On some classes of expansions of ideals in $MV$-algebras

Document Type : Research Paper


1 Faculty of Mathematics and computing, Higher Education Complex of Bam, Kerman, Iran.

2 Department of Mathematics , Shahid Bahonar University Kerman, Iran.


In this paper, we introduce the notions of expansion of ideals in $MV$-algebras, $ (\tau,\sigma)- $primary, $ (\tau,\sigma)$-obstinate  and $ (\tau,\sigma)$-Boolean  in $ MV- $algebras. We investigate the relations of them. For example, we show that every $ (\tau,\sigma)$-obstinate ideal of an $ MV-$ algebra is $ (\tau,\sigma)$-primary  and $ (\tau,\sigma)$-Boolean. In particular, we define an expansion $ \sigma_{y} $ of ideals in an $ MV-$algebra. A characterization of expansion ideal with respect to $ \sigma_{y} $ is given. Finally, we show that the class $ C(\sigma_{y}) $ of all constant ideals relative to $ \sigma_{y} $ is a Heyting algebra.


[1] A. Filipoiu, G. Georgescu, A. Lettieri, Maximal MV -algebras, Mathware, soft comput., 4 (1997), pp. 53–62.
[2] C. C. Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc., 88 (1958), pp. 467–490.
[3] C. C. Chang, A new proof of the completeness of the Lukasiewicz axioms,Trans. Amer. Math. Soc., 93
(1959), pp. 74–80.
[4] R. Cignoli, I. M. L. D’Ottaviano, D. Mundici, Algebraic foundations of many valued reasoning, Kluwer
Academic, Dordrecht, (2000).
[5] F. Forouzesh, E. Eslami, A. Borumand saeid, On obstinate ideals in MV −algebras, Politehn. Univ.
Bucharest Sci. Bull. Ser. A Appl. Math. Phys. Vol. 76, (2014), pp. 53–62.
[6] C. S. Hoo, S. Sessa, Implicative and Boolean ideals of MV-algebras, Math. Japon. 39 (1994), pp. 215-219.
[7] S. Motamed, J. Moghaderi, Expansions of filters in Residuated lattices, International Journal of Contem-
porary Mathematical siences, Vol. 11 (2016), pp. 9-15.
[8] D. Piciu, Algebras of fuzzy logic, Ed. Universitaria Craiova (2007).