This document introduces S-anti-fuzzy soft right R-subgroups of near-rings. It defines S-anti-fuzzy soft right R-subgroups and sensible S-anti-fuzzy soft right R-subgroups. It proves several properties of S-anti-fuzzy soft right R-subgroups, including properties related to homomorphic images and pre-images. The document concludes that S-anti-fuzzy soft right R-subgroups generalize previous concepts and could be further explored in other algebraic structures.
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contents
Introduction
Objectives of the work
Introduce S- anti-fuzzy soft right
R-subgroups of near-rings
Properties
Summary and conclusion
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Introduction
Zaid [1981] introduced the concept of R-subgroups of a near-rings and
Hokim [2001] introduced the concept of fuzzy R- subgroups of a near-ring. Then Zhan
[2010] introduced the properties of fuzzy hyper ideals in hyper near-rings with t-norm.
Recently, Cho et. al [2005] introduced the notion of fuzzy subalgebras with respect to S-
norm of BCK algebras and Akram [2006] introduced the notion of sensible fuzzy ideals
of BCK algebras with respect to t-norm.
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Objectives of the work
We introduce the notion of S- anti-fuzzy soft right R-subgroups
of near-rings and its basic properties are investigated.
We also study the homomorphic image and pre image of S- anti-
fuzzy soft right R- subgroups.
Using S-norm, we introduce the notion on sensible anti-fuzzy
soft right R-subgroups in near- rings and some related properties of
a near-rings ‘R’ are discussed.
7. Definition : Let ‘S’ be a s- norm. A function μ : R→ [0,1] is
called a S-anti fuzzy soft right (resp. left) R- subgroup of ‘R’
with respect to ‘S’ if
(C1) μ(x-y) ≤ S( μ(x), μ(y) )
(C2) μ(x r) ≤ μ(x) (resp. μ(r x) ≤ μ(x) for all r , x ∊ R.
If a S-anti fuzzy soft right R-subgroup ‘μ’ of R with respect
to ‘S’ is sensible , we say that ‘μ’ is a sensible S-anti fuzzy
soft right R- subgroup of R with respect to ‘S’.
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Definition : Let ‘f’ be a mapping defined on R. If ‘ψ’ is a
fuzzy soft subset in f(R) , then the fuzzy soft subset μ in R
(ie) μ(x) = ψ(f(x)) for all x in R is called the pre -image of
‘ψ’ under ‘f’.
Theorems :
An onto homomorphic pre image of a S- anti fuzzy soft
right R- subgroups of a near- ring is S-anti fuzzy soft right
R- subgroups of R.
Let f: R→ R1 be a homomorphism of near-rings. If ‘μ’ is
a S- anti fuzzy soft right R- subgroups of R1, then μf is S-
anti fuzzy soft right R- subgroup of R.
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If ‘μ’ is a S- anti fuzzy soft right R-subgroups of a near
ring R and ‘θ’ is an endomorphism of R, then μ[θ] is a S-
anti fuzzy soft right R- subgroups of R.
Let ‘S’ be a s-norm. Then every sensible S-anti fuzzy
soft right R- subgroups ‘μ’ of R is an anti- fuzzy soft right
R- subgroups of R.
Let f : R→ R1 be a homomorphism of near-rings. If ‘μf’
is a S- anti fuzzy soft right R- subgroups of R, then μ is
S- anti fuzzy soft right R- subgroup of R1.
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Let ‘S’ be a continuous S- norm and let ‘f’ be a
homomorphism on a near-ring R. If ‘μ’ is a S- anti fuzzy soft
right R- subgroup of R, then μf is a S- anti fuzzy soft right
R- subgroups of f(R).
If a fuzzy soft subset ‘μ’ of R is a S- anti fuzzy soft right
R- subgroup of R ,then ‘μc’ is ‘S’ anti fuzzy soft right R-
subgroup of R.
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Conclusion
Fuzzy set theory, rough set theory and soft set theory are all
mathematical tools for dealing with uncertainties.In this paper
we introduce the notion of S- anti-fuzzy soft right R-subgroups
of near-rings . We also study the homomorphic image and pre
image of S- anti-fuzzy soft right R- subgroups and proved some
properties. Further research could be done in other algebraic
structures such as fields ,modules.
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References
[1] S. Abou-Zoid , On fuzzy sub near rings and ideals , Fuzzy sets. Syst. 44
(1991) , 139-146.
[2] Antony, J.M and Sherwood, H., Fuzzy groups redefined, J. Math. Anal. Appl.
69, 124-130, 1979.
[3] M. T. Abu Osman, On some product of fuzzy subgroups, Fuzzy sets and
systems 24 ,(1987), pp 79-86.
[4] M. Akram and J. Zhan , On sensible fuzzy ideals of BCK algebras with
respect to a t- norm , int. J.math. Sci.(2006), pp 1-12
[5] Atagun AO,Sezgin A, Soft substructures of near ring modules (submitted).
[6]U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Computers and
Mathematics with Applications, 59 (2010) 3458-3463.
[7] H. Aktas and N. Cagman, Soft sets and soft group, Information Science 177
(2007), pp. 2726–2735.
13. ACKNOWLEDGEMENT
The authors like to thank the supported works from
various research centers such as Kamaraj College
(Tuticorin), M.S.University (Tirunelveli). Also the authors
are thankful to the management of Sri K.G.S. Arts
College(Srivaikuntam) for the encouragement given to us
to carry out the research.
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