The Localization of Commutative Bounded BCK-Algebras


In this paper we develop a theory of localization for bounded commutative BCK-algebras. We try to extend some results from the case of commutative Hilbert algebras (see [1]) to the case of commutative BCK-alge- bras.

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D. Piciu and D. Tascau, "The Localization of Commutative Bounded BCK-Algebras," Advances in Pure Mathematics, Vol. 1 No. 6, 2011, pp. 367-377. doi: 10.4236/apm.2011.16066.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] D. Piciu and C. Busneag, “The Localization of Commutative (Unbounded) Hilbert Algebras,” Mathematical Reports, Vol. 12, No. 3, 2010, pp. 285-300.
[2] Y. Imai and K. Iséki, “On Axiom Systems of Propositional Calculi XIV,” Proceedings of the Japan Academy, Vol. 42, No. 1, 1966, pp. 19-22. doi:10.3792/pja/1195522169
[3] A. N. Prior, “Formal logic, 2nd Ed,” Oxford, 1962.
[4] G. Georgescu, “F-Multipliers and the Localization of Distributive Lattices,” Algebra Universalis, Vol. 21, No. 2-3, 1985, pp. 181-197. doi:10.1007/BF01188055
[5] S. Rudeanu, “Localizations and Fractions in Algebra of Logic,” Journal of Multi-Valued Logic & Soft Computing, Vol. 16, No. 3-5, 2010, pp. 465-467.
[6] R. Cignoli and A. Torens, “Glivenko Like Theorems in Natural Expansions of BCK-Logic,” Mathematical Logic Quarterly, Vol. 50, No. 2, 2004, pp. 111-125. doi:10.1002/malq.200310082
[7] J. Gispert and A. Torrens, “Boolean Representation of Bounded BCK-Algebras,” Soft Computing, Vol. 12, No. 10, 2008, pp. 941-954. doi:10.1007/s00500-007-0261-0
[8] A. Iorgulescu, “Algebras of Logic as BCK-Algebras,” ASE, Bucharest, 2008.
[9] K. Iséki and S. Tanaka, “An Introduction to the Theory of BCK-Algebras,” Mathematica Japonica, Vol. 23, No. 1, 1978, pp. 1-25.
[10] F. M. G. Olmedo and A. J. R. Salas, “Negation and BCK- Algebras,” Mathematical Logic Quarterly, Vol. 49, No. 4, 2003, pp. 336-346. doi:10.1002/malq.200310035
[11] J. Schmid, “Multipliers on Distributive Lattices and Rings of Quotients,” Houston Journal of Mathematics, Vol. 6, No. 3, 1980, pp. 401-425.
[12] J. Schmid, “Distributive Lattices and Rings of Quotients,” Colloquia Mathematica Societatis János Bolyai, Vol. 33, Szeged, Hungary, 1980.
[13] J. Lambek, “Lectures on Rings and Modules,” Blaisdell Publishing Company, New York, 1966.
[14] H. Cornish, “The Multiplier Extension of a Distributive Lattice,” Journal of Algebra, Vol. 32, No. 2, 1974, pp. 339-355. doi:10.1016/0021-8693(74)90143-4
[15] R. Balbes and Ph. Dwinger, “Distributive Lattices,” University of Missouri Press, Columbia, 1974.
[16] N. Popescu, “Abelian Categories with Applications to Rings and Modules,” Academic Press, New York, 1973.
[17] B. Strenstr?m, “Platnes and Localization over Monoids,” Mathematische Nachrichten, Vol. 48, No. 1-6, 1971, pp. 315-334. doi:10.1002/mana.19710480124

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