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

Document Type : Research Paper

Authors

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

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

Abstract

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.

Keywords

#### References

[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