Original Article

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=x²(xpp)=x²(qxxpp)=x²(qxp)=x²q(qxx)p=x² q²(xxp)=x² q²x=x² q²qxx=x² q³x² 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

1. Los J. On the extending of models I. Fund Math. 1955;42:38-54.
2. Sioson FM. Ideal theory in ternary semigroups. Math Japonica. 1965;10:63-84.
3. Santiago ML. Some contributions to the study of ternary semigroups and semiheaps. Ph.D. Thesis. University of Madras 1983.
4. Dutta TK, Kar S. On regular ternary semirings. Advances in Algebra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics, World Scientific 2003;343-55.
5. Dutta TK, Kar S. A note on regular ternary semirings. Kyungpook Mathematical Journal 46:357-65.
6. Jayalalitha G, Sarala Y, Srinivasa Kumar B, et al. Filters in ternary semigroups. Int J Chem Sci. 2016;14:3190-4.