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Concepts on Ordered Ternary Semirings IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015. www.ijiset.com ISSN 2348 – 7968 Concepts on Ordered Ternary Semirings Dr.D.Madhusudanarao 1, G. Srinivasarao 2, P. Siva Prasad3 1 Head, Department of Mathematics, V.S.R. & N.V.R. College, Tenali, A.P., India. 2 Assistant Professor of Mathematics, Tirumala Engineering College, Narasaraopet, A.P., India 3Asstistant Professor of Mathematics Universal College of Engineering & Technology, Perecherla, Guntur, A.P. India. ABSTRACT In this paper, we study the properties of ordered Ternary semi- 2 2. Preliminaries rings satisfying the identity a + ab = a. It is proved that , let (T , + , . , ) be a totally ordered ternary semiring satisfying the Definition 2.1 : A non-empty set T together with a condition a + ab2 = a , a , b T. If ( T , + , ) is binary addition and a ternary multiplication denoted by positively totally ordered ( negatively totally ordered ) , then ( T , justaposition, is said to be ternary semiring if T is an . , ) is non-positively ordered ( non-negatively ordered). additive commutative semigroup satisfying the following conditions: Key Words: Ordered ternary semiring, positively totally (i) (abc)de = a(bcd)e = ab(cde) ordered, non-positively ordered. (ii) (a+b)cd = acd + bcd (iii) a(b+c)d = abd + acd 1. Introduction (iv) ab(c + d) = abc + abd, for all a , b , c , d , e T. Algebraic structures play a prominent role in Mathematics Example 2.2 : Let T = { 0 , 1 , 2 , 3 , 4 } is a ternary with wide ranging applications in many disciplines such as semiring with respect to addition modulo 5 and theoretical physics, computer sciences, control multiplication modulo 5 as ternary operation is defined as engineering, information sciences, coding theory, follows: topological spaces and the like. This provides sufficient motivation to researchers to review various concepts and results. +5 01234 x5 0 1 2 3 4 The theory of ternary algebraic systems was 0 01234 0 0 0 0 0 0 studied by LEHMER [9] in 1932, but earlier such 1 12340 1 0 1 2 3 4 structures were investigated and studied by PRUFER in 2 0 2 4 1 3 2 23401 1924, BAER in 1929. 3 0 3 1 4 2 3 34012 Generalizing the notion of ternary ring introduced 4 0 4 3 2 1 4 40123 by Lister [10], Dutta and Kar [6] introduced the notion of ternary semiring. Ternary semiring arises naturally as Definition 2.3 : A ternary semigroup ( T , . ) is said to follows, consider the ring of integers Z which plays a vital be role in the theory of ring. The subset Z+ of all positive 2 integers of Z is an additive semigroup which is closed (i) left regular, if it satisfies the identity ab = a a, under the ring product, i.e. Z+ is a semiring. Now, if we b T consider the subset Z− of all negative integers of Z, then (ii) right regular, if it satisfies the identity b2 a = a a, - we see that Z is an additive semigroup which is closed b T under the triple ring product (however, Z− is not closed under the binary ring product), i.e. Z− forms a ternary (iii) lateral regular, if it satisfies the identity aba = a a semiring. Thus, we see that in the ring of integers Z, Z+ , b T forms a semiring whereas Z− forms a ternary semiring. (iv) two-sided regular, if it is both left as well as right regular. 435 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015. www.ijiset.com ISSN 2348 – 7968 (v) regular, it is left, lateral and right regular. Definition 3.5 : An element x in a partial ordered semigroup (T, +, ) is non-negative (non-positive) if Definition 2.4 : An element a of a ternary multiplicative x + x x ( x + x x ). semigroup ‘ T ‘ is called an E-inverse if there exist an element x such that ( axa )( axa )( axa ) = axa , i.e., Definition 3.6: A partial ordered semi-group (T, +, ) axa E (.), where E(.) is the set of all ternary is non-negatively (non-positively) ordered if every multiplicative idempotent elements of T. element in T is non-negative ( non-positive). Definition 2.5 : A ternary semigroup T is called an Definition 3.7 : A ternary semiring (T, +, .) is said to be E-inverse ternary semigroup if every element of T is an a positive rational domain (PRD) if and only if (T, . ) is E-inverse. an ternary abelian group. Definition 2.6 : An element a of an additive semigroup Theorem 3.8 : Let (T, +, ., ) be a totally ordered ‘ T ‘ is called an E-inverse if there is an element x in T ternary semiring and satisfying the identity ab2 + a = such that axa + axa = axa i.e. axa E (+) , where a T . If (T, +, ) is non-negatively E (+) is the set of all additive idempotent elements of T . ordered ( non-positively ordered ), then (T, ., ) is Definition 2.7 : A semigroup T is called an E-inverse non-positively ( non-negatively) ordered. semigroup if every element of T is an E-inverse. Proof : We know that a + a3 = a a T . Since (T, +, 3 3 Definition 2.8 : A ternary semigroup (T, +) is said to ) is non-negatively ordered, we have a = a + a a, 3 3 satisfy quasi separative if x = xyx = yxy = y implies a3 a a T . Hence (T, ., ) is non-negatively x = y T. ordered. Suppose (T, +, ) is non-positively ordered, a3 = a + a3 a a3 a a T . Therefore (T, ., ) 3. Ordering on Ternary Semiring Satisfying is non-positively ordered. the Identity ab2 + a = a Definition 3.9: In a totally ordered ternary semiring (T, Definition 3.1 : A ternary semiring ( T , + , . ) is said to +, ., ). be totally ordered ternary semiring if there exist a (i) (T, +, ) is negatively totally ordered if a + b a partially order “ “ on T such that and b , a , b T and (i) (T, +, ) is a totally ordered semigroup (ii) (T, ., ) is negatively totally ordered if ab2 a and (ii) (T, ., ) is a totally ordered ternary semiring. b, a , b T. It is denoted by (T , + , . , . Definition 3.10: An element x in totally ordered ternary semi-ring is minimal (maximal) if x a (x a), Example 3.2 : Consider the set T = { 1, 2 , 3 , 4 } with the order 1 < 2 < 3 < 4 and with the following addition and a T. multiplication. Theorem 3.11: Let (T, +, ., ) be a totally ordered positive rational domain ternary semiring satisfying Hence (T, +, . , ) is a . 1 2 3 4 the identity ab2 + a = a T. If (T, +, ) is totally ordered ternary 1 2 4 4 4 positively totally ordered ( negatively totally ordered), semiring. 2 4 4 4 4 then 1 is minimum (maximum) element. 3 4 4 4 4 Definition 3.3 : An 4 4 4 4 4 We know that 1 + a = a a T. Suppose element ‘ x ‘ in a partially Proof: ordered ternary semigroup (T, ., ) is non-negative (T, +, ) is positively totally ordered a = 1 + a a (non-positive) if x3 x ( x3 . and 1 a 1 ‘ 1 ‘ is the minimal element. Suppose ( T , + , ) negatively totally ordered Definition 3.4 : A partial ordered ternary semi-group a = a + 1 a and 1 , a 1 1 is the maximal + 1 2 3 4 (T , ., is non-negatively element. 1 2 3 4 4 (non-positively) ordered if every 2 3 4 4 4 element in T is non-negative ( 3 4 4 4 4 non-positive ). 4. Ordered Ternary Semiring 4 4 4 4 4 436 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015. www.ijiset.com ISSN 2348 – 7968 Definition 4.1 : A totally ordered ternary semigroup 5. Conclusions (T, ., ) is said to be non-negatively (non–positively) ordered if every one of its element is non-negative (non – In this paper mainly we studies about some properties of positive ). ordered ternary semi rings. Definition 4.2 : A ternary semigroup (T, . ) is positively (negatively) ordered in strict sense if ab2 a and a2b Acknowledgments b ( ab2 a and a2b b ) a , b T. The authors would like to thank the referee(s) for Theorem 4.3 : Suppose (T, +, ., ) is a totally careful reading of the manuscript. ordered ternary semiring satisfying the identity a + ab2 = a , a , b T and (T, +, ) is positively totally ordered ( negatively totally ordered), then (T, ., ) is References negatively totally ordered (positively totally ordered ). [1] Arif Kaya and Satyanarayana M. Semirings satisfying Proof : Suppose a + ab2 = a T . properties of distributive type, Proceeding of the a = a + ab2 ab2, a ab2 American Mathematical Society, Volume 82, Number 3, July 1981. Suppose b a2b , a + b a + a2b. Since (T, +) is positively totally ordered a + b b. It is a [2] Chinaram, R., A note on quasi-ideal in ¡¡semirings, contradiction to the fact that (T, .) is positively totally Int. Math. Forum, 3 (2008), 1253{1259. ordered. Then b a2b. Therefore ab2 a & a2b b. [3] Daddi. V. R and Pawar. Y. S. Ideal Theory in Hence (T , . ) is negatively totally ordered. Commutative Ternary A-semirings, International Mathematical Forum, Vol. 7, 2012, no. 42, 2085 – Similarly we can prove that (T , . ) is positively totally 2091. ordered. [4] Dixit, V.N. and Dewan, S., A note on quasi and bi- Example 4.4 [12 ] : 1 < b < a ideals in ternary semigroups, Int. J. Math. Math. Sci. 18, no. 3 (1995), 501{508. + 1 a b ൈ 1 a b [5] Dutta, T.K. and Kar, S., On regular ternary 1 1 a b 1 1 a b semirings, Advances in Algebra, Proceedings a a a a a a a a of the ICM Satellite Conference in Algebra and b b a b b b a b Related Topics, World Scienti¯c, New Jersey, 2003, 343-355.
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