Article
Full entry |
PDF
(0.4 MB)
Feedback
Keywords:
Lattice effect algebra; CI-lattice; Sasaki arrow; (strong; fantastic; implicative; positive implicative) filter; Riesz ideal; D-ideal; MV-effect algebra; orthomodular lattice
Lattice effect algebra; CI-lattice; Sasaki arrow; (strong; fantastic; implicative; positive implicative) filter; Riesz ideal; D-ideal; MV-effect algebra; orthomodular lattice
Summary:
In this paper, we define some types of filters in lattice effect algebras, investigate some relations between them and introduce some new examples of lattice effect algebras. Then by using the strong filter, we find a CI-lattice congruence on lattice effect algebras, such that the induced quotient structure of it is a lattice effect algebra, too. Finally, under some suitable conditions, we get a quotient MV-effect algebra and a quotient orthomodular lattice, by this congruence relation.
References:
[1] Avallone, A., Vitolo, P.: Congruences and ideals of effect algebras. Kluwer Academic Publishers 20 (2003), 1, 67-77. DOI 10.1023/a:1024458125510 | MR 1993411
[2] Bennett, M. K., Foulis, D. J.: Phi-symmetric effect algebras. Found. Physics 25 (1995), 12, 1699-1722. DOI 10.1007/bf02057883 | MR 1377109
[3] Borzooei, R. A., Dvurečenskij, A., Sharafi, A. H.: Material implications in lattice effect algebras. Inform. Sci. 433-434 (2018), 233-240. DOI 10.1016/j.ins.2017.12.049 | MR 3759022
[4] Borzooei, R. A., Shoar, S. Khosravi, Ameri, R.: Some types of filters in MTL-algebras. Fuzzy Sets Systems 187 (2012), 1, 92-102. DOI 10.1016/j.fss.2011.09.001 | MR 2851998
[5] Chajda, I., Halaš, R., Kühr, J.: Many-valued quantum algebras. Algebra Univers. 60 (2009), 1, 63-90. DOI 10.1007/s00012-008-2086-9 | MR 2480632
[6] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Springer Netherlands, 2000. DOI 10.1007/978-94-017-2422-7 | MR 1861369 | Zbl 0987.81005
[7] Farahani, H., Zahiri, O.: Algebraic view of MTL-filters. Ann. Univ. Craiova 40 (2013), 1, 34-44. MR 3078957
[8] Foulis, D. J.: MV and Hyting effect algebras. Found. Physics 30 (2000), 10, 1687-1706. DOI 10.1023/a:1026454318245 | MR 1810197
[9] Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logics. Found. Physics 24 (1994), 10, 1331-1352. DOI 10.1007/bf02283036 | MR 1304942 | Zbl 1213.06004
[10] Foulis, D. J., Pulmannová, S.: Logical connectives on lattice effect algebras. Studia Logica 100 (2012), 6, 1291-1315. DOI 10.1007/s11225-012-9454-3 | MR 3001058
[11] Haveshki, M., Saeid, A. Borumand, Eslami, E.: Some types of filters in BL-algebras. Soft Computing 10 (2006), 8, 657-664. DOI 10.1007/s00500-005-0534-4
[12] Jenča, G., Marinová, I., Riečanová, Z.: Central elements, blocks and sharp elements of lattice effect algebras. In: Proc. Third Seminar Fuzzy Sets and Quantum Structures 2002, pp. 28-33.
[13] Jenča, G., Pulmannová, S.: Ideals and quotients in lattice ordered effect algebras. Soft Computing 5 (2001), 5, 376-380. DOI 10.1007/s005000100139
[14] Cignoli, R., D'Ottaviano, I. M. L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Springer Science and Business Media, 2000. DOI 10.1007/978-94-015-9480-6 | MR 1786097 | Zbl 0937.06009
[15] Pulmannová, S., Vinceková, E.: Congruences and ideals in lattice effect algebras as basic algebras. Kybernetika 45 (2009), 6, 1030-1039. MR 2650081
[16] Rad, S. Rafiee, Sharafi, A. H., Smets, S.: A Complete axiomatisation for the logic of lattice effect algebras. Int. J. Theoret. Physics (2019). DOI 10.1007/s10773-019-04074-y
[17] Riečanová, Z.: Generalization of blocks for D-lattices and lattice-ordered effect algebras. Int. J. Theoret. Physics 39 (2000), 2, 231-237. DOI 10.1023/a:1003619806024 | MR 1762594
Search
Partner of