CODEN (USA): IJCRGG, ISSN: 0974-4290, ISSN(Online):2455-9555 Vol.10 No.7, pp 87-93, 2017

Abstract : Weakly prime and weakly semiprime ideals in ordered semigroups have been introduced and studied by N. Kehayopulu in [4]. In this paper we introduced and studied the left weakly prime and weakly semiprime ideals of ternary semigroups. We also prove that any left ideal of ternary semigroup is the intersection of all irreducible left weakly prime ideals of ternary semigroup containing it. Key Words: Ternary semigroup, left weakly prime, left weakly semiprime, left strongly prime, irreducible, strongly irreducible, left semi regular.

Weakly prime and weakly semiprime ideals in ordered semigroups have been introduced and studied by

N. Kehayopulu in [4] who gave the characterizations of Weakly prime and weakly semiprime ideals. In this paper, we introduced and studied the Weakly prime and weakly semiprime ideals of ternary semigroups.

MSC 2010 Subject classification: 20M12, 20M17.

Definition 2.1:Let T be a ternary semigroup and . S is called left (lateral, right) weakly prime if for any left(lateral, right) ideals A,B,C of T such that we have or or .

Definition 2.3: Let T be a ternary semigroup and . S is called left (lateral, right) weakly semiprime if

for any left(lateral, right) ideals A of T such that we have .

S is called left (lateral, right) weakly semiprime ideal of T if S left (lateral, right) ideal which is left (lateral, right) weakly semiprime.

Definition 2.4:A ternary semigroup is called a fully left (lateral, right) weakly prime ternary semigroup if all its left (lateral, right) ideals are left (lateral, right) ) weakly prime left (lateral, right) ideals.

A ternary semigroup is called a fully left (lateral, right) weakly semi prime ternary semigroup if all its left (lateral, right) ideals are left (lateral, right) ) weakly semi prime left (lateral, right) ideals.

P. Bindu et al /International Journal of ChemTech Research, 2017,10(7):87-93.

One can easily see that the concepts of the left weakly prime, left strongly prime and the left weakly semiprime are the extension of the concepts of weakly prime and weakly semiprime of ternary semigroup.

the left weakly prime ideals are left

Clearly T is a ternary semigroup. | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

The right ideals of | T are the sets | and T. | ||||||||||

The | left | ideals | of | T | are | the sets and T. | , | , | , | , | , | , |

The ideals of T are the sets and T. The left weakly prime ideals of T are

, , , , , , , and T. The left idealsare not left weakly prime ideals. In fact,

.

The left strongly prime ideals of T are

, , andAll left ideals of Tare left weakly semiprime,so Tis a fully left weakly semiprime ternary semigroup. However , left ideals {r,t}and {p,r,t} are left weakly semiprime ideals but they are not left weakly prime ideals and left strongly prime ideals.

Infact, But

, and,

P. Bindu et al /International Journal of ChemTech Research, 2017,10(7):87-93.

But

, and .We now introduce an irreducible left (lateral, right) ideal, strongly irreducible left (lateral, right) ideal and characterize irreducible, strongly irreducible left (lateral, right) ideals.

Definition2.6: A left (lateral, right) ideal L of a ternary semigroup T is called an irreducible left(lateral, right) ideal if implies either

for every left ideal .

Definition2.7: A left(lateral, right) ideal L of a ternary semigroup T is called an strongly irreducible left(lateral, right) ideal if implies either

or or for every left ideal.

Every strongly irreducible left ideal of a ternary semigroup is an irreducible left ideal but the converse is not

a ternary semigroup may not be a strongly

defined by multiplication and the order below.

All strongly irreducible left ideals are the sets:

andT.

Theorem 2.9 : Every strongly irreducible left weakly semiprime ideal of a ternary semigroup T is left strongly prime ideal and the concepts of left strongly prime ideals and left weakly prime ideals of T are the same.

P. Bindu et al /International Journal of ChemTech Research, 2017,10(7):87-93.

Proof: Let S be a left weakly semiprime ideal of a ternary semigroup T. Let A,B,C be any three left ideals of T such that . By known theorem , we have

.

Since S be a left weakly semiprime ideal,then . AsS is a strongly irreducible left ideal, hence either . Thus S is a left strongly prime ideal.

Obviously, left strongly prime ideals are left weakly semiprime ideals. Conversely, let S be a left weakly semiprime ideal of a ternary semigroup T, A,B,C be any three left ideals of T such that .

Since is a left ideal of Tand .

By hypothesis either or

or , So either or or . Since S is a left weakly prime ideal of T , then either . That is S is a left strongly prime ideal of T.

Theorem 2.10 : Let T be a ternary semigroup. Then the following conditions are equivalent:

i) The left ideals of T form a chain under inclusion. ii) Every left ideal of T is strongly irreducible. iii) Every left ideal of T is irreducible

Let S be a left ideal of T and let A,B,C are three left ideals of T such that . Since the left ideals of T form a chain under inclusion, so we have . Thus either or or .

Hence .implies either . i.e, T is strongly irreducible.

It implies either .

That is, the left ideals Tform a chain under inclusion.

Definition 2.11: An element a of a ternary semigroup T is a left(lateral, right) semi regular element if (a=paqrastau, a=apqarsatu) for some .

Definition 2.12:T is called left(lateral, right) semiregular if all elements of T are left(lateral, right) semiregular.

T is called quasi-completely regular if T is a left or right semiregular.

Equivalently definition for every .

P. Bindu et al /International Journal of ChemTech Research, 2017,10(7):87-93.

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, defined by multiplication and the order below

Clearly T is a ternary semigroup.

Remark 2.14 :Every left weakly semiprime ideal of a ternary semigroup T is left semiregular.

Theorem 2.15 : Every left ideal of a ternary semigroup T is left strongly prime if and only if T is left semi regular and the left ideals of T form a chain under inclusion. Proof :Let any left ideal of T be left strongly prime, then every left ideal of T is left weakly semiprime. So T is

left semiregular.(By remark 2.14).

Now we prove that the left ideal of T form a chain under inclusion. Let A,B,C are any three ideals of T, then

Since every left ideal of T is left strongly prime and is a left ideal of T.

So is left strongly prime.

Conversely, let S be an arbitrary left ideal of T and let A,B,C be three left ideals of T such that

Since T is left semiregular, we have . Since the left ideals of T form a chain under inclusion. So we have . Then either

P. Bindu et al /International Journal of ChemTech Research, 2017,10(7):87-93.

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Hence .implies either . i.e, S is stronglyprime.

Theorem 2.16:Let S be a left weakly prime ideal of a ternary semigroup T with 0, defined by 0.0.a= o.a.0 =

a.0.0 , then (S,a)= {x T/ xaa } is a left weakly prime ideal of T, for any a T\S.

Proof :since S be a left weakly prime ideal of a ternary semigroup T, then obviously 0T, and hence 0 (S,a). Therefore (S,a) . Let x (S,a); b,c T. We have (bcx)aa=b(cxa)a=bc(xaa) S. Infact xaa Sand S is a left ideal of T. Thus bcx (S,a). Therefore (S,a) is a left ideal of T. Let A,B,C be three left ideals of T such that !" (S,a) because Aaa,Baa,Caa are left ideal of T.

Theorem 2.17:Let T be a ternary semigroup with identity element e as e.e.a = e.a.e = a.e.e = a.S is a left weakly prime ideal of T , if J= {a T/aTT

S} is a non empty subset of T , then J is the maximal ideal of T contained in S and J is left weakly prime.Proof :First we prove J= {a T/aTT

S} is an ideal of T. LetTherefore J is an ideal of T.Clearly . Let Kbe an ideal of T such that

HenceJ is the maximal ideal of T contained in S. For all left ideals A,B,C of T such that SinceS is a left weakly prime ideal of T and .So

. IfA J orB J orC J,then are left ideal of T, and or or . The maximality of J implies that .

Then we have J=T impossible. Remark2.18:Every left ideal of a ternary semigroup T is a left semiregular. Theorem2.19:Let T be a ternary semigroup. If T is fully left weakly prime, then T is left semiregular and the

ideals of T form a chain under set inclusion. Proof :Let T be fully left weakly prime and L be any left ideal of T. Thus L = L3 and hence T is a left semiregular. Let A,B,C be three ideals of T , then or

Theorem2.20:If T is a left semiregular ternary semigroup such that the left ideals of T form a chain under inclusion , then every left ideal of T is left weakly prime.

P. Bindu et al /International Journal of ChemTech Research, 2017,10(7):87-93.

Proof :.Let A,B,C be three ideals of T, such that . Since the left ideals of T form a chain, so without loss of generality, we assume that . By T is a left semiregular and , we have

Hence T is left weakly prime.

Theorem2.21:Let T be a ternary semigroup such that the left ideals of T form a chain under set inclusion, then T is fully left weakly prime if and only if T is left semiregular.

Proof :The proof of the theorem follows by theorem 2.19 and 2.20.

Theorem 2.22 : Let a be a left semi regular element of a ternary semigroup T . If S is a left ideal not containing a, then there exist an irreducible left weakly prime ideal Pof T containing Sand not containing a.

Proof :Let be a chain of left ideals of T containing S and not containing a,then

is a left idealof T containing Sand not containing a. Therefore, by Zorn’s lemma, the set of all left ideals of T containing Sand not containing ahas a maximal element M. Suppose , where B,C and D are left ideals of T properly containing M. Then by the choice of M , .Thus =M,

Which is a contradiction. Hence , M is an irreducible left ideal of T. Let be three left ideals such

that ;

M, Mand M.Since are left ideals of T , by the choice of M, we have. sincea be a left semiregular element of T, then

Hence which is a contradiction. Hence M is a left weakly prime ideal of T containing S and not containing a.

We can get easily the following corollary from theorem 2.22

Corollary2.23: Let a be a left semi regular element of a ternary semigroup T. If S is a left ideal not containing

for any odd positive integer n, then there exist an irreducible left weakly prime ideal P of T containing S and not containing for any odd positive integer n.Theorem 2.24:Let T be a left semiregular ternary semigroup. Then any left ideal S of T is the intersection of all irreducible left weakly prime ideals of T containing S .

Proof :Let S be a left ideal of T and be the collection of irreducible left weakly prime ideals of T

containing S , then . For the reverse inclusion, let

then by theorem2.22, then there exists anirreducible left weakly prime ideal P of T containing S but not containing a. Thus Hence, .

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