Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 214 https://internationalpubls.com Generalization of Bi- Г-Ideals in Ternary Г-Semirings K. Maha lakshmi,1,2 ,P. Siva Prasad3,D.Madhusudana Rao4, 1 Research Scholar, Department of Mathematics, VFSTR Deemed to be University, Vadlamudi, Guntur ,Andhra Pradesh ,India, mailid: mhlakahmi@gmail.com. 2Assistant Professor, Department of Mathematics, Vignan Nirulla Engineering College, Guntur, Andhra Pradesh, India ,mailid: mhlakahmi@gmail.com. 3Associate Professor, Department of Computer Science & Engineering, School of Computing & Informatics, VFSTR Deemed to University, Vadlamudi,Guntur,A.P,India, mailid:pusapatisivaprasad@gmail.com 4Professor of Mathematics, Government College For Women(A), Samba Siva Peta Rd, Opp: AC College, Samba Siva Pet, Guntur, Andhra Pradesh, India,mailid:dmrmaths@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: Murali Krishna Rao was presented the idea of Г-semirings as a speculation of the thought of Г-rings as well as of semirings. We have realized that the thought of Г-semirings is a generality of the idea of semirings. During this paper we sum up the thought of summed up bi-Г-ideals of Ternary Г-semirings and examine a few related properties of summed up bi-Г- ideals. Keywords: Ternary Г-Ideal, generalized bi- ternary Г-ideal, bi- ternary Г-ideal Mathematics Subject Classification. 16Y30, 16Y99 1. Introduction and Preliminaries The possibility of Г-semirings was laid out and concentrated in 1995 by Murali Krishna Rao [10] as a speculation of the thought of Г-rings as well as of semiring, and summed up bi-ideals was first presented for rings in 1970 by Szasz[12, 13] and afterward for semigroups by Lajos[8]. Many sorts of ideals on the mathematical designs were portrayed by a few creators, for example, In 2000, concentrated on the portrayal of semiprime ideals and irreducible ideals of Г-semirings concentrated by Dutta and Sardar[3] and primitive ideals of Г-semirings, Pianskool, Sangwirotjanapat and Tipyota[9] presented and concentrated on valuation of Г-semirings and valuation of Г-ideals of a Г- semiring, and Chinram[1] gave a few properties of quasi-ideals in Г-semirings. Jagatap and Pawar [6] presented the idea of minimal quasi-ideals in Г-semirings in 2009. A few properties of minimal quasi-ideals in Г-semirings are given. In 2010, Ghosh and Samanta[5] concentrated on the connection between the fuzzy left (separately, right) ideals of Г-semirings and that of operator semiring. In 2011, Dutta, Sardar and Goswami[4] presented various kinds of procedure on fuzzy ideals of Г-semirings and demonstrated consequently that these activities bring about various designs like complete lattice, modular lattice on some limited class of fuzzy ideals of Г-semirings. In 2012, Bektas, Bayrak and Ersoy[2] presented and concentrated on the portrayal of soft Г-semirings and soft sub-Г-semiring. The view of ideals for some kinds of Г-semirings is the truly significant and intrigued thing with regards to Г-semirings. Hence, we will start and concentrate on summed up bi-Ternay Г-ideals of Г- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 215 https://internationalpubls.com semirings similarly as of bi-Г-ideals of Ternary Г-semirings which was concentrated by Kaushik, Moin and Khan[7]. “Throughout this paper generalized bi- ternary Г-ideal- GBTΓI, smallest generalized bi- ternary Г- ideal- SGBTΓI, Ternary Г-semiring- TГS, minimal generalized bi- Γ-ideal- MGBTΓI, Quasi ternary Г-Ideal – QTГI, partial ternary Γ-ideal- PTΓI” Definition 1.1: Let U,  be two additive commutative semigroups. Then U is entitled a ternary gamma semiring presented  a mapping U U U U  → suiting the next conditions: (1) ( ) ( ) ( )x y z p q x y z p q x y z p q           = = (2) [( ) ] [( ] [ ]p q r s p r s q r s     + = + (3) [ ( ) ] [( ] [ ]p q r s p q s p r s     + = + (4) [ ( )] [( ] [ ] , , , & , , , .p q r s p q r p q s p q r s U         + = +    Example: 1.2 Set N of natural numbers &= {1, 2, 3}. Then ( , max) & (N, max) are commutative semigroups. Characterize the mapping N N N N  → , by min{ , , , , } , , & , .b c d b c d b c d N     =    Next N is a TΓS. Example: 1.3 Rational numbers set (Q) & N= the set of natural numbers. Afterward (N, +), (Q, +) are commutative semigroups. Identify the mapping Q Q Q Q  → by b c d  usual product , , , , , , & , .b c d b c d Q      Followed by Q is a TΓS. Example: 1.4 Rational numbers set (Q). The commutative semigroup (S, +) of all 2 3 matrices over Q & ( , +) commutative semigroup of all 3 2 matrices over Q. Classify W Y X  standard matrix product of , , , , , , & , .W Y X W Y X S       Afterward S is not a semiring & a TS. Definition 1.5: A partial ternary  -semiring known to have a L (La, R) unity element afford , : , e :i i ii I i I    of M & of   ( , )i i i i i i i i i i i ie e c c e c e c c e e c     = = =   for any cM. Definition 1.6: Let M be a partial ternary Γ-semiring. ( )A M  is known to be L (La, R) partial ternary Γ-ideal of M afford (i) ( : )iv i I is a sum able family & vi∈ A i ii I v A    (ii) , & A ( A, A)u v M q A q u v u q v u v q            If A is L (La, R) PTΓI of M. Definition 1.7: Let M be a TΓS. ( )A M  is known as a L (La, R) TΓI of M, if it satisfy the next: (i) A is a L (La, R) PTΓI of M. (ii) &v M w A v w    then v∈ A. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 216 https://internationalpubls.com If A is L (La, R) TΓI of M, afterward A is recognized as TΓI of M. Definition1.8: A ( )O    is entitled (1) a BTΓI of N if O is a sub- Γ-semiring of N & .O N O N O O     (2) a GBTΓI of N if .O N O N O O     (3) a QTΓI of N if O is a sub- Γ-semiring of N & ( ) ( ) ( ) +    Remark1.9: Let M be a TΓS. Then: (1)Each QTΓI of M is a BTΓI. (2)Each BTΓI of M is a GBTΓI. Definition1.10: A TΓS is named a GB-simple TΓ-semiring if M is the unique GBTΓI of M. 2.Properties of GBTΓI’s: Before the portrayals of generalized bi-Γ-ideals of Γ-semirings for the primary outcomes, we give a few helper results which are vital in what follows. Lemma2.1: Let U be a TΓS and .l U Then , &l l U l U l U l l      are GBTΓI’s of U. Lemma2.2: Let M be a TΓS, { / }iD i I a non-empty family of GBTΓI’s of M with .i i I D    Then i i I D  is a GBTΓI of M. Proof: i I  , we have ( ) ( ) ( )i i i i I i I i I i i i i D M D M D D M D M D D              ( ) ( ) ( ) .i i i i i I i I i I i I Thus D M D M D D          Hence i i I D  is a GBTΓI of M. Lemma 2.3: Let M be a TΓS & E S   . Then E E S E S E    is the SGBTΓI of S containing E. Proof: Let D E E S E S E=     . Then .E D Therefore ( ) ( ) ( ) [ ( )( ( ) )] [ ( ) ( ) ] . D S D S D S S S S S U S S S S S S S S S S S S S S S S S S S S S S D     =                                              =         = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 217 https://internationalpubls.com Thus D S S=    is a GBTΓI of S. We shall explain that D is the SGBTΓI of S containing E. Let C be a GBTΓI of S. Then .S S C S C S C C        Thus .D S S C=    Hence D is the SGBTΓI of S having E. ( ) S S  =     It is also denoted the SGBTΓI of S containing {e} as (e). Lemma 2.4: Let T be a sub TΓ-semiring of a TΓS M, a M & ( )a a a     . Then ( )a a a   is a GBTΓI of T. Proof: Believe ( ) ( ) [( ) ] ( ) [( ) ] ( ) [[( ) ( )] [ ( )]] [( ) ( )] ( ) . a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a                                                Hence ( )a a a   is a generalized bi- Γ-ideal of T. Lemma 2.5: Let J be a TΓS & j J . Then j j J j J j     is a GBTΓI of J. Proof: Consider ( ) ( ) ( ) ( ) j J j J j I j J j J j J j J j J j j J j J j J j J j J j J j J j J j j J J j                 =                     Hence j J j J j    is a GBTΓI of J. Proposition2.6: Let M be a TΓS & T a sub-TΓS of M. Then each subset of T having M M T  is a sub- TΓS of M. Proof: Let .J T T J   .J J J T J     Consequently J is a sub- TΓS of M. Proposition 2.7: Let Z be a TΓS & T is TΓI of M. Next each subset of T including T T T    is a TΓI of M. Proof: Let .B B   Then B B    B B    Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 218 https://internationalpubls.com B B   Thus B is a TΓI of Z. Proposition 2.8: Let L be a TΓS & T is TΓI of L. Afterward each subset of T containing ( ) ( ) ( )L L L L L L L L L L   +      is a QTΓI of L. Proof: Allow ( ) ( ) ( ) .C V V V V V V V V V V C      +       Then ( ) ( ) ( )C C C L L L L L L L L L L C      +       And ( ) ( ) ( ) ( ) ( ) ( ) . C L L L C L L L C L L L L C T L L L T L L L T L L L L T C     +            +        So C is a QTΓI of L. Proposition 2.9: Let M be a TΓS & T be BTΓI of M. Afterward each subset of T holding T T T  and all of its images is a BTΓI of M. Proof: Permit & .T T T T       .T T T    Consequently  is a BTΓI of M. Proposition 2.10. Let M be a TΓS & T is a GBTΓI of M. Then each subset of T holding T M T M T    is a GBTΓI of M. Proof: Permit .T T T T     Then .M M T T T      E is a GBTΓI of M. Theorem2.11. Let M be a TΓS. Then the next declarations are correspondent. (1)M is a GB-simple Γ-semiring. (2) .h h h h  =  (3) ( ) .h h=   Proof: (1) (2) Believe that M is a GB-simple TΓS & .h h h h  is a GBTΓI of M by lemma 2.5, M is a GB-simple TΓ-semiring, .h h h=   (2) (3) Believe that h h h h=     & let .h Then, by (2.2), we cover ( ) { } { } .h h h h h h=   =  = (3) (1) Assume that (h)=M for all ,h and let H be a GBTΓI of M & .h H ( ) .h H  With statement, ( ) .h H =   Thus M=H.  M is a GB-simple TΓ-semiring. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 219 https://internationalpubls.com Lemma 2.12: Let B be a GBTΓI of a TΓS M & T a sub-TΓ-semiring of M. If T is a GB-simple TΓ-semiring ,T B   then .T B Proof: Presume T is a GB-simple TΓ-semiring T B   & let .h T  Via Lemma 2.3, { } .T h h T h T h M M=               Hence .T  Theorem 2.13: Let V be a TΓS, B be a GBTΓI of V & ( )G V  then , &G G B G B G B G G      are GBTΓI of V. Proof: B is a GBTΓI of V, ( ) ( ) ( ( ) )B G G V B G G B G G V B G B G G       =         ( ) ( ) ( ( ) )G B G V G B G G B G V G B G G B G      =          ( ) ( )Similarly G G B V G G B G G B         are GBTΓI of V. Theorem2.14: Let M be a TΓS. Subsequently i i i i =  iff M is a GB-simple TΓ- semiring. Proof: Presume i i i i = & let B is a GBTΓI of M & .b By hypothesis, .b b b=    Hence M=B, so M is a GB- simple TΓ-semiring. On the contrary, assume that M is a GB- simple TΓ-semiring .i By Lemma 2.1 and Theorem 2.13, we have i i i is a GBTΓI of M. M is a GB-simple TΓ-semiring , .i i i  = Theorem2.15: Let Q be a TΓS & D a BTΓI of Q. Then D is a MGBTΓI of Q iff D is a GB- simple TΓ-semiring. Proof: Presume that D is a MGBTΓI of Q. Via supposition, D is a TΓS. Let B be a GBTΓI of D. Next .B D B D B B D      D is a GBTΓI of Q & by Th 2.13, cover B D B D B    is a GBTΓI of Q. As D is a MGBTΓI of Q, we get .B D B D B D    = Via (2.3), D=B. So D is GB-simple TΓ-semiring. Permit B be a GBTΓI of Q .B Q  .B D B D B B Q B Q B B           Consequently C is a GBTΓI. D is a GB-simple TΓ-semiring, enclose D=B.  D is a MGBTΓI of Q. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 220 https://internationalpubls.com Theorem2.16: Let M be a TΓS having a proper GBTΓI. Afterward each proper GBTΓI of M is minimal iff the intersection of any two distinct proper GBTΓI’s is empty. Proof: Similar to the above theorem References: [1] R.Chinram, A note on quasi-ideals in Γ-semirings, International Mathematical Forum 26(2008), 1253-1259. [2] O.Bektas, N.Bayrak, and A.Ersoy, Soft Γ-semirings, arXiv:1202.1496[math.RA], 7 Feb 2012. [3] T.K.Dutta and S.K.Sardar, Semiprime ideals and irreducible ideals of Γ-semirings, Novi Sad Journal of Mathematics 30(2000), 97-108. 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