Original Article
Int. J. Chem. Sci, Volume: 15( 4)

Regular Ternary Semigroups

*Correspondence:
Jaya Lalitha G Department of Mathematics, KL University, Guntur, Andhra Pradesh, India, Tel: 040 2354 2127; E-mail: [email protected]

Received Date: May 27, 2017 Accepted Date: August 29, 2017 Published Date: September 04, 2017

Citation: Jaya Lalitha G, Sarala Y, Madhusudhana R. Regular Ternary Semigroups. Int J Chem Sci. 2017;15(4):191

Abstract

Intriguing properties of regular ternary semigroups and completely regular ternary semigroups were discussed in the article.

Keywords

Regular ternary semigroup; Completely regular ternary semigroup

Introduction

Los [1] concentrated a few properties of ternary semigroups and demonstrated that each ternary semigroup can be installed in a semigroup. Sioson [2] concentrated ideal theory in ternary semigroups. He likewise presented the thought of regular ternary semigroups and characterized them by utilizing the thought of quasi ideals. Santiago [3] built up the theory of ternary semigroups and semiheaps. Dutta and Kar [4,5] presented and concentrated the thought of regular ternary semirings. Jayalalitha et al. [6] presented and learned about the filters in ternary semigroups. As of late, various mathematicians have taken a shot at ternary structures. In this paper, we concentrate some intriguing properties of regular ternary semigroups and completely regular ternary semigroups.

Definition 1

An element x in a ternary semigroup T is said to be a regular if Ǝ an element equation [2].

A ternary semigroup is said to be regular if all of its elements are regular.

Theorem 1

The following conditions in a ternary semigroup T are equivalent:

(i) T is regular.

(ii) For any right ideal R, lateral ideal M and left ideal L of T, equation

(iii) For equation

(iv) For equation

Proof

(i) ⇒ (ii) Suppose T is a regular ternary semigroup. Let R, M and L be a right ideal, a lateral ideal and a left ideal of T.

Then clearly, equation. Now for equation we have x=xax for some aequationT . This implies that

equation

Thus, we have equation So we find that equation

Clearly, (ii) ⇒ (iii) and (iii) ⇒ (iv) .

It remains to show that (iv) ⇒ (i) .

Let x equationT .Clearly,equation

Then we have, equation

So we find that x equationxTa and hence there exists an elements aequationT such that x=xax. This implies that x is regular and hence T is regular.

We note that every left and right ideal of a regular ternary semigroup may not be a regular ternary semigroup.

However, for a lateral ideal of a regular ternary semigroup, we have the following result:

Lemma

Every lateral ideal of a regular ternary semigroup T is a regular ternary semigroup.

Proof

Let L be a lateral ideal of regular ternary semigroup T. Then for each x equationL there exists a equationT such that x=xax. Now

x=xax=xaxax=x(axa)x=xpx where p= axa equation L. This implies that L is a regular ternary semigroup.

Definition 2

An ideal A of a ternary semigroup T is said to be a regular ideal if equation for any right idealequation

lateral ideal equation and left ideal equation

Remark 1

From Definition 2, it follows that T is always a regular ideal and any ideal that contains a regular ideal is also a regular ideal.

Now if for any right ideal R, lateral ideal M and left ideal L; RML contains a regular ideal, then equation

Proposition

A ternary semigroup T is a regular ternary semigroup if and only if {0} is a regular ideal of T.

Proof

Let P be the nuclear ideal of a ternary semigroup T. i.e., the intersection of all non-zero ideals of T, Pr is the intersection of all non-zero right ideals of T, Pm is the intersection of all non-zero lateral ideals of T and Pl is the intersection of all non-zero left ideals of T. Now if P={0}, then clearly P=Pr=Pm=Pl.

Theorem 2

Let T be a ternary semigroup and P=Pr=Pm=Pl. Then T is a regular ternary semigroup if and only if P is a regular ideal of T.

Proof

If P=Pr=Pm=Pl={0}, then proof follows from proposition. So we suppose that,

P=Pr=Pm=Pl ≠ {0}. Let T be a regular ternary semigroup. Then from proposition, it follows that {0} is a regular ideal of T.

Now, equation implies that P is a regular ideal of T, by using Remark 1.

Conversely, let P be a regular ideal of T. Then equation for any right ideal equation lateral ideal equation and left ideal equation of T. Since PPP is a right ideal of T and P=Pr, we have equation

Consequently, equation So equation and hence from Theorem 2, it follows that T is a regular ternary semigroup.

Corollary 1

Let T be a ternary semigroup and P=Pr=Pm=Pl. Then T is a regular ternary semigroup if and only if every ideal of T is regular.

Proof

Suppose T is a regular ternary semigroup. Then from Theorem 2, it follows that P is a regular ideal of T. Now P=Pr=Pm=Pl implies that every non-zero ideal of T contains the regular ideal P of T. Consequently, by using Remark 1, we find that every ideal of T is regular.

Conversely, if every ideal of T is regular, then P is a regular ideal of T and hence from Theorem 2, it follows that T is a regular ternary semigroup.

Theorem 3

The following conditions in a ternary semigroup T are equivalent:

(i) A is a regular ideal of T.

(ii) For equation

(iii) For equation

(iv) For each equation for some equation and equation

Proof

(i) ⇒(ii) Suppose A is a regular ideal of T. We note that for x, y, z equation T ,

equation

Now equation (since A is regular).

equationequation

Again equation implies that equation

So we find that equation

ii) ⇒ (iii) Put y=z=x in (ii) we get (iii).

(iii) ⇒ (iv)We first note that equationequation

Similarly we have, equation and equation

Now equation

equation

Since, equation there exists equation and equation such that

equation

(iv) ⇒ (i) Let R, M and L be any right, lateral and left ideal of T respectively such that equation Then clearly,

equation Again, let equation Then by using condition (iv), we have equation for some equation and equation Since equation and hence equation Thus equation Consequently, A is a regular ideal.

Theorem 4

Let A be a regular ideal of a ternary semigroup T. For any right ideal R, lateral ideal M and left ideal L of T, if equation then equation

Proof

Suppose for any right ideal R, lateral ideal M and left ideal L of T, equation where A is a regular ideal of T. Then

equation

Now equationequation [Since A is regular]

equation

From Theorem 4, we have the following results:

Corollary 2

A regular and strongly irreducible ideal of a ternary semigroup T is a prime ideal of T.

Corollary 3

Every regular ideal of a ternary semigroup T is a semi prime ideal of T.

Theorem 5

Proof

Let T be a regular ternary semigroup and A be any ideal of T. Then equation Then there exists equation such that x=xax=xaxax. Since A is an ideal andequation Thusequation

Consequently, equation and hence equation is idempotent.

Conversely, suppose that every ideal of T is idempotent. Let P, Q and R be three ideals of T. Then equationequation This implies that equation Also,equation Again, since equation is an ideal of T,equation Thus equation and hence equation Therefore, by Theorem 2, T is a regular ternary semigroup.

Theorem 6

A ternary semigroup T is left (resp. right) regular if and only if every left (resp. right) ideal of T is completely semiprime.

Proof

Let T be a left regular ternary semigroup and L be any left ideal of T. Suppose equation Since T is left regular, there exists an element equation such that equation Thus L is completely semiprime.

Conversely, suppose that every left ideal of T is completely semiprime. Now for any equation is a left ideal of T. Then by hypothesis, Taa is a completely semiprime ideal of T. Now equation Since Taa is completely semiprime, it follows that equation So there exists an element equation such that a=xaa. Consequently, a is left regular. Since a is arbitrary, it follows that T is left regular.

Equivalently, we can prove the Theorem for right regularity.

Completely Regular Ternary Semigroup

Definition 3

A pair (p, q) of elements in a ternary semigroup T is known as an idempotent pair if pq(pqx)=pqx and (xpq)pq=xpq for all equation [3].

Definition 4

Two idempotent pairs (p, q) and (r, s) of a ternary semigroup T are known as an equivalent, if pqx=rsx and xpq=xrs for all equation [3].

In notation we write (p, q) ~ (r, s).

Definition 5

An element x of a ternary semigroup T is said to be completely regular if Ǝ an element equation idempotent pairs (a, x) and (x, a) are equivalent.

If all the elements of T are completely regular, then T is called completely regular [3].

Definition 6

An element x of a ternary semigroup T is known as a left regular if Ǝ an element equation

Definition 7

7 An element x of a ternary semigroup T is said to be right regular if Ǝ an element equation

Theorem 7

A ternary semigroup T is completely regular then T is left and right regular. equation

Proof

Suppose T is a completely regular ternary semigroup. Let equation Then Ǝ an element equation and the idempotent pairs (x, a) and (a, x) are equivalent i.e., xab=axb and bxa=bax for all equation Now in particular, putting b=x we find that xax=axx and xaa=xax. This implies that equation Hence T is left and right regular.

Theorem 8

A ternary semigroup T is left and right regular then equation

Proof

Suppose that T is both left and right regular. Let equation Then equation and and x=qxx. This implies that

equation

Now x=xxp=x(xxp)p=x2(xpp)=x2(qxxpp)=x2(qxp)=x2q(qxx)p=x2 q2(xxp)=x2 q2x=x2 q2qxx=x2 q3x2 equation Hence equation for all equation

Theorem 9

If T is ternary semigroup equation for all equation then T is completely regular.

Proof

Suppose equation for all equation Then equation

Now equation where equation This implies that T is regular. Also equation This shows that the idempotent pairs (x, b) and (b, x) are equivalent.

Consequently, T is a completely regular ternary semigroup.

Definition 8

A sub semigroup S of a ternary semigroup T is said to be a bi-ideal of T if equation

Theorem 10

A ternary semigroup T is completely regular ternary semigroup if and only if every bi-ideal of T is completely semiprime.

Proof

Let T is a completely regular ternary semigroup. Let P be any bi-ideal of T. Let equation Since T is completely regular, from Theorem 10, it follows that equation This implies that there exists equation such thatequation This shows that P is completely semiprime.

Conversely, assume that every bi-ideal of T is completely semiprime. Since every left and right ideal of a ternary semigroup T is a bi-ideal of T, it follows that every left and right ideal of T is completely semiprime. Consequently, we have from Theorem 6 that T is both left and right regular. Now by using Theorem 9, we find that T is a completely regular ternary semigroup.

Theorem 11

If T is a completely regular ternary semigroup, then every bi-ideal of T is idempotent.

Proof

Let T be a completely regular ternary semigroup and P be a bi-ideal of T. Clearly T is a completely regular ternary semigroup. Let equation Then there exists equation such that p=pxp. This implies that equation and hence equation Also equation Thus we find that P=PTP. Again, we have from Theorem 11 that equation This implies that equation Hence equation Therefore every bi-ideal of P is idempotent.

Conclusion

Ternary structures and their speculation, the purported n-ary structures bring certain expectations up in perspective of their conceivable applications in organic chemistry.

References