Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 357 https://internationalpubls.com Investigation on Anti-Fuzzy Tssemiring of a Ter.semi-ring *1 G. Srinivasa Rao, 2 R.Venkata Aravinda Raju, 3 V. Savithri, 4 S. Vinoth *1 Associate Professor, Department of Mathematics, School of Applied Sciences & Humanities, VFSTR Deemed to be University, Vadlamudi, Guntur (Dt.), A.P., India.Email:gsrinulakshmi77@gmail.com 2 Assistant Professor, DVR & DR. HS MIC College of Technology, Kanchikacherla, Krishna District- 521180, A.P, INDIA. Email: ravindaraju.1@gmail.com 3 Assistant Professor, Department of Mathematics, Karpagam Academy of Higher Education, (Deemed to be University), Eachanari, Coimbatore, Tamilnadu, India. Email:savithri.vijayakumar@kahedu.edu.in 4 Assistant Professor, Department of Mathematics, School of Applied Sciences & Humanities, VFSTR Deemed to be University, Vadlamudi, Guntur (Dt.), A.P., India.Email:vinomaths6@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: In this article, we scrutinise the algebraic characteristics of the anti-fuzzy ternary sub- semi-ring of a ternary semi-ring and explore many theorems inside it. Keywords: Fuzzy set, fuzzy ternary sub-semi-ring, anti-fuzzy ternary sub-semi-ring, anti- fuzzy normal ternary sub-semi-ring, homomorphism, anti-homomorphism, isomorphism, anti-isomorphism. 2000 AMS subject classification: 03F55, 06D72, 08A72. 1. INTRODUCTION Numerous ideas exist for universal algebras that extend an associative ring (R, +,.. ). Several near-rings and semi-ring types, in particular, have shown to be quite beneficial. If both (T, +) and (T,.) are commutative semi-groups with + being a binary operation and. being a ternary multiplication fulfilling a(b+c)d = abd+acd, ab(c + d)= abc + abd, (a+b)cd = abd + bcd, a, b, c, d T, then an algebra (T, +,.) is considered a ter.semi-ring. Elaborately ternary semi-ring was researched by G. Srinivasa Rao et al. [10–14]. This paper delves into a few theorems related to the anti-fuzzy ter.sub-semi-ring of a semi-ring. 2. PRELIMINARIES Def.2.1: Let P . A fuzzy subset  of P is a mapping  : P → [0, 1]. Def.2.2: Let T be a ter.semi-ring. A fuzzy subset S of T is said to be a fuzzy ter.sub-semi-ring (FTSSR) of T if (i) S (a + b) ≥ min{ S (a), S (b)},(ii) S (abc) ≥ min{ S (a), S (b), S (c)},  a, b, cT. Ex.:2.3: Let Z be a ring of integers and S = (Z\N) Z , set of all negative integers with zero. Then (Z\N, +, .) forms a ternary semi-ring S with zero with respect to binary addition and ternary multiplication. Define a fuzzy subset A: Z [0, 1], we have A (x) = { Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 358 https://internationalpubls.com Then A(x) is a fuzzy ternary sub-semi-ring of S. Ex.2.4: Consider the set of integers modulo 5, non-positive integers = {0, -1, -2, -3, -4}under the usual addition and ternary multiplication, we have Clearly ( , +, .) is a ternary semi ring. Let a fuzzy set : [0, 1] be defined as (0) = 1, (−1) = 0.3, (−2) = 1, (−3) = 0.3, (−4) = 1 and (−5) = 0.3. Thus ( , +, ., ) is a fuzzy ternary semi-ring. Ex.2.5: Let R = N, be the set of all-natural numbers with zero and = {0, 1}. Define a mapping R × × R × × R R as aαbβc by ternary multiplication of a, α, b, β, c for all a, b, c R and α, β . Clearly R is a ternary gamma semi-ring. Define : R [0, 1] as = { Clearly is a fuzzy ternary gamma semi-ring. Ex.2.6: Let R = [0, 1], = N. Define + and ternary multiplication ‘. ‘ defined as a + b = max {a, b} and aαbβc = min {aαbβc} Def.2.7: Let T be a ter.semi-ring. A fuzzy subset S of T is said to be an anti-fuzzy ter.sub-semi-ring (AFSSR) of T when (i) S ( a + b) ≤ max{ S (a), S (b)},(ii) S (abc) ≤ max{ S (a), S (b), S (c) }, a, b, cT. Def.2.8: Let T be a ter.semi-ring. An anti-fuzzy ter.sub-semi-ring (AFTSSR) S of T is said to be an anti-fuzzy normal ter.sub-semi-ring (AFNSSR) of T if it satisfies the following conditions: (i) S ( a + b) = S (b+a), (ii) S (abc) = S (cba),  a, b, cT. Def.2.9: Let (T, +, .) and (T1, +, .) be any two ter.semi-rings(TSR). Let h: T →T1 be any function and P be an AFTSSR in T, U be an AFTSSR in h(T) = T1, defined by U (b) =    aU bha  1 inf  , for all a in T and b in T1. Then P is called a pre-image of U with respect to h and is denoted by h -1 (U). Def.2.10: Let (T, +, .) and (T1, +, .) be any two TSRs. A mapping from h: T →T1is said to be a TSR homomorphism if h(a + b) = h(a) + h(b), h(abc) = h(a) h(b)h(c),  a, b, cT. Def.2.11: Let (T, +,.) and (T1, +, .) be any two TSRs. A mapping from h: T →T1 is said to be a TSR anti-homomorphism if h(a + b) = h(a) + h(b), h(abc) = h(c) h(b) h(a),  a, b, c T. + 0 -1 -2 -3 -4 0 0 -1 -2 -3 -4 -1 -1 -2 -3 -4 0 -2 -2 -3 -4 0 -4 -3 -3 -4 0 -1 -2 -4 -4 0 -1 -2 -3 . 0 -1 -2 -3 -4 0 0 0 0 0 0 -1 0 1 2 3 4 -2 0 2 4 1 3 -3 0 3 1 4 2 -4 0 4 3 2 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 359 https://internationalpubls.com Def.2.12: Let (T, +,.) and (T1, +, .) be any two TSRs. Function f: T →T1 is a TSR isomorphism, if f is homomorphism, one-to-one and onto. Def.2.13: Let (T, +,.) and (T1, +, .) be any two TSRs. Then the function f: T →T1 is a TSR is said to be a ter.semi-ring anti-isomorphism, if f is anti-homomorphism, one-to-one and onto. Def.2.14: Let P be an AFTSSR of a ter.semi-ring (T, +, ∙) and x in T. Then the pseudo anti-fuzzy coset (xP) p is defined by ( (x P )p)(u) = p(x P (u), for every u in T and for some p in P. 3. PROPERTIES OF ANTI-FUZZY TERNARY TER.SUBSEMIRING OF A TERNARY SEMIRING Th.3.1: Union of any two AFTSSR of a ter.semi-ring T is an AFTSSR of T. Pf.: Let P and Q be any two AFTSSRs of a ter.semi-ring T and p and q in T. Let P={( p, P (α) )/α T}and Q={(q, Q (α) )/αT}and also let U = PQ = {( α, U (α)) /αT}, where max{ P (α), Q (α)} = U (α). Now, U (α + β) = max{ P (α + β), Q (α + β)} ≤ max{max { P (α), P (β) }, max{ Q (α), Q (β)}} = max{max{ P (α), Q (α) }, max{ P (β), Q (β) } } = max{ U (α), U (β)}. Therefore, U (α + β) ≤max{ U (α), U (β)}, . And,  U =max{  P ,  Q )} ≤ max{{max P (α), P (β),   P }, max{max Q (α), Q (β),  Q }}=max{max{ P (α), Q (α)},max{ P (β), Q (β)}, max{ P (γ), Q (γ)}}= max{ U (α), U (β), U (γ)}. Therefore, U (αβγ) ≤ max{ U (α), U (β), U (γ)},  α, β, γT. Therefore, U is an AFTSSR of a TSRT. Hence the union of any two AFTSSRs of a TSRT is an AFTSSR of T. Th.3.2: The arbitrary union of a family of AFTSSRs of TSRT is an AFTSSR of T. Pf.: Let {Si :i  } be an arbitrary family of AFTSSRs of a TSRT and let P =  i iS . Let α, β and γ in T. Then, P (α + β) =     iS i Sup ≤ i Sup max{ iS (α), iS (β)} =max{ i Sup {   iS , i Sup {   iS }= max{   P ,   P }. Therefore, P (α + β) ≤ max{   P ,   P },  α, β, γ T. And,  P =   iS i Sup  ≤ i Sup max{ iS (α), iS (β), iS (γ)}}= max{ i Sup {   iS , i Sup {   iS }, i Sup {   iS }}= max{   P ,   P ,   P }, Therefore,  P ≤ max{   P ,   P ,   P },  α, β, γ T. That is, P is an AFTSSR of a TSRT. Thus, the union of a family of AFTSSRs of R is an AFTSSR of T. Th.3.3: If P, Q and R, any AFTSSRs of the TSRs T1, T2 and T3 respectively, then anti-product P × Q × R is an AFTSSR of T1 x T2 x T3. Proof: Let P, Q and R are any AFTSSRs of the TSRs T1, T2 and T3 respectively. Let a1, a2 and a3 be in T1, b1, b2 and b3 be in T2, c1, c2 and c3 be in T3. Then (a1, b1 , c1), (a2, b2 , c2) and (a3, b3 , c3) are in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 360 https://internationalpubls.com T1 x T2 x T3. Now,       333222111 ,,,,,, cbacbacbaRQP  =   321321321 ,, cccbbbaaaRQP  = max{  321 aaaP  ,  321 bbbQ  ,  321 cccR  } ≤ max{max{  1aP ,  2aP ,  3aP },max{  1bQ ,  2bQ ,  3bQ )}, max{  1cR ,  2cR ,  3cR } = max{max{  1aP ,  1bQ ,  1cR },max{  2aP ,  2bQ ,  2cR }, max{  3aP ,  3bQ ,  3cR } = max{  111 ,, cbaRQP  ,  222 ,, cbaRQP  ,  333 ,, cbaRQP  }. Therefore,   321321321 ,, cccbbbaaaRQP  ≤ max{  111 ,, cbaRQP  ,  222 ,, cbaRQP  ,  333 ,, cbaRQP  }. Also,     333222111 ,,,,,, cbacbacbaRQP  =  321321321 ,, cccbbbaaaRQP  = max{  321 aaaP ,  321 bbbQ ,  321 cccR } ≤ max{max{  1aP ,  2aP ,  3aP },max{  1bQ ,  2bQ ,  3bQ )}, max {  1cR ,  2cR ,  3cR }} = max{max{  1aP ,  1bQ ,  1cR }, max{ max{  2aP ,  2bQ ,  2cR }, max{  3aP ,  3bQ ,  3cR }}= max{  111 ,, cbaRQP  ,  222 ,, cbaRQP  ,  333 ,, cbaRQP  }. Therefore,     333222111 ,,,,,, cbacbacbaRQP  ≤ max{  111 ,, cbaRQP  ,  222 ,, cbaRQP  ,  333 ,, cbaRQP  }. Hence P × Q × R is an AFTSSR of TSR of T1 x T2 x T3. Th.3.4: P is an AFTSSR of T if and only if U is an AFTSSR of T×T×T, when P is a fuzzy subset of a TSRT and U is a strongest anti-fuzzy relation of T. Pf.: Given that P is an AFTSSR of a TSR T. Then for any a = (a1, b1, c1), b = (a2, b2 , c2) and c=(a3, b3 , c3), are in T x T x T. We have,  baU  = U [( a1, b1 , c1)+(a2, b2 , c2)] =  212121 ,, ccbbaaU  = max  212121 ,, ccbbaaU  )} ≤ max{max{  1aU ,  2aU },max{  1bU ,  2bU )} max{  1cU ,  2cU }}= max{max{  1aU ,  2aU },max{  1bU ,  2bU )}, max{  1cU ,  2cU }} = max{  111 ,, cbaU ,  222 ,, cbaU )} = max{  aU ,  bU }. Therefore,  baU  ≤ max{  aU ,  bU },  a, b, c  T × T × T. And,  abcU =     333222111 ,,,,,, cbacbacbaU =  321321321 ,, cccbbbaaaU = max {  321 aaaU ,  321 bbbU ,  321 cccU } ≤ max {max{  1aU ,  2aU ,  3aU }, max{  1bU ,  2bU ,  3bU }, max{  1cU ,  2cU ,  3cU }} = max{{max{  1aU ,  2aU ,  3aU }, max{  1bU ,  2bU ,  3bU }, max{  1cU ,  2cU ,  3cU }} = max{  111 ,, cbaU ,  222 ,, cbaU ,  333 ,, cbaU }= max{  aU ,  bU ,  cU }. Therefore,  abcU ≤ max{  aU ,  bU ,  cU },  a, b, cT×T×T. This proves that P is an AFTSSR of T×T×T. Conversely assume that U is an AFTSSR of T×T×T, then for any a = ( a1, b1 , c1), b=(a2, b2 , c2) and c=(a3, b3 , c3),are in T x T x T, we have max{  21 aaU  ,  21 bbU  ,  21 ccU  } =   222111 , cbacbaU  = U [( a1, b1 , c1)+( a2, b2 , c2)] =  baU  ≤ {  aU ,  bU } = max{  111 ,, cbaU ,  222 ,, cbaU ) }= max{max{  1aU ,  1bU ,  1cU }, max{  2aU ,  2bU ,  2cU }. If  21 aaU  ≥  21 bbU  ,   1aU ≥  1bU ,  2aU ≥  2bU , we get,  21 aaU  ≤ max{  1aU ,  2aU },  a1 and a2 in T. And, max {  321 aaaU ,  321 bbbU ,  321 cccU }=  333222111 ,, cbacbacbaU = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 361 https://internationalpubls.com     333222111 ,,,,,, cbacbacbaU =  abcU ≤ max{  aU ,  bU ,  cU }= max{  111 ,, cbaU ,  222 ,, cbaU ,  333 ,, cbaU } =max{ max{  1aU ,  2aU ,  3aU }, max{  1bU ,  2bU ,  3bU }, max{  1cU ,  2cU ,  3cU }}.If  321 aaaU ≥  321 bbbU ,  1aU ≥  1bU ,  2aU ≥  2bU ,  3aU ≥  3bU , we get  321 aaaU ≤ max{  1aU ,  2aU ,  3aU },  a1, a2, a3 in T. Therefore, P is an AFTSSR of T. Th.3.5: P is an AFTSSR of a TSR (T, +, ∙ ) if and only if   U ≤ max{  U ,  U },  U ≤ max{  U ,  U ,  U },  α, β and γ in T. Proof: It is trivial. Th.3.6: If P is an AFTSSR of a TSR (T, +, ∙), then H = {α/ αT:  P =0} is either H= or a ter.sub-semi-ring(TSSR) of T. Pf.: If each element doesn’t satisfies this condition, then H = . If α, β H, then   U ≤ max {  U ,  U }= maximum{0, 0}=0.    U = 0. And,  U ≤ maximum{  U ,  U ,  U } = maximum{0,0,0} = 0.   U =0  α + β, αβγ H. Therefore, H is a TSSR of T. Hence H= or a TSSR of T. Th.3.7:If P be an AFTSSR of a TSR (T, +, ∙), then if   U = 1, then either  U =1 or  U = 1,  α, β in T. Pf.: Let α and β in T. By the definition   U ≤ max {  U ,  U },  1 ≤ max {  U ,  U }. Therefore, either  U = 1 or  U = 1. Th.3.8: Let P be an AFTSSR of a TSR T and f, an isomorphism from a TSRT onto S. Then P◦f is an AFTSSR of T. Pf.: Let α, β and γ in T and P be an AFTSSR of a TSRT. Then we have, ( fP  )(α + β) = P (f(α + β)) = P (f(α)+ f(β)) ≤ max { P (f(α)), P (f(β))} ≤ max{( fP  ) (α),( fP  )(β)},  ( fP  ) (α + β) ≤ max {( fP  )(α),( fP  )(β)}. And ( fP  )(αβγ) = P (f(αβγ)) = P (f(α)f(β)f(γ)) ≤ max{ P (f(α)), P (f(β)), P (f(γ))}≤ max {( fP  )(α), ( fP  )(β), ( fP  )(γ)} ( fP  )(αβγ) ≤ max{( fP  )(α),( fP  )(β), ( fP  )(γ) }. Thus ( fP  ) is an AFTSSR of a TSRT. Th.3.9: Let P be an AFTSSR of a TSRT and h be an anti-isomorphism from a TSRT onto S. Then hP  is an AFTSSR of T. Pf.: Let a, b and c in T and P be an AFTSSR of a TSRT. Then we have, ( hP  )( α + β)= P ( h(α + β) ) = P (h(α) + h(β) ) ≤ max{ P (h(α)), P (h(β)) } ≤ max{( hP  ) (α),( hP  )(β)},  ( hP  )(α + β) ≤ max{( hP  )(α),( hP  )(β)}. And ( hP  )(αβγ)= P (h(αβγ))= P (h(γ)h(β)h(α)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 362 https://internationalpubls.com ≤ max{ P (h(α)), P ( h(β), P (h(γ))} ≤ max {( hP  )(α),( hP  )(β), ( hP  )(γ) },  ( hP  ) (αβγ) ≤ max{( hP  )(α),( hP  )(β), ( hP  )(γ)}. Therefore, hP  is AFTSSR of a TSRT. Th.3.10: Suppose P be an AFTSSR of a TSR (T, +, .). The pseudo anti-fuzzy coset ( P ) p is an AFTSSR of a TSRT, for a in T. Pf.: Let P be an AFTSSR of a TSR T. For every α, β and γ in T, we have,((x P ) p )(α + β) = p(x) P (α + β) ≤ p(x)max{ P (α), P (β)} = max{p(x) P (α), p(x) P (β)} = max{( (x P ) p ) (α), ((x P ) p )(β)}. ((x P ) p )(α + β) ≤ max {((x P ) p ) (α), ((x P ) p )(β) }. Now, ((x P ) p )( αβγ) = p(α) P (αβγ) ≤ p(x) max { P (α), P (β), P (γ)}= max {p(x) P (α) ), p(x) P (β)} = max {((x P ) p )(α), ((x P ) p ) (β), ((x P ) p )(γ)}.  ((x P ) p ) (αβγ) ≤ max { ((x P ) p ))(α), ((x P ) p )(β), (((x P ) p )(γ)}. Hence ((x P ) p )is an AFTSSR of a TSRT. Th.3.11: Let (T, +, .) and (T1, +, .) be any two TSRs. The homomorphic image of an AFTSSR of T is an AFTSSR of T1. Pf.: Given (T, +, .) and (T1, +, .) are two TSRs. Let f :TT1 be a homomorphism. Then, f(α + β) = f(α) + f(β) and f(αβγ) = f(α)f(β)f(γ),  α, β and γ in T. Let Q = f(P), where P is an AFTSSR of T. To prove, Q is an AFTSSR of T1. Now, for f(α), f(β), f(γ) in T1, Q ( f(α) + f(β)) = Q ( f(α + β) )≤ P (α + β) ≤ max{ P (α), P (β)} Q (f(α) +f(β)) ≤ max{ Q (f(α)), Q (f(β))}. Again, Q (f(α)f(β)f(γ)) = Q (f(αβγ)) ≤ Q ( αβγ) ≤ max{ Q (α), Q (β), Q (γ)}  Q (f(α)f(β)f(γ)) ≤ max{ Q (α), Q (β), Q (γ)}. Hence Q is an AFTSSR of T1. Th.3.12: Let (T, +, .) and (T1, +, .) be any two TSRs. The homomorphic pre-image of an AFTSSR of T1 is an AFTSSR of T. Pf.: Given (T, +, .) and (T1, +, .) are two TSRs. Let f:TT1 be a homomorphism. Then, f(α + β) = f(α) + f(β) and f(αβγ) = f(α)f(β)f(γ), . Let Q = f(P), where P is an AFTSSR of T. We have to prove that P is AFTSSR of T. Let α, β and γ in T. Then, P (α + β)= Q (f(α + β))= Q ( f(α) + f(β)) ≤ max{ Q (f(α)), Q (f(β))} = max { P (α), P (β)}  P (α + β) ≤ max { P (α), P (β)}. Again, P (f(αβγ)) = Q (f(αβγ)) = Q (f(α)f(β)f(γ)) ≤ max{ Q (α), Q (β), Q (γ)}}= max{ P (α), P (β), P (γ)} P (f(αβγ)) ≤ max{ P (α), P (β), P (γ)}. Hence P is an AFTSSR of T. Th.3.13: If (T, +, .) and (T1, +, .) are two TSRs, then anti-homomorphic image of an AFTSSR of T is an AFTSSR of T1. Pf.: Given (T, +, .) and (T1, +, .) are two TSRs. Let f:TT1 a homomorphism. Then, f(α + β) = f(β) + f(α) and f(αβγ) = f(γ)f(β)f(α), . Let Q = f(P), where P is an AFTSSR of T. We prove that Q is an AFTSSR of T1. Now, for f(α), f(β), f(γ)in T1, Q (f(α) + f(β))= Q (f(α + β))≤ P (β + α) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 363 https://internationalpubls.com ≤ max{ P (β), P (α)}= max{ P (α), P (β)}  Q (f(α) + f(β))≤ max{ P (α), P (β)}. Again, Q (f(α)f(β)f(γ))= Q (f(γβα)) ) ≤ P (γβα) ≤ max{ P (γ), P (β), P (α)}= max{ P (α), P (β), P (γ)},  Q (f(α)f(β)f(γ)) ≤ max{ Q (f(α)), Q (f(β)), Q (f(γ))}. Hence Q is an AFTSSR of T1. Th.3.14: Let (T, +, .) and (T1, +, .) be any two TSRs. The anti-homo-morphic pre-image of an AFTSSR of T1 is an AFTSSR of T. Pf.: Given (T, +, .) and (T1, +, .) are two TSRs. Let f :TT1 be a homomorphism. Then, f(α + β) = f(β) + f(α) and f(αβγ) = f(γ)f(β)f(α), . Let Q = f(P), where P is an AFTSSR of T. We have to prove that P is an AFTSSR of T. Let α, β and γ in T. Then P (α + β) = Q (f(α + β)) = Q (f(α) + f(β)) ≤ max{ Q (f(γ)), Q (f(β)), Q (f(α))}= max{ Q (f(α)), Q (f(β)), Q (f(γ))}= max{ P (α), P (β), P (γ)},  P (α + β)≤ max{ P (α), P (β), P (γ)}. Again, P ((αβγ))= Q (f(αβγ))= Q (f(γ)f(β)f(α))≤max{{ Q (f(γ)), Q (f(β)), Q (f(α))}= max{ Q (f(α)), Q (f(β)), Q (f(γ))}= max{ P (α), P (β), P (γ))}  P (( αβγ))≤ max{ P (α), P (β), P (γ))}. Hence P is an AFTSSR of T. REFERENCES [1] A Zaid. S, On fuzzy sub-near rings and ideals, fuzzy sets and systems, 44(1991), 139-146. 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