The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.
@article{bwmeta1.element.doi-10_2478_s11533-013-0302-0,
author = {Wolfgang Rump},
title = {Quantum B-algebras},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1881-1899},
zbl = {1326.03077},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0302-0}
}
Wolfgang Rump. Quantum B-algebras. Open Mathematics, Tome 11 (2013) pp. 1881-1899. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0302-0/
[1] van Alten C.J., On varieties of biresiduation algebras, Studia Logica, 2006, 83(1–3), 425–445 http://dx.doi.org/10.1007/s11225-006-8312-6 | Zbl 1099.03057
[2] Bigard A., Keimel K., Wolfenstein S., Groupes et Anneaux Réticulés, Lecture Notes in Math., 608, Springer, Berlin-New York, 1977
[3] Borceux F., Rosický J., Van den Bossche G., Quantales and C*-algebras, J. London Math. Soc., 1989, 40(3), 398–404 http://dx.doi.org/10.1112/jlms/s2-40.3.398
[4] Bosbach B., Komplementäre Halbgruppen, Ein Beitrag zur instruktiven Idealtheorie kommutativer Halbgruppen, Math. Ann., 1965, 161, 279–295 http://dx.doi.org/10.1007/BF01359910 | Zbl 0154.01702
[5] Bosbach B., Concerning cone algebras, Algebra Universalis, 1982, 15(1), 58–66 http://dx.doi.org/10.1007/BF02483708
[6] Connes A., Noncommutative Geometry, Academic Press, San Diego, 1994
[7] Dudek W.A., Yun Y.B., Pseudo-BCI algebras, East Asian Mathematical Journal, 2008, 24(2), 187–190 | Zbl 1149.06010
[8] Dunn J.M., Gehrke M., Palmigiano A., Canonical extensions and relational completeness of some substructural logics, J. Symbolic Logic, 2005, 70(3), 713–740 http://dx.doi.org/10.2178/jsl/1122038911 | Zbl 1101.03021
[9] Dvurečenskij A., Vetterlein T., Pseudoeffect algebras. I. Basic properties, Internat. J. Theoret. Phys., 2001, 40(3), 685–701 http://dx.doi.org/10.1023/A:1004192715509 | Zbl 0994.81008
[10] Dvurečenskij A., Vetterlein T., Pseudoeffect algebras. II. Group representations, Internat. J. Theoret. Phys., 2001, 40(3), 703–726 http://dx.doi.org/10.1023/A:1004144832348 | Zbl 0994.81009
[11] Dvurečenskij A., Vetterlein T., Algebras in the positive cone of po-groups, Order, 2002, 19(2), 127–146 http://dx.doi.org/10.1023/A:1016551707476 | Zbl 1012.03064
[12] Galatos N., Tsinakis C., Generalized MV-algebras, J. Algebra, 2005, 283(1), 254–291 http://dx.doi.org/10.1016/j.jalgebra.2004.07.002
[13] Gehrke M., Jónsson B., Bounded distributive lattices with operators, Math. Japon., 1994, 40(2), 207–215 | Zbl 0855.06009
[14] Gehrke M., Jónsson B., Bounded distributive lattice expansions, Math. Scand., 2004, 94(1), 13–45 | Zbl 1077.06008
[15] Gehrke M., Priestley H.A., Canonical extensions and completions of posets and lattices, Rep. Math. Logic, 2008, 43, 133–152 | Zbl 1147.06005
[16] Georgescu G., Iorgulescu A., Pseudo-MV algebras: a noncommutative extension of MV algebras, In: Information Technology, Bucharest, May 6–9, 1999, Inforec, Bucharest, 1999, 961–968 | Zbl 0985.06007
[17] Georgescu G., Iorgulescu A., Pseudo-BCK algebras: an extension of BCK algebras, In: Combinatorics, Computability and Logic, Constanţa, July 2–6, 2001, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 2001, 97–114 http://dx.doi.org/10.1007/978-1-4471-0717-0_9 | Zbl 0986.06018
[18] Giles R., Kummer H., A non-commutative generalization of topology, Indiana Univ. Math. J., 1971, 21(1), 91–102 http://dx.doi.org/10.1512/iumj.1972.21.21008 | Zbl 0219.54003
[19] Goldblatt R., Varieties of complex algebras, Ann. Pure Appl. Logic, 1989, 44(3), 173–242 http://dx.doi.org/10.1016/0168-0072(89)90032-8
[20] Jónsson B., Tarski A., Boolean algebras with operators. I, Amer. J. Math., 1951, 73(4), 891–939 http://dx.doi.org/10.2307/2372123 | Zbl 0045.31505
[21] Jónsson B., Tarski A., Boolean algebras with operators. II, Amer. J. Math., 1952, 74(1), 127–162 http://dx.doi.org/10.2307/2372074 | Zbl 0045.31601
[22] Jun Y.B., Kim H.S., Neggers J., On pseudo-BCI ideals of pseudo-BCI algebras, Mat. Vesnik, 2006, 58(1–2), 39–46 | Zbl 1119.03068
[23] Kondo M., Relationship between ideals of BCI-algebras and order ideals of its adjoint semigroup, Int. J. Math. Math. Sci., 2001, 28(9), 535–543 http://dx.doi.org/10.1155/S0161171201010985 | Zbl 1007.06014
[24] Kruml D., Resende P., On quantales that classify C*-algebras, Cah. Topol. Géom. Différ. Catég., 2004, 45(4), 287–296 | Zbl 1076.46054
[25] Kühr J., Pseudo BCK-algebras and residuated lattices, In: Contributions to General Algebra, 16, Dresden, June 10–13, Malá Morávka, September 5–11, 2004, Heyn, Klagenfurt, 2005, 139–144
[26] Kühr J., Pseudo-BCK Algebras and Related Structures, Habilitation Thesis, University of Olomouc, 2007 | Zbl 1140.06008
[27] Mulvey C.J., &, In: Second Topology Conference, Taormina, April 4–7, 1984, Rend. Circ. Mat. Palermo, 1986, Suppl. 12, 99–104
[28] Mulvey C.J., Pelletier J.W., On the quantisation of points, J. Pure Appl. Algebra, 2001, 159(2–3), 231–295 http://dx.doi.org/10.1016/S0022-4049(00)00059-1 | Zbl 0983.18007
[29] Mulvey C.J., Pelletier J.W., On the quantisation of spaces, J. Pure Appl. Algebra, 2002, 175(1–3), 289–325 http://dx.doi.org/10.1016/S0022-4049(02)00139-1 | Zbl 1026.06018
[30] Mulvey C.J., Resende P., A noncommutative theory of Penrose tilings, Internat. J. Theoret. Phys., 2005, 44(6), 655–689 http://dx.doi.org/10.1007/s10773-005-3997-2 | Zbl 1087.52509
[31] Nguyen T.Z., Torus actions and integrable systems, In: Topological Methods in the Theory of Integrable Systems, Camb. Sci. Publ., Cambridge, 2006, 289–328 | Zbl 1329.37054
[32] Ono H., Semantics for substructural logics, In: Substructural Logics, Tübingen, October 7–8, 1990, Stud. Logic Comput., 2, Oxford University Press, New York, 1993, 259–291 | Zbl 0941.03522
[33] Ono H., Komori Y., Logics without the contraction rule, J. Symbolic Logic, 1985, 50(1), 169–201 http://dx.doi.org/10.2307/2273798 | Zbl 0583.03018
[34] Protin M.C., Resende P., Quantales of open groupoids, J. Noncommut. Geom., 2012, 6(2), 199–247 http://dx.doi.org/10.4171/JNCG/90 | Zbl 1253.06019
[35] Resende P., Étale groupoids and their quantales, Adv. Math., 2007, 208(1), 147–209 http://dx.doi.org/10.1016/j.aim.2006.02.004 | Zbl 1116.06014
[36] Rump W., Yang Y.C., Non-commutative logical algebras and algebraic quantales (manuscript) | Zbl 1322.03049
[37] Ward M., Dilworth R.P., Residuated lattices, Trans. Amer. Math. Soc., 1939, 45(3), 335–354 http://dx.doi.org/10.1090/S0002-9947-1939-1501995-3