Non-commutative effect algebras, L-algebras, and local duality

References

[1] Artin, E.: Theory of braids, Ann. of Math. 48 (1947), 101–126.10.2307/1969218Suche in Google Scholar

[2] Baxter, R. J.: Partition function of the eight-vertex lattice model, Ann. Physics 70 (1972), 193–228.10.1016/0003-4916(72)90335-1Suche in Google Scholar

[3] Beran, L.: Orthomodular Lattices. Algebraic Approach. Mathematics and its Applications, Vol. 18, D. Reidel, Dordrecht, 1985.10.1007/978-94-009-5215-7Suche in Google Scholar

[4] Bigard, A.—Keimel, K.—Wolfenstein, S.: Groupes et Anneaux Réticulés, Lecture Notes in Math., Vol. 608, Springer-Verlag, Berlin-New York, 1977.10.1007/BFb0067004Suche in Google Scholar

[5] Blok, W. J.—Ferreirim, I. M. A.: Hoops and their implicational reducts (abstract). In: Algebraic Methods in Logic and in Computer Science, Banach Center Publ., Vol. 28, Polish Acad. Sci., Warsaw, 1993, pp. 219–230.10.4064/-28-1-219-230Suche in Google Scholar

[6] Blok, W. J.—Ferreirim, I. M. A.: On the structure of hoops, Algebra Universalis 43 (2000), 233–257.10.1007/s000120050156Suche in Google Scholar

[7] Bosbach, B.: Komplementäre Halbgruppen. Axiomatik und Arithmetik, Fund. Math. 64 (1969), 257–287.10.4064/fm-64-3-257-287Suche in Google Scholar

[8] Bosbach, B.: Concerning cone algebras, Algebra Universalis 15 (1982), 58–66.10.1007/BF02483708Suche in Google Scholar

[9] Brieskorn, E.—Saito, K.: Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271.10.1007/BF01406235Suche in Google Scholar

[10] Busch, P.—Lahti, P. J.—Mittelstaedt, P.: The Quantum Theory of Measurements, Springer-Verlag, Berlin, 1991.10.1007/978-3-662-13844-1Suche in Google Scholar

[11] Busch, P.—Grabowski, M.—Lahti, P. J.: Operational Quantum Physics, Springer-Verlag, Berlin, 1995.10.1007/978-3-540-49239-9Suche in Google Scholar

[12] Chang, C. C.: Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467–490.10.1090/S0002-9947-1958-0094302-9Suche in Google Scholar

[13] Chang, C. C.: A new proof of the completeness of the Łukasiewicz axioms, Trans. Amer. Math. Soc. 93 (1959), 74–80.10.1090/S0002-9947-1959-0122718-1Suche in Google Scholar

[14] Cīrulis, J.: (H)-Hilbert algebras are not same as Hertz algebras, Bull. Sect. Logic Univ. Łódź 32(3) (2003), 107–108.Suche in Google Scholar

[15] Cīrulis, J.: Hilbert algebras as implicative partial semilattices, Cent. Eur. J. Math. 5(2) (2007), 264–279.10.2478/s11533-007-0008-2Suche in Google Scholar

[16] Cornish, W. H.: BCK-algebras with a supremum, Math. Japon. 27(1) (1982), 63–73.Suche in Google Scholar

[17] Darnel, M. R.: Theory of Lattice-Ordered Groups. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 187, Marcel Dekker, Inc., New York, 1995.Suche in Google Scholar

[18] Dehornoy, P.—Digne, F.—Godelle, E.—Krammer, D.—Michel, J.: Foundations of Garside Theory. EMS Tracts in Math., Vol. 22, European Mathematical Society, 2015.10.4171/139Suche in Google Scholar

[19] Deligne, P.: Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302.10.1007/BF01406236Suche in Google Scholar

[20] Diego, A.: Les Algèbres de Hilbert. Collect. de Logique Math. Sér. A, Vol. 21, Gauthier-Villars, Paris, 1966.Suche in Google Scholar

[21] Dietzel, C.—Rump, W.: The structure group of a non-degenerate effect algebra, Algebra Universalis 81 (2020), Art. No. 27.10.1007/s00012-020-00657-7Suche in Google Scholar

[22] Dietzel, C.—Rump, W.: The paraunitary group of a von Neumann algebra, Bull. London Math. Soc. 54 (2022), 1220–1231.10.1112/blms.12621Suche in Google Scholar

[23] Dietzel, C.—Rump, W.—Zhang, X.: One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups, J. Algebra 526 (2019), 51–80.10.1016/j.jalgebra.2019.02.012Suche in Google Scholar

[24] Dvurečenskij, A.: Pseudo MV-algebras are intervals in l-groups, J. Austral. Math. Soc. 72 (2002), 427–445.10.1017/S1446788700036806Suche in Google Scholar

[25] Dvurečenskij, A.—Pulmannová, S.: New Trends in Quantum Structures. Mathematics and its Applications, Vol. 516, Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava, 2000.10.1007/978-94-017-2422-7Suche in Google Scholar

[26] Dvurečenskij, A.—Vetterlein, T.: Pseudoeffect algebras, I, Basic properties, Internat. J. Theoret. Phys. 40(3) (2001), 685–701.10.1023/A:1004192715509Suche in Google Scholar

[27] Dvurečenskij, A.—Vetterlein, T.: Pseudoeffect algebras, II, Group representations, Internat. J. Theoret. Phys. 40(3) (2001), 703–726.10.1023/A:1004144832348Suche in Google Scholar

[28] Dvurečenskij, A.—Vetterlein, T.: Algebras in the positive cone of po-groups, Order 19(2) (2002), 127–146.10.1023/A:1016551707476Suche in Google Scholar

[29] Dvurečenskij, A.—Vetterlein, T.: Non-commutative algebras and quantum structures, Internat. J. Theoret. Phys. 43 (2004), 1599–1612.10.1023/B:IJTP.0000048806.69906.11Suche in Google Scholar

[30] Etingof, P.—Schedler, T.—Soloviev, A.: Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169–209.10.1215/S0012-7094-99-10007-XSuche in Google Scholar

[31] Ferreirim, I. M. A.: On a conjecture by Andrzej Wroński for BCK-algebras and subreducts of hoops, Sci. Math. Jpn. 53(1) (2001), 119–132.Suche in Google Scholar

[32] Foulis, D. J.—Bennett, M. K.: Effect algebras and unsharp quantum logics, Special issue dedicated to Constantin Piron on the occasion of his sixtieth birthday, Found. Phys. 24(10) (1994), 1331–1352.10.1007/BF02283036Suche in Google Scholar

[33] Foulis, D. J.—Pulmannová, S.: Spectral order on a synaptic algebra, Order 36 (2019), 1–17.10.1007/s11083-018-9451-xSuche in Google Scholar

[34] Fuchs, L.: Riesz groups, Ann. Scuola Norm. Sup. Pisa, 3e série, 19 (1965), 1–34.Suche in Google Scholar

[35] Georgescu, G.—Iorgulescu, A.: Pseudo MV-algebras, Multiple-Valued Logic 6 (2001), 95–135.Suche in Google Scholar

[36] Gispert, J.—Mundici, D.: MV-algebras: a variety for magnitudes with Archimedean units, Algebra Universalis 53 (2005), 7–43.10.1007/s00012-005-1905-5Suche in Google Scholar

[37] Giuntini, R.—Greuling, H.: Toward a formal language for unsharp properties, Found. Phys. 19 (1989), 931–945.10.1007/BF01889307Suche in Google Scholar

[38] Glass, A. M. W.: Partially Ordered Groups, World Scientific, 1999.10.1142/3811Suche in Google Scholar

[39] Gudder, S.: Sharp and unsharp quantum effects, Adv. Appl. Math. 20 (1998), 169–187.10.1006/aama.1997.0575Suche in Google Scholar

[40] Gudder, S.: Convex structures and effect algebras, Int. J. Theor. Physics 38 (1999), 3179–3187.10.1023/A:1026678114856Suche in Google Scholar

[41] Gudder, S.: A histories approach to quantum mechanics, J. Math. Phys. 39 (1998), 5772–5788.10.1063/1.532592Suche in Google Scholar

[42] Gudder, S.—Nagy, G.: Sequentially independent effects, Proc. Amer. Math. Soc. 130 (2001), 1125–1130.10.1090/S0002-9939-01-06194-9Suche in Google Scholar

[43] Hamhalter, J.: Spectral order of operators and range projections, J. Math. Anal. Appl. 331 (2007), 1122–1134.10.1016/j.jmaa.2006.09.045Suche in Google Scholar

[44] Hamhalter, J.: Spectral order on AW^∗-algebras and its preservers, Lobachevskii J. Math. 37 (2016), 439–448.10.1134/S1995080216040107Suche in Google Scholar

[45] Kadison, R.: Order properties of bounded self-adjoint operators, Proc. Amer. Math. Soc. 34 (1951), 505–510.10.1090/S0002-9939-1951-0042064-2Suche in Google Scholar

[46] Kalmbach, G.: Orthomodular Lattices. London Mathematical Society Monographs, Vol. 18, Academic Press, Inc., London, 1983.Suche in Google Scholar

[47] Kôpka, F.—Chovanec, F.: D-posets, Math. Slovaca 44 (1994), 21–34.Suche in Google Scholar

[48] Kowalski, T.: A syntactic proof of a conjecture of Andrzej Wroński, Rep. Math. Logic 28 (1994), 81–86.Suche in Google Scholar

[49] Lu, J.-H.—Yan, M.—Zhu, Y.-C.: On the set-theoretical Yang-Baxter equation, Duke Math. J. 104 (2000), 1–18.10.1215/S0012-7094-00-10411-5Suche in Google Scholar

[50] Molnár, L.—Šemrl, P.: Spectral order automorphisms of the spaces of Hilbert space effects and observables, Lett. Math. Phys. 80 (2007), 239–255.10.1007/s11005-007-0160-4Suche in Google Scholar

[51] Molnár, L.—Nagy, G.: Spectral order automorphisms on Hilbert space effects and observables: The 2-dimensional case, Lett. Math. Phys. 106 (2016), 535–544.10.1007/s11005-016-0830-1Suche in Google Scholar

[52] Monteiro, A.: Cours sur les algèbres de Hilbert et de Tarski, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca, 1960.Suche in Google Scholar

[53] Mundici, D.: Interpretation of AFC^∗ algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15–63.10.1016/0022-1236(86)90015-7Suche in Google Scholar

[54] Navara, M.: An orthomodular lattice admitting no group-valued measures, Proc. Amer. Math. Soc. 122 (1994), 7–12.10.1090/S0002-9939-1994-1191871-XSuche in Google Scholar

[55] Olson, M. P.: The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice, Proc. Amer. Math. Soc. 28 (1971), 537–544.10.1090/S0002-9939-1971-0276788-1Suche in Google Scholar

[56] Polakovič, M.: Generalized effect algebras of bounded positive operators defined on Hilbert spaces, Rep. Math. Phys. 68 (2011), 241–250.10.1016/S0034-4877(12)60007-XSuche in Google Scholar

[57] Pulmannova, S.: Effect algebras with the Riesz decomposition property and AF C^∗-algebras, Found. Phys. 29 (1999), 1389–1401.10.1023/A:1018809209768Suche in Google Scholar

[58] Pulmannová, S.—Vinceková, E.: Congruences and ideals in lattice effect algebras as basic algebras, Kybernetika 45 (2009), 1030–1039.Suche in Google Scholar

[59] Rachůnek, J.: A non-commutative generalization of MV-algebras, Czechoslovak Math. J. 52 (2002), 255–273.10.1023/A:1021766309509Suche in Google Scholar

[60] Rees, D.: On the group of a set of partial transformations, J. London Math. Soc. 22 (1948), 281–284.10.1112/jlms/s1-22.4.281Suche in Google Scholar

[61] Riečanová, Z.: Subalgebras, intervals, and central elements of generalized effect algebras, Internat. J. Theoret. Phys. 38(2) (1999), 3209–3220.Suche in Google Scholar

[62] Riečanová, Z.: Generalization of blocks for D-lattices and lattice-ordered effect algebras, Internat. J. Theoret. Phys. 39(2) (2000), 231–237.Suche in Google Scholar

[63] Rump, W.: L-Algebras, self-similarity, and ℓ-Groups, J. Algebra 320 (2008), 2328–2348.10.1016/j.jalgebra.2008.05.033Suche in Google Scholar

[64] Rump, W.: Right l-groups, geometric Garside groups, and solutions of the quantum Yang-Baxter equation, J. Algebra 439 (2015), 470–510.10.1016/j.jalgebra.2015.04.045Suche in Google Scholar

[65] Rump, W.: Von Neumann algebras, L-algebras, Baer^∗-monoids, and Garside groups, Forum Math. 30(4) (2018), 973–995.10.1515/forum-2017-0108Suche in Google Scholar

[66] Rump, W.—Zhang, X.: L-effect algebras, Studia Logica 108(4) (2020), 725–750.10.1007/s11225-019-09873-2Suche in Google Scholar

[67] Rump, W.: Commutative L-algebras and measure theory, Forum Math. 33(6) (2021), 1527–1548.10.1515/forum-2020-0317Suche in Google Scholar

[68] Rump, W.: L-Algebras and three main non-classical logics, Ann. Pure Appl. Logic 173(7) (2022), Art. ID 103121.10.1016/j.apal.2022.103121Suche in Google Scholar

[69] Sherman, S.: Order in operator algebras, Amer. J. Math. 73 (1951), 227–232.10.2307/2372173Suche in Google Scholar

[70] Wroński, A.: An algebraic motivation for BCK-algebras, Math. Japon. 30(2) (1985), 187–193.Suche in Google Scholar

[71] Wu, Y.—Wang, J.—Yang, Y.: Lattice-ordered effect algebras and L-algebras, Fuzzy Sets Systems 369 (2019), 103–113.10.1016/j.fss.2018.08.013Suche in Google Scholar

[72] Weinstein, A.—Xu, P.: Classical solutions of the quantum Yang-Baxter equation, Comm. Math. Phys. 148(2) (1992), 309–343.10.1007/BF02100863Suche in Google Scholar

[73] Yang, C. N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312–1315.10.1103/PhysRevLett.19.1312Suche in Google Scholar