Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 437 https://internationalpubls.com A Research on Bipolar Valued Vague Normal Subrings of a Ring 1 B.Deeba, 2 S. Naganathan & 3 K.Arjunan 1. Department of Mathematics, Idhaya College for Women (affiliated to Alagappa University, Karaikudi), Sarugani – 630411, Tamilnadu, India. Email:bdeepa85@gmail.com 2. Department of Mathematics, Sethupathy Government Arts College (affiliated to Alagappa University, Karaikudi), Ramanathapuram -623 502, Tamilnadu, India. Email: nathanaga @yahoo.com 3. Department of Mathematics, Alagappa Government Arts college (affiliated to Alagappa University, Karaikudi), Karaikudi – 630003, Tamilnadu, India. Email: arjunan.karmegam@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: This paper introduces and discusses certain properties of bipolar valued vague normal subring of a ring. Keywords: INTRODUCTION. [ ] Succeeding years, fuzzy set was grown in different ways. The following are extension of fuzzy set, they are vague set, intuitionistic fuzzy set, bipolar valued fuzzy set and etc. V [ ] Rosenfeld [2]; Bipolar valued fuzzy subset by W.R.Zhang[16]; Vague group by RanjitBiswas [12]; Bipolar vague set by Cicily Flora. S and Arockiarani.I [4]; Bipolar valued fuzzy subgroup by Anitha.M.S., et.al.[1]; In similar way, [3], [11], [13], [14], [5], [6], [8], [9] and [10] were useful to write this paper. 1.PRELIMINARIES. Definition 1.1 [15] [ ] Definition 1.2 [7] { [ ] } a [ ] map and [ ] is a false membership map, such that . Definition 1.3 [7] [ ] [ ] Example 1.4. = { < , [0.5, 0.6] >, < , [0.7, 0.8] >, < , [0.4, 0.9] >} is a vague set of { } Definition 1.5 [16] {( ) } a bipolar [ ] map and [ ] is a negative membership map. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 438 https://internationalpubls.com Definition 1.6 [4] { [ ] [ ] } [ ] [ ] [ ] [ ] {( ) }, where =[ ] and = [ ] It is denoted as Example 1.7. = { < , [0.5, 0.75], [0.55, 0.32] >, < , [0.7, 0.8], [0.45, 0.23] >, < , [0.4, 0.9], [ 0.005, 0.002] >} is a of { } Definition 1.8 [4] Let =  ,  and =  ,  be . (i) and (ii) = { rmin( , , rmax( , )  / }. Definition 1.9 [5]  ,  valued (i) { } (ii) { } (iii) { } (iv) { } where {[ ] [ ]} [ { } { }] and {[ ] [ ]} [ { } { }] Example 1.10. { [ ] [  ] [ ] [  ] [ ] [  ] } { } Definition 1.11  ,  valued (i) (ii) Definition 1.12. [5]  ,  the strongest that is a on {( , ), ( , ), ( , ) / for all ,  }, where ( , ) = rmin{ ( ), ( )} and ( , ) = rmax{ ( ), ( )}, for all ,  . Definition 1.13. [5]  ,  and   and , denoted by , is defined as = {( , ), ( × ) + ( , ), ( × )  ( , ) / for all ( , ) }, where ( × ) + ( , ) = rmin{ + ( ), + ( )} and ( × )  ( , ) = rmax{ ( ), ( )}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 439 https://internationalpubls.com Definition 1.14.   be a of a set   which is defined as and for all . Definition 1.15.   be a of a set   is defined as [ ] and [ ] for all . 2 – THEOREMS. Theorem 2.1. [5]  ,  is a where is an first operation identity element of Theorem 2.2.  ,  is a where is an first operation identity element of Proof. By the theorem 2.1, it can be easily shown. Theorem 2.3. [5]  ,  be a (  [ ] ( [ ] ( [ ]  ( [ ] ( [ ] ( [ ]  (  [ ] ( [ ] ( [ ]  ( [ ] ( [ ] ( [ ]  Theorem 2.4.  ,  be a (  [ ] ( [ ] ( [ ]  ( [ ] ( [ ] ( [ ]  (  [ ] ( [ ] ( [ ]  ( [ ] ( [ ] ( [ ]  Proof. By the theorem 2.3, it can be easily shown. Theorem 2.5. [5]  ,  is a { [ ] [ ]} is either empty or Theorem 2.6.  ,  is a { [ ] [ ]} is either empty or Proof. By the theorem 2.5, it can be easily shown. Theorem 2.7. [5]     Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 440 https://internationalpubls.com Theorem 2.8.     Proof. Let be in . Let Then ( ) = rmin{ ( ), ( )} = rmin{ ( ), ( )} = ( ),  in . And ( ) = rmax{ ( ), ( )} = rmax{ ( ), ( )} = ( ),  in . Hence Theorem 2.9. [5] , , … and … Theorem 2.10. , , … and … Proof. By the theorem 2.9, it can be easily shown. Theorem 2.11. [5] , , … intersection … Theorem 2.12. , , … … Proof. By the theorem 2.11, it can be easily shown. Theorem 2.13. [5] and be the stronget relation of . Then is a of if and only if is a of × Theorem 2.14. and be the stronget relation of . Then is a of if and only if is a of × Proof. Let be in . Then ( , ) and ( , ) are in × . By Theorem 2.13, is a of × then [( , )( , )] = ( , ) = rmin{ + ( ), + ( )} = rmin { + ( ), + ( )}= ( , ) = [( , )( , )],  ( , ), ( , ) × . And [( , )( , )] = ( , ) = rmax{ ( ), ( )} = rmax{ ( ), ( )} = ( , ) = [( , )( , )],  ( , ), ( , ) × . Hence is a of × . Conversely, assume that is a of × . By Theorem 2.13, is a of rmin{ + ( ), + ( )} = ( , ) = [( , )( , )] = [( , )( , )] = ( , ) = rmin{ + ( ), + ( )}, put and , where is an first operation identity element of , then + ( ) = + ( ),  ,  . And rmax{ ( ), ( )} = ( , ) = [( , )( , )] = [( , )( , )] = ( , ) = rmax{ ( ), ( )}, put and , where is an first operation identity element of , then ( ) = ( ),  ,  . Hence is a of . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 441 https://internationalpubls.com Theorem 2.15. [5] , , … , be and be the strongest n- dimensional relation of . Then , , …, are × …× (m times). Theorem 2.16. , , … , be and be the strongest n-dimensional relation of . Then , , …, are × …× (m times). Proof. By the theorem 2.15, it can be easily shown. Theorem 2.17. [5] and is a Theorem 2.18. and is a Proof. Let be in 1 and be in 2. Then ( , ), ( , ) 1× 2. By Theorem 2.17, is a then ( × ) + [( , )( , )] = ( × ) + ( , ) = rmin{ + ( ), + ( )}= rmin{ + ( ), + ( )} = ( × ) + ( , )= ( × ) + [( , )( , )],  ( , ), ( , ) 1× 2. And ( × )  [( , )( , )] = ( × )  ( , ) = rmax{ ( ), ( )} = rmax{ ( ), ( )}= ( × )  ( , ) = ( × )  [( , )( , )],  ( , ), ( , ) 1× 2. Hence × is a of 1× 2. Theorem 2.19. [5] , , … , is a Theorem 2.20. , , … , is a Proof. By the theorem 2.19, it can be easily shown. Theorem 2.21. [5] If is a , then ( ) is a of . Theorem 2.22.If is a , then ( ) is a of . Proof. Let be in 1. By Theorem 2.21, ( ) is a of , ( ) = )+[1] ) = )+[1] ) = ( ),  ,  1. And ( ) = )+[1] ) = )+[1] ) = ( ),  ,  1. Hence ( ) is a of . Theorem 2.23. [5] Let is a . (i) Then ( [ ] ( [ ] . (ii) [ ] [ ] ( . (iii) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 442 https://internationalpubls.com ( [ ] ( [ ] . (iv) [ ] [ ] ( [ ] ( [ ]. (v) ( . (vi) (vii) (viii) Theorem 2.24. Let is a . (i) Then ( [ ] ( [ ] . (ii) [ ] [ ] ( . (iii) ( [ ] ( [ ] . (iv) [ ] [ ] ( [ ] ( [ ]. (v) ( . (vi) (vii) (viii) Proof. By the theorem 2.23, it can be easily shown. CONCLUSION Using the above theorems, we can find more results. It can be extended into different types of algebra. REFERENCES [1] Anitha.M.S., Muruganantha Prasad & K.Arjunan, Notes on bipolar valued fuzzy subgroups of a group, Bulletin of Society for Mathematical Services and Standards, Vol. 2 No. 3 (2013), pp.52-59. [2] Azriel Rosenfeld, fuzzy groups, Journal of mathematical analysis and applications 35(1971), 512-517. 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