Radical of $\cdot$-ideals in $PMV$-algebras

Document Type : Research Paper

Author

Faculty of Mathematics and computing‎, ‎Higher Education complex of Bam‎, Bam‎, ‎Iran.

Abstract

‎In this paper‎, ‎we introduce the notion of the radical of a $PMV$-algebra $A$ and we charactrize radical $A$ via elements of $A$‎. ‎Also‎, ‎we introduce the notion of the radical of a $\cdot$-ideal in $PMV$-algebras‎. ‎Several characterizations of this radical is given‎. ‎We define the notion of a semimaximal $\cdot$-ideal in a $PMV$-algebra‎. ‎Finally we show that $A/I$ has no nilpotent elements if and only if $I$ is a semi-maximal $\cdot$-ideal of $A$.

Keywords

Main Subjects


A. Bigard, K. Keimel and S. Wolfenstein, Groupes et Anneaux Reticules, Lecture Notes in Math., 608, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
C. C. Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc. 88 (1958) 467--490.
R. Cignoli, I. M. L. D'Ottaviano and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht, 2000.
A. Di Nola and A. Dvurecenskij, Product MV -algebras, Mult.-Valued Log. 6 (2001), no. 1-2, 193--215.
A. Di Nola, P. Flondor and I. Leustean, MV -modules, J. Algebra, 267 (2003), no. 1, 21--40.
A. Dvurecenskij, On partial addition in pseudo MV -algebras, Information technology (Bucharest, 1999), 952--960, Inforec, Bucharest, 1999.
A. Filipoiu, G. Georgescu and , A. Lettieri, Maximal MV -algebras, Mathware Soft Comput. 4 (1997), no. 1, 53--62.
F. Forouzesh, E. Eslami and A. Borumand Saeid, On prime A-ideals in MV -modules, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 76 (2014), no. 3, 181--198.
F. Forouzesh, E. Eslami and A. Borumand Saeid, Radical of A-ideals in MV -modules, An. Stiint. Univ. Al. I. Cuza Iasi Inform., Accepted.
F. Forouzesh, Some results in PMV -algebras, U.P.B. Sci. Bull., Series A, Submitted.
A. Iorgulescu, Algebras of logic as BCK algebras, Academy of economic studies Bucharest, Romania, 2008.
F. Montagna, An algebraic approach to propositional fuzzy logic, J. Logic Lang. Inform. 9 (2000), no. 1, 91--124.
D. Mundici, Interpretation of AFC-algebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), no. 1, 15--63.
D. Piciu, Algebras of fuzzy logic, Ed. Universitaria, Craiova, 2007.