Original Article

, 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 [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,

(iii) For

(iv) For

**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, . Now for we have *x*=*xax* for some aT . This implies that

Thus, we have So we find that

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

It remains to show that (iv) ⇒ (i) .

Let x T .Clearly,

Then we have,

So we find that x xTa and hence there exists an elements aT 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 L there exists a T such that *x*=*xax*. Now

*x=xax=xaxax=x(axa)x=xpx* where p= axa 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 for any right ideal

lateral ideal and left ideal

**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

**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, P_{r} is the intersection of all non-zero right ideals of T, P_{m} is the intersection of all non-zero lateral ideals of T and P_{l} is the intersection of all non-zero left ideals of T. Now if P={0}, then clearly P=P_{r}=P_{m}=P_{l}.

**Theorem 2**

Let T be a ternary semigroup and P=P_{r}=P_{m}=P_{l}. Then T is a regular ternary semigroup if and only if P is a regular ideal of T.

**Proof**

If P=P_{r}=P_{m}=P_{l}={0}, then proof follows from proposition. So we suppose that,

P=P_{r}=P_{m}=P_{l} ≠ {0}. Let T be a regular ternary semigroup. Then from proposition, it follows that {0} is a regular ideal of T.

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

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

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

**Corollary 1**

Let T be a ternary semigroup and P=P_{r}=P_{m}=P_{l}. 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=P_{r}=P_{m}=P_{l} 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

(iii) For

(iv) For each for some and

**Proof**

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

Now (since A is regular).

Again implies that

So we find that

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

(iii) ⇒ (iv)We first note that

Similarly we have, and

Now

Since, there exists and such that

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

Again, let Then by using condition (iv), we have for some and Since and hence Thus 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 then

**Proof**

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

Now [Since A is regular]

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 Then there exists such that *x*=*xax*=xaxax. Since A is an ideal and Thus

Consequently, and hence is idempotent.

Conversely, suppose that every ideal of T is idempotent. Let P, Q and R be three ideals of T. Then This implies that Also, Again, since is an ideal of T, Thus and hence 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 Since T is left regular, there exists an element such that Thus L is completely semiprime.

Conversely, suppose that every left ideal of T is completely semiprime. Now for any is a left ideal of T. Then by hypothesis, Taa is a completely semiprime ideal of T. Now Since Taa is completely semiprime, it follows that So there exists an element 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 [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 [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 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

**Definition 7**

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

**Theorem 7**

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

**Proof**

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

**Theorem 8**

A ternary semigroup T is left and right regular then

**Proof**

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

Now *x=xxp=x(xxp)p=x ^{2}(xpp)=x^{2}(qxxpp)=x^{2}(qxp)=x^{2}q(qxx)p=x^{2} q^{2}(xxp)=x^{2} q^{2}x=x^{2} q^{2}qxx=x^{2} q^{3}x^{2}* Hence for all

**Theorem 9**

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

**Proof**

Suppose for all Then

Now where This implies that T is regular. Also 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

**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 Since T is completely regular, from Theorem 10, it follows that This implies that there exists such that 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 Then there exists such that p=pxp. This implies that and hence Also Thus we find that *P=PTP*. Again, we have from Theorem 11 that This implies that Hence 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

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- Jayalalitha G, Sarala Y, Srinivasa Kumar B, et al. Filters in ternary semigroups. Int J Chem Sci. 2016;14:3190-4.