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

Transcription

1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 11, 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

2 528 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.

3 A note on quasi and bi-ideals in ordered ternary semigroups 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.

4 530 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,

5 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.

6 532 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), [2] V. N. Dixit, S. Dewen, A note on quasi and bi-ideals in ternary semigroups, Int. J. Math. Sci., 18 (1995), [3] A. Iampan, On Ordered Ideal Extensions of Ordered Ternary Semigroups, Lobachevskii Journal of Mathematics, 31(1) (2010), [4] D. H. Lehmer, A ternary analogue of abelian groups, Am. J. Math., 54(2) (1932), [5] M. Petrich, Introduction to Semigroups, Merrill, Columbus, [6] F. M. Siomon, Ideal theory in ternary semigroups, Math. Jap., 10 (1965), [7] O. Steinfeld, Quasi-ideals in rings and semigroups, Akadémiai Kiadó, Budapest, [8] J. Los, On the extending of models I, Fundam. Math., 42 (1995), Received: October, 2011