. MR 1861369; reference:[4] Greechie R.J.: Orthomodular lattices admitting no states.J. Combin. Theory Ser. A 10 (1971), 119-132. Zbl 0219.06007, MR 0274355; reference:[5] Harding J., Pták P.: On the set representation of an orthomodular poset.Colloquium Math. 89 (2001), 233-240. Zbl 0984.06005, MR 1854706; reference:[6] Kallus M., Trnková V.: Symmetries and retracts of quantum logics.Internat. J. Theoret. Phys. 26 (1987), 1-9. MR 0890206; reference:[7] Katrnoška F.: A representation of orthoposets.Comment. Math. Univ. Carolinae 23 (1982), 489-498. MR 0677857; reference:[8] Navara M.: An orthomodular lattice admitting no group-valued measure.Proc. Amer. Math. Soc. 122 (1994), 7-12. Zbl 0809.06008, MR 1191871; reference:[9] Navara M., Pták P., Rogalewicz V.: Enlargements of quantum logics.Pacific J. Math. 135 (1988), 361-369. MR 0968618; reference:[10] Navara M., Rogalewicz V.: The pasting constructions for orthomodular posets.Math. Nachr. 154 (1991), 157-168. Zbl 0767.06009, MR 1138377; reference:[11] Ovchinnikov P.G.: Exact topological analogs to orthoposets.Proc. Amer. Math. Soc. 125 (1997), 2839-2841. Zbl 0880.06003, MR 1415360; reference:[12] Pták P.: Weak dispersion-free states and the hidden variables hypothesis.J. Math. Phys. 24 (1983), 839-840. MR 0700618; reference:[13] Pták P., Pulmannová S.: Orthomodular Structures as Quantum Logics.Kluwer Academic Publishers, Dordrecht, 1991. MR 1176314; reference:[14] Sultanbekov F.F.: Set logics and their representations.Internat. J. Theoret. Phys. 32 (1993), 11 2177-2186. Zbl 0799.03081, MR 1254335; reference:[15] Tkadlec J.: Partially additive states on orthomodular posets.Colloquium Math. 62 (1991), 7-14. Zbl 0784.03037, MR 1114613; reference:[16] Trnková V.: Automorphisms and symmetries of quantum logics.Internat. J. Theoret. Phys. 28 (1989), 1195-1214. MR 1031603; reference:[17] Varadarajan V.: Geometry of Quantum Theory I, II.Van Nostrand, Princeton, 1968, 1970.; reference:[18] Weber H.: There are orthomodular lattices without non-trivial group valued states; a computer-based construction.J. Math. Anal. Appl. 183 (1994), 89-94. Zbl 0797.06010, MR 1273434 |
| الاتاحة: | http://hdl.handle.net/10338.dmlcz/119527 |
| Rights: | access:Unrestricted ; rights:DML-CZ Czech Digital Mathematics Library, http://dml.cz/ ; rights:Institute of Mathematics AS CR, http://www.math.cas.cz/ ; conditionOfUse:http://dml.cz/use |
| رقم الانضمام: | edsbas.C8E21FE0 |
| قاعدة البيانات: | BASE |
| تدمد: | 00102628 12137243 |
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