# ON ( $\alpha, \beta$ )-FUZZY Hv-IDEALS OF H_{v}-RINGS

Document Type : Research Paper

Authors

1 Department of Mathematics, Yazd University, Yazd, Iran

2 Dipartimento di Matematica e Informatica, Via delle Scienze 206, 33100 Udine, Italy

Abstract

Using the notion of “belongingness ($\epsilon$)” and “quasi-coincidence
(q)” of fuzzy points with fuzzy sets, we introduce the concept of an ($\alpha, \beta$)-
fuzzyHv-ideal of an Hv-ring, where , are any two of {$\epsilon$, q,$\epsilon$ $\vee$ q, $\epsilon$ $\wedge$ q} with
$\alpha$ $\neq$ $\epsilon$ $\wedge$ q. Since the concept of ($\epsilon$, $\epsilon$ $\vee$ q)-fuzzy Hv-ideals is an important and
useful generalization of ordinary fuzzy Hv-ideals, we discuss some fundamental
aspects of ($\epsilon$, $\epsilon$ $\vee$ q)-fuzzy Hv-ideals. A fuzzy subset A of an Hv-ring R is an
($\epsilon$, $\epsilon$ $\vee$ q)-fuzzy Hv-ideal if and only if an At, level cut of A, is an Hv-ideal
of R, for all t$\epsilon$(0, 0.5]. This shows that an($\epsilon$, $\epsilon$ $\vee$ q)-fuzzy Hv-ideal is a
generalization of the existing concept of fuzzy Hv-ideal. Finally, we extend
the concept of a fuzzy subgroup with thresholds to the concept of a fuzzy
H_{v}-ideal with thresholds.

Keywords

#### References

[1] S. K. Bhakat, ($epsilon$ $vee$ q)-fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and
Systems,112 (2000), 299-312.
[2] S. K. Bhakat and P. Das,Fuzzy subrings and ideals, Fuzzy Sets and Systems, 81 (1996),383-393.
[3] S. K. Bhakat and P. Das, ($epsilon$ $vee$ q)-fuzzy subgroup, Fuzzy Sets and Systems, 80 (1996),
359-368.
[4] P. Corsini,Prolegomena of hypergroup theory, Second Edition, Aviani Editor, 1993.
[5] P. Corsini and V. Leoreanu,Applications of hyperstructures theory, Advanced in Mathematics,
[6] B. Davvaz, ($epsilon$ $vee$ q)-fuzzy subnear-rings and ideals, Soft Computing 10 (2006), 206-211.
[7] B. Davvaz,A brief survey of the theory of H_{v}-structures, in: Proc. 8th International Congress
on Algebraic Hyperstructures and Applications, 1-9 Sep., 2002, Samothraki, Greece, Spanidis
Press, 2003, 39-70.
[8] B. Davvaz,Fuzzy H_{v}-groups, Fuzzy Sets and Systems, 101 (1999), 191-195.
[9] B. Davvaz,Fuzzy H_{v}-submodules, Fuzzy Sets and Systems, 117 (2001), 477-484.
[10] B. Davvaz,T-fuzzy H_{v}-subrings of an H_{v}-ring, J. Fuzzy Math., 11 (2003), 215-224.
[11] B. Davvaz,On Hv-rings and fuzzy H_{v}-ideals, J. Fuzzy Math., 6 (1998), 33-42.
[12] B. Davvaz,Product of fuzzy H_{v}-ideals in Hv-rings, Korean J. Compu. Appl. Math., 8 (2001),
685-693.
[13] Y. B. Jun,On ( $alpha, beta$ )-fuzzy subalgebra of BCK/BCI-algebras, Bull. Korean Math. Soc., 42
(2005), 703-711.
[14] W. J. Liu,Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982),
133-139.
[15] F. Marty,Sur une generalization de la notion de group, 8th Congress Math. Scandenaves,
Stockholm, 1934, 45-49.
[16] P. M. Pu and Y. M. Liu,Fuzzy topology I, Neighborhood structure of a fuzzy point and
Moore-Smith convergence, J. Math. Anal. Appl., 76 (1980), 571-599.
[17] A. Rosenfeld,Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
[18] T. Vougiouklis,Hyperstructures and their representations, Hadronic Press, Inc, 115, Palm
Harber, USA, 1994.
[19] T. Vougiouklis,The fundamental relation in hyperrings. The general hyperfield, in: Proc. 4th
International Congress on Algebraic Hyperstructures and Applications, Xanthi, 1990, World
Sci. Publishing, Teaneck, NJ, (1991), 203-211.
[20] X. Yuan, C. Zhang and Y. Ren,Generalized fuzzy groups and many-valued implications,
Fuzzy Sets and Systems,138 (2003), 205-211.
[21] L. A. Zadeh,Fuzzy sets, Inform. Control, 8 (1965), 338-353.