### On Rough Sets and Hyperlattices

Ali Akbar Estaji, Fereshteh Bayati

#### Abstract

In this paper, we introduce the concepts of upper and lower rough hyper fuzzy ideals (filters) in a hyperlattice and their basic properties are discussed. Let $\theta$ be a hyper congruence relation on $L$. We show that if $\mu$ is a fuzzy subset of $L$, then $\overline{\theta}(<\mu>)=\overline{\theta}(<\overline{\theta}(\mu)>)$ and $\overline{\theta}(\mu^*) =\overline{\theta}((\overline{\theta}(\mu))^*)$, where $<\mu>$ is the least hyper fuzzy ideal of $L$ containing $\mu$ and $$\mu^*(x) = sup\{\alpha \in [0, 1]: x \in I( \mu_{\alpha} )\}$$ for all $x \in L$. Next, we prove that if $\mu$ is a hyper fuzzy ideal of $L$, then $\mu$ is an upper rough fuzzy ideal. Also, if $\theta$ is a $\wedge-$complete on $L$ and $\mu$ is a hyper fuzzy prime ideal of $L$ such that $\overline{\theta}(\mu)$ is a proper fuzzy subset of $L$, then $\mu$ is an upper rough fuzzy prime ideal. Furthermore, let $\theta$ be a $\vee$-complete congruence relation on $L$. If $\mu$ is a hyper fuzzy ideal, then $\mu$ is a lower rough fuzzy ideal and if $\mu$ is a hyper fuzzy prime ideal such that $\underline{\theta}(\mu)$ is a proper fuzzy subset of $L$, then $\mu$ is a lower rough fuzzy prime ideal.

#### Keywords

rough set, upper and lower approximations, hyperlattice, hyper fuzzy prime ideal, hyper fuzzy prime filter.

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#### References

S. M. Anvariyeh, S. Mirvakili and B. Davvaz, Pawlaks approximations in semihypergroups, Computers & Mathematics with Applications, Volume 60, Issue 1, July 2010, 45-53.

M. Asghari-Larimi, Hyperstructures with fuzzy subgroup and Fuzzy Ideal, International Mathematical Forum, 5, 2010, no. 10, 467-476.

K. Chakrabarty, R. Biswas and S. Nanda, Fuzziness in rough sets, Fuzzy Sets Syst. 110 (2) (2000) 247-251.

B. Davvaz, Approximations in hyperrings, Journal of Multiple-Valued Logic and Soft Computing 15 (2009) 471-488.

C. Degang, Z.Wenxiu, D. Yeung and E. C. C. Tsang, Rough approximations on a complete completely distributive lattice with applications to generalized rough sets, Information Sciences 176 (2006) 1829-1848.

A. A. Estaji, S. Khodaii and S. Bahrami, On rough set and fuzzy sublattice, Information Sciences, Volume 181, Issue 18, 15 September 2011, 3981-3994.

A. A. Estaji, M. R. Hooshmandasl and B. Davvaz, Rough set theory applied to lattice theory, Information Sciences, Volume 200, 1 October 2012, 108-122.

Y. Feng and L. Zou, On Product of Fuzzy Subhyperlattice and Interval-Valued Fuzzy Subhyperlattice, Discrete Mathematics, Algorithms and Applications Vol. 2, No. 2 (2010) 239-246.

X. Z. Guo and X.L. Xin, On hyperlattice, Pure and Applied Mathematics 20 (1) (2004) 40-43.

O. Kazancı, S. Yamak and B. Davvaz, The lower and upper approximations in a quotient hypermodule with respect to fuzzy sets, Information Sciences 178 (2008) 2349-2359.

B. B. N. Koguep, C. Nkuimi and C. Lele, On fuzzy prime ideal of lattice, Samsa Journal of Pure and Applied Mathematics, 3

(2008) 1-11.

B. B. N. Koguep, C. Nkuimi and C. Lele, On Fuzzy Ideals of Hyperlattice, International Journal of Algebra, Vol. 2, 2008, no. 15, 739-750.

M. Kondo, On the structure of generalized rough sets, Information Sciences 176 (2006) 589-600.

F. Marty, Sur une generalization de la notion de group, 8th Congress Math. Scandenaves, Stockholm, (1934), 45-49.

J. N. Mordeson, Rough set theory applied to (fuzzy) ideal theory, Fuzzy Sets Syst. 121 (2) (2001) 315-324.

A. D. Lokhande and Aryani Gangadhara, Congruences in hypersemilattices, International Mathematical Forum, Vol. 7, 2012, no. 55, 2735 - 2742.

Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (1982) 341-356.

A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets Syst. 126 (6) (2002) 137-156.

A. Rahnamai-Barghi, The prime ideal theorem for distributive hyperlattices, Ital. Journal of Pure and Applied Math., vol. 10, 2001 , 75-78.

M. Sambasiva Rao, Multipliers of Hypersemilattices, International Journal of Mathematics and Soft Computing Vol.3, No.1 (2013), 29 - 35.

S. Yamak, O. Kazancı, and B. Davvaz, Soft hyperstructure, Computers & Mathematics with Applications, Volume 62, Issue 2, July 2011, Pages 797-803.

L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965) 338-353.

DOI: http://dx.doi.org/10.23755/rm.v34i0.350

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