A Note on Quasi and Bi-Ideals in Ordered Ternary Semigroups

Int. Journal of Math. Analysis, Vol. 6, 2012, no. 11, 527-532 A Note on Quasi and Bi-Ideals in Ordered Ternary Semigroups Thawhat Changphas 1 Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen 40002, Thailand Centre of Excellence in Mathematics CHE, Si Ayuttaya Rd. Bangkok 10400, Thailand thacha@kku.ac.th Abstract In this paper, we study the properties of quasi-ideals and bi-ideals in ordered ternary semigroups. Mathematics Subject Classification: 20N99 Keywords: semigroup, ordered ternary semigroup, quasi-ideal, bi-ideal, ternary group 1 Preliminaries Let S be a nonempty set. Then S is called a ternary semigroup if there exists a ternary operation S S S S, written as (x 1,x 2,x 3 ) [x 1 x 2 x 3 ], such that [[x 1 x 2 x 3 ]x 4 x 5 ]=[x 1 [x 2 x 3 x 4 ]x 5 ]=[x 1 x 2 [x 3 x 4 x 5 ]] for all x 1,x 2,x 3,x 4,x 5 S. Hereafter, let [ ] denotes the ternary operation on S if S is a ternary semigroup. Ternary algebraic systems, called triplexes, have been introduced by Lehmer in 1932 (see [4]). This turns out to be commutative ternary groups. Ternary 1 This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand

528 T. Changphas semigroups were first introduced by Banach who showed by an example that a ternary semigroup does not necessarily reduce to an ordinary semigroup (see [8]). Indded, the set S = { i, 0,i} is a ternary semigroup under the multiplication over complex number while S is not a binary semigroup under the multiplication over complex number. Let S be a semigroup. For x 1,x 2,x 3 S, define a ternary operation on S by [x 1,x 2,x 3 ]=x 1 x 2 x 3. Then S a ternary semigroup. For nonempty subsets A 1,A 2 and A 3 of a ternary semigroup S, let [A 1 A 2 A 3 ]={[x 1 x 2 x 3 ] x 1 A 1,x 2 A 2,x 3 A 3 }. For x S, let [xa 1 A 2 ] = [{x}a 1 A 2 ]. For any other cases can be defined analogously. The author [3] gave the definition of an ordered ternary semigroup as follows: A ternary semigroup S is called an ordered ternary semigroup if there is an ordered relation on S such that x y [xx 1 x 2 ] [yx 1 x 2 ], [x 1 xx 2 ] [x 1 yx 2 ], [x 1 x 2 x] [x 1 x 2 y] for all x, y, x 1,x 2 S. ForA S, let (A] ={x S x a for some a A}. For A, B S, we have the following: 1) A (A] 2)(A] (B] ifa B and 3) (A B] =(A] (B]. Let S be an (ordered) ternary semigroup. A nonempty subset T of S is said to be a ternary subsemigroup of S if [TTT] T. In [7], the author gave the definitions of ideals as follows: Let S be a ternary semigroup and I a nonempty subset of S. Then I is said to be a left (right, middle) ideal of S if the following hold: (i) [SSI] I,([ISS] I,[SIS] I). (ii) If x I and y S such that y x, then y I. If I is a left, right and middle ideal of S, then I is called an ideal of S. A nonempty subset Q of a ternary semigroup S is called a quasi-ideal of S if the following conditions holds: (i) [SSQ] [SQS] [QSS] Q. (ii) [SSQ] [SSQSS] [QSS] Q. (iii) If x Q and y S such that y x, then y Q. Note that every left, right and middle ideal is a quasi-ideal.

A note on quasi and bi-ideals in ordered ternary semigroups 529 2 Main Results In this section, let S be an ordered ternary semigroup. Proposition 2.1 Let Q be a quasi-ideal of S and T a ternary subsemigroup of S. IfQ T, then Q T is a quasi-ideal of T. Proof. Assume that Q 1 = Q T. Since Q 1 Q, it follows that [TTQ 1 ] [TQ 1 T ] [Q 1 TT] [SSQ] [SQS] [QSS] Q. Since Q 1 T and T is a ternary subsemigroup of S, we have [TTQ 1 ] [TQ 1 T ] [Q 1 TT] T. Then [TTQ 1 ] [TQ 1 T ] [Q 1 TT] Q 1. Let x Q 1 and y T be such that y x. Since x Q, y Q. So, y Q 1. Therefore, Q 1 is a quasi-ideal of T. Proposition 2.2 Let {Q i i I} be a nonempty family of quasi-ideals of S. If i I Q i, then i I Q i is a quasi-ideal of S. Proof. Assume that Q = i I Q i. Since [SSQ] [SQS] [QSS] [SSQ i ] [SQ i S] [Q i SS] Q i for all i I, we obtain [SSQ] [SQS] [QSS] Q. Similarly, we have that [SSQ] [SSQSS] [QSS] Q. Let x i I Q i and y S be such that y x. Let i I. Since y x and x Q i, y Q i. Thus y i I Q i. Therefore, i I Q i is a quasi-ideal of S. Let X be a nonempty subset of S. Then the intersection of all quasi-ideal of S containing X, denoted by (X) q, is a quasi-ideal of S containing X. This is called the quasi-ideal of S generated by X. Proposition 2.3 The intersection of a left, a middle and a right ideal of S is a quasi-ideal of S. Proof. Let L, M and R be a left, a middle and a right ideals of S, respectively. Let Q = L M R. Choose l L, m M and r R. Since [lmr] L M R, Q is not empty. Since [SSQ] L, [SQS] M and [QSS] R, it follows that [SSQ] [SQS] [QSS] L M R = Q. Similarly, [SSQ] [SSQSS] [QSS] Q. Let x L M R and y S be such that y x. Since x L M R, y L M R. Therefore, L M R is a quasi-ideal of S.

530 T. Changphas Proposition 2.4 If Q is a quasi-ideal of S, then there exist a left ideal L, a middle ideal M and a right ideal R of S such that Q = L M R. Proof. Assume that Q is a quasi-ideal of S. Let L =(Q [QSS]], R =(Q [SSQ]] and M =(Q [SQS] [SSQSS]]. Since Q, L. Let x [LSS], then x =[lss ] for some l L and s, s S. Let l p for some p Q [QSS]. Then [lss ] [pss ]. There are two cases to consider. Case 1. p Q. Then [pss ] [QSS] Q [QSS]. So, x L. Case 2. p [QSS]. Let p =[qs 1 s 2 ] for some q Q and s 1,s 2 S. Since [pss ]=[[qs 1 s 2 ]ss ]=[q[s 1 s 2 s]s ] [QSS] Q [QSS], we have x L. Then [LSS] L. Let x L and y S be such that y x. Then x z for some z Q [QSS]. Since y z, y L. Hence L is a left ideal of S. Similarly, R is a right ideal of S. To show that M is a middle ideal of S, let x [SMS]. Then x =[s 3 ms 4 ] for some m M and s 3,s 4 S. Let m n for some n Q [SQS] [SSQSS]. Then [s 3 ms 4 ] [s 3 ns 4 ]. There are three cases to consider. Case 1. n Q. Since [s 3 ns 4 ] [SQS], x M. Case 2. n [SQS]. Since [s 3 ns 4 ] [SSQSS], x M. Case 3. n [SSQSS]. Since [s 3 ns 4 ] [SQS], x M. Therefore, [SMS] M. Let x M and y S be such that y x. Then x z for some z Q [SQS] [SSQSS]. Thus y M. We shall show that Q = L M R. It is clear that Q L M R. Since Q is a quasi-ideal of S, we have that L M R = (Q [QSS]] (Q [SQS] [SSQSS]] (Q [SSQ]] = (Q] ([QSS]] ([SQS] [SSQSS]] ([SSQ]] Q. This completes the proof. Then Theorem 2.5 Let A S and (A) q = {Q i Q i is a quasi-ideal of S containing A}. (A) q =(A [SSA]] (A [SAS] [SSASS]] (A [ASS]]. Proof. By Theorem 2.2, (A) q is a quasi-ideal of S containing A. Since (A [SSA]], (A [SAS] [SSASS]] and (A [ASS]] are left, middle and right ideals of S, respectively, by Theorem 2.3,

A note on quasi and bi-ideals in ordered ternary semigroups 531 (A [SSA]] (A [SAS] [SSASS]] (A [ASS]] is a quasi-ideal of S containing A. Thus (A) q (A [SSA]] (A [SAS] [SSASS]] (A [ASS]]. For each i I, since A Q i, we have Then (A [SSA]] (A [SAS] [SSASS]] (A [ASS]] = (A] {([SSA]] ([SAS] [SSASS]] ([ASS]]} (Q i ] ([SSQ i ]] ([SQ i S] [SSQ i SS]] ([Q i SS]] Q i. (A) q (A [SSA]] (A [SAS] [SSASS]] (A [ASS]]. We conclude that (A) q =(A [SSA]] (A [SAS] [SSASS]] (A [ASS]]. A ternary semigroup S is said to be quasi-simple if S is the unique quasiideal of S. A quasi-ideal Q of S is called a minimal quasi-ideal of S if it contains no proper quasi-ideal of S. Theorem 2.6 A ternary semigroup S is quasi-simple if and only if for all x S. S =([xss]] ([SxS]] ([SSxSS]] ([SSx]] Proof. Assume that S is quasi-simple. Let x S. We shall show that L =([xss]] is a left ideal of S. Let y [LSS]. Then y =[lss ] for some l L and s, s S. Since l L, l =[xs 1 s 2 ] for some s 1,s 2 S. Thus y =[lss ]=[[xs 1 s 2 ]ss ]=[x[s 1 s 2 s]s ] [xss] L. Let y L and z S be such that z y. Since y L, y w for some w [xss]. Since z w, z ([xss]] = L. Therefore, [xss] is a left ideal of S. Similarly, [SSx] is a right ideal of S. To show that M =[SxS] [SSxSS] is a middle ideal of S, let y [SMS]. Then y =[sms ] for some m M and s, s S. If m =[s 1 xs 2 ] for some s 1,s 2 S, then y =[sms ]=[s[s 1 xs 2 ]s ] [SSxSS] M. If m =[s 3 s 4 xs 5 s 6 ] for some s 3,s 4,s 5,s 6 S, then y =[sms ]=[s[s 3 s 4 xs 5 s 6 ]s ] [SxS] M.

532 T. Changphas Then [SMS] M. Let y M and z S be such that z y. If y w for some w [SSxSS], then z ([SSxSS]] M. Ify w for some w [SxS], then z ([SxS]] M. Therefore, M is a middle ideal of S. By Theorem 2.3, S =([xss]] ([SxS]] ([SSxSS]] ([SSx]]. Conversely, assume that S =([xss]] ([SxS]] ([SSxSS]] ([SSx]] for all x S. Let Q be a quasi-ideal of S. Let q Q. By assumption we get S = ([qss]] ([SqS]] ([SSqSS]] ([SSq]] ([QSS]] ([SQS]] ([SSQSS]] ([SSQ]] Q. Thus S = Q. This completes the proof. Theorem 2.7 Every quasi-simple ideals of S is a minimal quasi-ideal of S. Proof. Let Q be a quasi-simple ideal of S. Let Q be a quasi-ideal of S such that Q Q. Since Q is a quasi-ideal of Q, Q = Q. Then Q is a minimal quasi-ideal of S. References [1] R. Chinram, On quasi-gamma-ideals in gamma-semigroups, Science Asia, 32 (2006), 351-353. [2] V. N. Dixit, S. Dewen, A note on quasi and bi-ideals in ternary semigroups, Int. J. Math. Sci., 18 (1995), 501-508. [3] A. Iampan, On Ordered Ideal Extensions of Ordered Ternary Semigroups, Lobachevskii Journal of Mathematics, 31(1) (2010), 13-17. [4] D. H. Lehmer, A ternary analogue of abelian groups, Am. J. Math., 54(2) (1932), 329-338. [5] M. Petrich, Introduction to Semigroups, Merrill, Columbus, 1973. [6] F. M. Siomon, Ideal theory in ternary semigroups, Math. Jap., 10 (1965), 63-84. [7] O. Steinfeld, Quasi-ideals in rings and semigroups, Akadémiai Kiadó, Budapest, 1978. [8] J. Los, On the extending of models I, Fundam. Math., 42 (1995), 38-54. Received: October, 2011