Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 414 https://internationalpubls.com A Research in Bipolar Valued Vague Subfields of a Field 1 K.Bala Bavithra, 2 M.Muthusamy & 3 K.Arjunan 1 . Department of Mathematics, Sonaimeenal Arts and Science College(Affiliated to Alagappa University, Karaikudi), Mudukulathur – 623704, Tamilnadu, India. Email:kpavi94pk@gmail.com 2 . Department of Mathematics, Dr.Zakir Husain College (Affiliated to Alagappa University, Karaikudi), Ilayangudi- 630702, Tamilnadu, India. Email: msamy0207@yahoo.com 3 .Department of Mathematics, Alagappa Govt. Arts College, Karaikudi – 630003, Tamilnadu, India. Email: arjunan.karmegam@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: Certain properties of bipolar valued vague subfield of a field are introduced and discussed. 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 [3]; Bipolar valued fuzzy subset by W.R.Zhang[15]; Vague group by RanjitBiswas [11]; Bipolar vague set by Cicily Flora. S and Arockiarani.I [5]; Bipolar valued fuzzy subgroup by Anitha.M.S., et.al.[2]; In similar way, [1], [4], [7], [8], [9], [10], [12] and [13] were useful to write this paper. 1.PRELIMINARIES. Definition 1.1 [14] , - Definition 1.2 [6] *( , ( ) ( )-) + a , - map and , - is a false membership map, such that ( ) ( ) . Definition 1.3 [6] , ( ) ( )- ( ) ( ) , ( ) ( )- Example 1.4. = { < , [0.04, 0.07] >, < , [0.02, 0.06] >, < , [0.03, 0.08] >} is a vague set of * + Definition 1.5 [15] {( ( ) ( )) } a bipolar , - map and , - is a negative membership map. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 415 https://internationalpubls.com Definition 1.6 [5] *( , ( ) ( )- , ( ) ( )-) + , - , - , - , - ( ) ( ) ( ) ( ) {( ( ) ( )) }, where ( ) =, ( ) ( )- and ( ) = , ( ) ( )- It is denoted as Example 1.7. = { < , [0.05, 0.07], [0.05, 0.02] >, < , [0.04, 0.08], [0.06, 0.03] >, < , [0.14, 0.19], [ 0.25, 0.22] >} is a of * + Definition 1.8 [5] Let =  ,  and =  ,  be . (i) ( ) ( ) and ( ) ( ) (ii) = { rmin( ( ), ( )), rmax( ( ), ( ))  / }. Definition 1.9 [5]  ,  valued ( ) (i) ( ) * ( ) ( )+ (ii) ( ) * ( ) ( )+ (iii) ( ) * ( ) ( )+ (iv) ( ) * ( ) ( )+ (v) ( ) ( ) (vi) ( ) ( ) where is an first operation identity element of *, - , -+ , * + * +- and *, - , -+ , * + * +- Example 1.10. * , - ,  - , - ,  - , - ,  - + * + Definition 1.11. [5]  ,  the strongest that is a on {( , ), ( , ), ( , ) / for all ,  }, where ( , ) = rmin{ ( ), ( )} and ( , ) = rmax{ ( ), ( )}, for all ,  . Definition 1.12. [5]  ,  and   and , denoted by , is defined as = {( , ), ( × ) + ( , ), ( × )  ( , ) / for all ( , ) }, where ( × ) + ( , ) = rmin{ + ( ), + ( )} and ( × )  ( , ) = rmax{ ( ),  ( )}. 2 – THEOREMS. Theorem 2.1.  ,  is a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 416 https://internationalpubls.com ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (iii) ( ) ( ) ( ) ( ) (iv) ( ) ( ) ( ) ( ) where are first, second operation identity elements of Proof. (i) Let Then ( )= ( ( )) ( ) ( ). That is ( ) ( ) And ( )= ( ( )) ( ) ( ). Thus ( ) ( ) (ii) Let Then ( ) = (( ) )  ( )  ( ). That is ( ) ( ) And ( ) = (( ) )  ( )  ( ). That is ( ) ( ) (iii) Also ( ) = ( )  rmin{ ( ) ( )+ = ( ), And ( ) = ( )  rmax{ ( ) ( )+ = ( ), (iv) Also ( ) = ( )  rmin{ ( ) ( )+ = ( ), And ( ) = ( )  rmax{ ( ) ( )+ = ( ), Theorem 2.2.  ,  and   Proof. Let be in . Let Then (  ) = rmin{ (  ), (  )}  rmin{rmin{ ( ), ( )}, rmin{ ( ), ( )}} = rmin{rmin{ ( ), ( )}, rmin{ ( ), ( )}} = rmin{ ( ), ( )},   . And ( ) = rmin{ ( ), ( )}  rmin{rmin{ ( ), ( )}, rmin{ ( ), ( )}} = rmin{rmin{ ( ), ( )}, rmin{ ( ), ( )}} = rmin{ ( ), ( )},   . Also (  ) = rmax{ (  ), (  )}  rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}} = rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}} = rmax{ ( ), ( )},   . And ( ) = rmax{ ( ), ( )}  rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}} = rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}} = rmax{ ( ), ( )},   . Hence Theorem 2.3.  , ,  , , … and  ,  … is also a of . Proof. By Theorem 2.2, it can be easily shown. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 417 https://internationalpubls.com Theorem 2.4.  , ,  , , … … is also a of . Proof. By Theorem 2.3, it can be easily shown. Theorem 2.5.  ,  and   Proof. It can be easily shown. Theorem 2.6.  ,  and   Proof. It can be easily shown. Theorem 2.7.  ,  and   is a Proof. Let  1 and  2. Then ( , ), ( , )  1× 2. Then ( × ) + [( , )( , )] = ( × ) + (  ,  ) = rmin{ + (  ), + (  )} rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}} = rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}} = rmin{( × ) + ( , ), ( × ) + ( , )},  ( , ), ( , ) 1× 2. And ( × ) + [( , )( ) ] = ( × ) + ( , ) = rmin{ + ( ), + ( )}  rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}} = rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}} = rmin {( × ) + ( , ), ( × ) + ( , )},  ( , ), ( , ) 1× 2. Also ( × )  [( , )( , )] = ( × )  (  ,  ) = rmax{  (  ),  (  )}  rmax{rmax{  ( ),  ( )}, rmax{  ( ),  ( )}} = rmax{rmax{  ( ),  ( )}, rmax{  ( ),  ( )}} = rmax{( × )  ( , ), ( × )  ( , )},  ( , ), ( , ) 1× 2. And ( × )  [( , ) ( ) ] = ( × )  ( , ) = rmax{  ( ),  ( )}  rmax {rmax{  ( ),  ( )}, rmax{  ( ),  ( )}} = rmax{rmax{  ( ),  ( )}, rmax{  ( ),  ( )}} = rmax{( × )  ( , ), ( × )  ( , )},  ( , ), ( , ) 1× 2. Hence × is a of 1× 2. Theorem 2.8.  , ,  , , … and  ,  … is also a of … . Proof. By Theorem 2.7, it can be easily shown. Theorem 2.9.  , ,  ,  be two is a of the field then atleast the following one holds, where are , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Proof. Let is a of the field ( ) ( ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 418 https://internationalpubls.com ( , ) 1× 2. Then ( ) + ( , ) = rmin{ + ( ), + ( )} > rmin{ + ( ), + ( )} = ( ) + ( , ), which is  to is a of the field . Hence atleast one of the two (i) and (ii) are true. Theorem 2.10.  , ,  ,  be two is a of the field ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) where are ; ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Proof. Let be in . Then ( , ) and ( , ) are in × . Then (i) + (  ) = rmin{ + (  ), + (  )} = ( × ) + (  ,  )= ( × ) + [( , )( , )]  rmin{( × ) + ( , ), ( × ) + ( , )}=rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}}= rmin{ + ( ), + ( )},  in . And + ( ) = rmin{ + ( ), + ( )} = ( × ) + ( , ) = ( × ) + [( , )( ) ]  rmin{( × ) + ( , ), ( × ) + ( , )} = rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}} = rmin{ + ( ), + ( )},  in . Also  (  ) = rmax{  (  ),  (  )} = ( × )  (  ,  ) = ( × )  [( , )( , )]  rmax{( × )  ( , ), ( × )  ( , )} = rmax {rmax{  ( ),  ( )}, rmax{  ( ),  ( )}}= rmax{  ( ),  ( )},  in . And  ( ) =rmax{  ( ),  ( )} = ( × )  ( , ) = ( × )  [( , )( ) )]  rmax{( × )  ( , ), ( × )  ( , )}= rmax{rmax{  ( ),  ( )}, rmax{  ( ),  ( )}}= rmax{  ( ),  ( )},   . Hence Let be in . Then ( ) and ( ) are in × . Then (ii) + (  ) = rmin{ + (  ), + (  )} = ( × ) + (   ) = ( × ) + [( )( )]  rmin{( × ) + ( ), ( × ) + ( )}= rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}} = rmin{ + ( ), + ( )},   . And + ( ) = rmin{ + ( ), + ( )} = ( × ) + ( ) = ( × ) + [( )( ) ]  rmin{( × ) + ( ), ( × ) + ( )} = rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}}= rmin{ + ( ), + ( )},  . Also  (  ) = rmax{  (  ),  (  )}= ( × )  (   ) = ( × )  [( )( )]  rmax{( × )  ( ), ( × )  ( )}= rmax{rmax{  ( ),  ( )}, rmax{  ( ),  ( )}}= rmax{  ( ),  ( )},   . And  ( ) = rmax{  ( ),  ( )} = ( × )  ( ) = ( × )  [( )( ) ]  rmax{( × )  ( ), ( × )  ( )} =rmax{rmax{  ( ),  ( )}, rmax{  ( ),  ( )}}= rmax{  ( ),  ( )},   . Hence Theorem 2.11.  , ,  , , ….,  ,  be is a of the field for each ( ) ( ) ( ) ( ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 419 https://internationalpubls.com ( ) ( ) ( ) ( ) where are . Proof. By Theorem 2.10, it can be easily shown. Theorem 2.12. and be the stronget relation of . Then is a of if and only if is a of × Proof. Let be in and be in . Then ( , ) and ( , ) are in × . If is a of then [( , )( , )] = (  ,  ) = rmin{ + (  ), + (  )}  rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}}= rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}}= rmin{ ( , ), ( , )},  ( , ), ( , ) × . And [( , )( ) )] = ( , ) = rmin{ + ( ), + ( )}  rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}} = rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}} = rmin{ ( , ), ( , )},  ( , ), ( , ) × . Also [( , )( , )] = (  ,  ) = rmax{ (  ), (  )}  rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}} = rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}}= rmax{ ( , ), ( , )},  ( , ), ( , ) × . And [( , )( ) )] = ( , ) = rmax{ ( ), ( )}  rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}} = rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}} = rmax{ ( , ), ( , )}, for all ( , ), ( , ) in × . Hence is a of × Conversely, assume is a of × rmin{ + (  ), + (  )} = (  ,  ) = [( , )( , )]  rmin{ ( , ), ( , )} = rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}}, put and , where is an first operation identity element of , then + (  )  rmin{ + ( ), + ( )},  ,  . And rmin{ + ( ), + ( )} = ( , ) = [( , ) ( ) ] rmin{ ( , ), ( , )} = rmin{rmin{ + ( ), + ( )}, rmin{ + ( ), + ( )}}, put and , where is an first operation identity element of , then + ( )  rmin{ + ( ), + ( )},  ,  . Also rmax{ (  ), (  )} = (  ,  ) = [( , )( , )]  rmax{ ( , ), ( , )} = rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}}, put and , where is an first operation identity element of , then (  )  max{ ( ), ( )},  ,  . And rmax{ ( ), ( )} = ( , ) = [( , ) ( ) ]  rmax{ ( , ), ( , )} = rmax{rmax{ ( ), ( )}, rmax{ ( ), ( )}}, put and , where is an first operation identity element of , then ( )  rmax{ ( ),  ( )},  ,  . CONCLUSION Using the above theorems, we can find more results. 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