The aim of this paper is to define the notions of pseudo-MV algebra of fractions and maximal pseudo-MV algebra of quotients for a pseudo-MV algebra (taking as a guide-line the elegant construction of complete ring of quotients by partial morphisms introduced by G. Findlay and J. Lambek-see [14], p.36). For some informal explanations of the notion of fraction see [14], p. 37. In the last part of this paper the existence of the maximal pseudo-MV algebra of quotients for a pseudo-MV algebra (Theorem 4.5) is proven and I give explicit descriptions of this MV-algebra for some classes of pseudo MV-algebras.
@article{bwmeta1.element.doi-10_2478_BF02476540,
author = {Dana Piciu},
title = {Pseudo-MV algebra of fractions and maximal pseudo-MV algebra of quotients},
journal = {Open Mathematics},
volume = {2},
year = {2004},
pages = {199-217},
zbl = {1059.06008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02476540}
}
Dana Piciu. Pseudo-MV algebra of fractions and maximal pseudo-MV algebra of quotients. Open Mathematics, Tome 2 (2004) pp. 199-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02476540/
[1] D. Buşneag: “Hilbert algebra of fractions and maximal Hilbert algebras of quotients”, Kobe Journal of Mathematics, Vol. 5, (1988), pp. 161–172. | Zbl 0676.06018
[2] D. Buşneag and D. Piciu: Meet-irreducible ideals in an MV-algebra, Analele Universitâţii din Craiova, Seria Matematica-Informatica, Vol. XXVIII, 2001, pp. 110–119. | Zbl 1058.06014
[3] D. Buşneag and D. Piciu: “On the lattice of ideals of an MV-algebra”, Scientiae Mathematicae Japonicae, Vol. 56, No. 2, (2002), pp. 367–372, e6, pp. 221–226. | Zbl 1019.06004
[4] D. Buşneag and D. Piciu: MV-algebra of fractions relative to an λ-closed system, Analele Universitâţii din Craiova, Seria Matematica-Informatica, Vol. XXX, 2003, pp. 1–6. | Zbl 1073.06501
[5] D. Buşneag and D. Piciu: “MV-algebra of fractions and maximal MV-algebra of quotients”, to appear in Journal of Multiple-Valued Logic and Soft Computing. | Zbl 1069.06006
[6] C.C. Chang: “Algebraic analysis of many valued logics”, Trans. Amer. Math. Soc., Vol. 88, (1958), pp. 467–490. http://dx.doi.org/10.2307/1993227 | Zbl 0084.00704
[7] R. Cignoli, I.M.L. D'Ottaviano and D. Mundici: Algebraic foundation of many-valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000.
[8] W.H. Cornish: “The multiplier extension of a distributive lattice”, Journal of Algebra, Vol. 32, (1974), pp. 339–355. http://dx.doi.org/10.1016/0021-8693(74)90143-4
[9] W.H. Cornish: “A multiplier approach to implicative BCK-algebras”, Mathematics Seminar Notes, Kobe University, Vol. 8, No. 1, (1980), pp. 157–169. | Zbl 0465.03029
[10] C. Dan: F-multipliers and the localisation of Heyting algebras, Analele Universitâţii din Craiova, Seria Matematica-Informatica, Vol. XXIV, 1997, pp. 98–109.
[11] A. Dvurečenskij: “Pseudo-MV-algebras are intervals in l-groups”, Journal of Australian Mathematical Society, Vol. 72, (2002), pp. 427–445. http://dx.doi.org/10.1017/S1446788700036806 | Zbl 1027.06014
[12] G. Georgescu and A. Iorgulescu: “Pseudo-MV algebras”, Multi. Val. Logic, Vol. 6, (2001), pp. 95–135. | Zbl 1014.06008
[13] P. Hájek: Metamathematics of fuzzy logic Kluwer Acad. Publ., Dordrecht, 1998.
[14] J. Lambek: Lectures on Rings and Modules, Blaisdell Publishing Company, 1966.
[15] I. Leuştean: “Local Pseudo-MV algebras”, Soft Computing, Vol. 5, (2001), pp. 386–395. http://dx.doi.org/10.1007/s005000100141 | Zbl 0992.06011
[16] D. Piciu: Pseudo-MV algebra of fractions relative to an λ-closed system, Analele Universitâţii din Craiova, Seria Matematica-Informatica, Vol. XXX, 2003, pp. 7–13.
[17] J. Schmid: “Multipliers on distributive lattices and rings of quotients”, Houston Journal of Mathematics, Vol. 6, No. 3, (1980), pp. 401–425. | Zbl 0501.06008
[18] J. Schmid: Distributive lattices and rings of quotients, Coll. Math. Societatis Janos Bolyai, Szeged, Hungary, 1980. | Zbl 0501.06008