SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v16i45.1937research articleSome Properties of Commutative Ternary Right Almost SemigroupsRameshG1MahendranSmahemahendran9785@gmail.com2Associate Professor, Department of Mathematics, Government Arts College (Autonomous), (Affiliated to Bharathidasan University, Tiruchirappalli)Kumbakonam , Tamil Nadu, 612 002IndiaResearch Scholar, Department of Mathematics, Government Arts College (Autonomous), (Affiliated to Bharathidasan University, Tiruchirappalli)Kumbakonam, Tamil Nadu, 612 002India612202316454255282023301020232023
<bold id="s-87bde98df682">Abstract</bold>
Objective/Background: In this paper, the concept of commutative ternary right almost semigroups is introduced. The properties of ternary right almost semigroups and commutative ternary right almost semigroups are also discussed. Finally, regular only and the regularity are also explored in ternary right almost semigroups. Methods: Properties of ternary right almost semigroup have been employed to carry out this research work to obtain all the characterizations of commutative ternary right almost semigroups, regular and normal corresponding to that ternary semigroup. Findings: We call an algebraic structure (S,+,.) is a ternary semigroup if (S,.)is a Semigroup, S is a ternary semigroup under ternary multiplication. Let S be a groupoid. Then it is a right almost semigroup (RA-semigroup), if we have a1a2a3=a3a2a1=a2a1a3, for all ai∈S,1≤i≤3. (i) RA-semigroup - R-cyclic if (a1a2)a3=(a3a1)a2=(a2a3)a1, for all ai∈S,1≤i≤3. (ii) RA-semigroup - L-cyclic if a1(a2a3)=a3(a1a2)=a2(a3a1), for all ai∈S,1≤i≤3. In this ternary structure we try to study commutative ternary semigroups concept and obtain their properties. Novelty:In this study, we define the notion of some properties of commutative ternary right almost semigroups, regular and normal. We also find some of their interesting results.
AMS Subject Classification code: 20M12, 20N10
KeywordsTernary semigroupsTernary right almost semigroupCommutative ternary right almost semigroupsQuasi- commutative ternary right almost semigroupsRegular ternary right semigroups and Normal ternary right almost semigroupsNone
<bold id="s-9a3cfd083fd8">Introduction</bold>
A. Anjaneyulu 1 extended the ideal theory of commutative semigroup to duo semigroup. D.D. Arlderson and E.W. Johnson 2 used the term semigroup to mean a commutative multiplicative semigroup with 0 and 1. The multiplicative theory of ideals in a commutative is a highly developed area of research in ternary semigroup. Ronnason Chinram, Wichayaporn Jantanan, Natee Raikham, Pattarawan Singavananda 3 introduced covered left ideals and covered right ideals of a ternary semigroup. We here study some results of a ternary semigroup containing covered left ideals and give the conditions for every proper left ideal of a ternary semigroup to be a covered left ideal. The theory of ternary algebraic system was introduced by D.H. Lehmer 4 in 1932, but earlier such structures were studied by Kasner who gave the idea of n-ary algebras.5 F.M. Sioson introduced the notion of regular ternary semigroup. Y. Sarala, A. Anjaneyulu and D. Madhusudhana Rao initiated the study of quasi commutative, pseudo commutative and normal ternary semigroups. We define the notion of ternary semigroup to some properties of commutative ternary right almost semigroups, regular and normal. We also find some of their interesting results.
<bold id="s-2da092798876">Methodology</bold>
In this article, some properties of commutative ternary right almost semigroups, some characterizations of the quasi commutative ternary right almost semigroups, regular ternary right almost semigroups and normal ternary right almost semigroups are discussed.
<bold id="s-d8bd28a96bb1">Results and Discussion</bold>
This section the deals preliminary concepts and some basic results of ternary right almost semigroups 6, 7, 8.
Definition3.1. A class S with an operation between triplets of elements is called a triplex if the following postulates hold.
provided a,b,c,d,e and all the expressions belong to S.
Postulate II. If a,b,c∈S, then there is an element xof S such that a.b.x=c.
The number of elements in S is called the order of triplex and is specified, when necessary, by adding one of the postulates:
Postulate III_{1}. S contains 'n' elements.
Postulate III_{2}.Scontains infinitely many elements.
According as III_{1}, or III_{2} holds, the triplex is called finite or infinite.
Definition 3.2. A ternary semigroup is a nonempty set S together with a ternary operation a1,a2,a3→a1a2a3, satisfying the associative law of the first kind
((a1a2a3)(a4a5))=(a1(a2a3a4)a5)=(a1a2(a3a4a5))
for all ai∈S,1≤i≤5.
Definition 3.3.Let S be a groupoid. Then it is a right almost semigroup (RA-semigroup), if we have
a1a2a3=a3a2a1=a2a1a3,
for all ai∈S,1≤i≤3.
(i) RA-semigroup -R-cyclic if (a1a2)a3=(a3a1)a2=(a2a3)a1, for all ai∈S,1≤i≤3.
(ii) RA-semigroup - L-cyclic if a1(a2a3)=a3(a1a2)=a2(a3a1), for all ai∈S,1≤i≤3.
Remark 3.4. A groupoid S is medial if for all ai∈S,1≤i≤4, S satisfies medial (or) bi-symmetry law,
(i.e)(a1a2)a3a4=(a1a3)a2a4,for all ai∈S,1≤i≤4.
Example 3.5. Let 000010011000010000100001.
Then S is a ternary semigroup under usual multiplication.
Example 3.6. Let S={0,1,2,3,4,5} and abc=(a*b)*cfor all a,b,c∈S, where '*' is defined in the following table:
*
0
1
2
3
4
5
0
0
0
0
0
0
0
1
0
1
1
1
1
1
2
0
1
2
2
1
1
3
0
1
1
1
2
2
4
0
1
4
5
1
1
5
0
1
1
1
4
5
Then (S,*) is a ternary semigroup.
Definition3.7. A ternary right almost semigroup S is said to be commutative if
abc=bca=cab=bac=cba=acb for all a,b,c∈S.
A ternary right almost semigroup S is said to be quasi commutative if for any a,b,c∈S, there exists a natural number'n' such that
abc=bnac=bca=cnba=cab=ancb.
Definition3.8.A ternary right almost semigroup S is said to be normal if abS=Sab for all a,b∈S.
Definition3.9. A ternary right almost semigroup S is said to be right pseudo commutative if
abcde=abdec=abecd—abdce=abedc=abced for all a,b,c,d,e∈S.
Example3.10. Let S={a,b,c,d,e} be a set. Define a ternary operation ‘.’ on S. where '.' is defined by the following table:
.
a
b
c
d
e
a
a
a
a
a
a
b
b
a
a
a
a
c
a
a
a
a
a
d
a
a
a
a
a
e
a
b
c
d
e
Then (S,.)is a right pseudo commutative ternary right almost semigroup.
Definition3.11. An element ‘a’ of a ternary right almost semigroup S is said to be right identity if saa=s for all s∈S.
Definition3.12. An element 'a' of ternary right almost semigroup S is said to be identity or unital if saa=s for all s∈S.
Example 3.13. Let Z0- be the set of all non-positive integers. Then with the usual ternary operation ‘.’, Z0- forms a ternary right almost semigroup with the identity element -1.
Theorem 3.14. Any ternary right almost semigroup S has almost one identity.
Note 2. The identity of ternary right almost semigroup is usually denoted by '1' (or) 'e'.
Definition 3.15.9 An element 'a' of a ternary right almost semigroup S is said to be right zero of S if bca=a for all b,c∈S.
A ternary right almost semigroupS is said to be right zero ternary right almost semigroup if every element of Sis right zero element.
Definition3.16. An element 'a' of a ternary right almost semigroup S is said to be zero of S if bca=a for all b,c∈S.
A ternary right almost semigroup S is said to be zero ternary right almost semigroup if every element of S is zero element.
Example3.17. Let 0∈S and ‖S‖ > 2. Then S with the ternary operation '.' defined by
x.y.z =- is ternary right almost semigroup with 0 (zero).
Result 3.18. Any ternary right almost semigroup S has at most one nonzero element.
Definition3.19. An element 'a' of a ternary right almost semigroup S is said to be an idempotent if a3=a.
Note 3. The set of all idempotent elements in a ternary right almost semigroup S is denoted by I(S).
Definition3.20. An element 'a'of a ternary right almost semigroup S is said to be a proper idempotent element provided 'a' is an idempotent and which is not an identity of S when identity exists.
Definition3.21. A ternary right almost semigroup Sis said to be an idempotent ternary right almost semigroup or a ternary band if every element of S is an idempotent.
Definition3.22. A ternary right almost semigroup S is said to be a right cancellative if
xab=yab=>x=y.
Definition3.23. An element 'a' of a ternary right almost semigroup S is said to be a regular if there exists x,y∈S such that axaya=a.
A ternary right almost semigroup Sis said to be a regular ternary right almost semigroup if every element of S is regular.
Result 3.24. Every idempotent element of a ternary right almost semigroup S is regular.
Definition3.25. An element'a' of a ternary right almost semigroup S is said to be a right regular if there exists x,y∈S such that a=xya2.
An element ‘a’ of a ternary right almost semigroup S is said to be intra regular if there exists x,y∈S such that a=xa5y.
Definition3.26. An element 'a' of a ternary right almost semigroup S is said to be completely regular if there exists x,y∈S such that axaya=a and axa=aax=xaa=aya=aay=yaa=axy=yxa=xay=yax.
A ternary right almost semigroup S is said to be completely regular ternary right almost semigroup if every element of Sis completely regular.
Result3.27. Let S be a ternary right almost semigroup and a∈S. If 'a'is a completely regular element in S, then ‘a’ is right regular in S.
Result3.28. If S is a commutative ternary right almost semigroup, then S is a quasi-commutative ternary right almost semigroup.
Result3.29. If S is a quasi-commutative ternary right almost semigroup, then S is a normal ternary right almost semigroup.
Result3.30. Every commutative ternary right almost semigroup S is a normal ternary right almost semigroup.
Result3.31. If S is a commutative ternary right almost semigroup, then S is a pseudo commutative ternary right almost semigroup.
<bold id="s-74d78888e679">Ternary right almost Semigroups Satisfying the Identity abc=ba</bold>
In this section we prove some properties of ternary right almost semigroups satisfying the identity abc=ba10, 11, 12.
Theorem 4.1.A quasi-commutative ternary right almost semigroup S is a commutative ternary right almost semigroup if all elements of S are idempotent.
Proof. Let S be a quasi-commutative ternary right almost semigroup.
Then
abc=bnac-bca=cnba=cab=ancb
for alla,b,c∈S,
where 'n' is a natural number.
Since a∈S⇒a3∈S
⇒aa3=aa⇒a4=a2⇒a5=a3=a⇒a5=a,a7=a,.…
In generally we write this a2n+l=a for n=1,2,3,….
From result 3.21, every idempotent element of S is regular.
Here 'a' is regular. Then there exists x,y∈S such that a=axaya.
Now, we have to prove that S is commutative,
i.e., abc=bca=cab=bac=cba=acb for all a,b,c∈S.
From Equation 1 it is enough to prove that
bnac=bac,cnba=cba,ancb=acb for n=1,2,3,....
Consider bnac=bbn-1ac,
= b2n+1bn-1ac, (since b=b2n+1)
= bb3n-1ac,
= bb3n-2bac,
= bb3n-2b2n+1ac,
= bb3n-2bb2nac,
= bb3n-2bb2n-1bac,
= bxbybac,(x=b3n-2andy=b2n-1)
=bac (Since ‘b’ is regular)
bnac=bac,
Similarly, cnba=cba,ancb=acb.
Therefore, abc=bca=cab=bac=cba=acb for all a,b,c∈S.
Hence, S is a commutative ternary right almost semigroup.
Theorem4.2.If a regular ternary right almost semigroup S satisfies the identity abc=ba for all a,b,c∈S,then S is a commutative ternary right almost semigroup.
Proof.Let S be a regular ternary right almost semigroup. Then for every a∈S, there exists x,y∈S such that a=axaya. Given that S satisfies the identity abc=ba for all a,b,c∈S, we have to prove that S is a commutative ternary right almost semigroup,
From , , , , we get abc=acb=cab=bac=cba=bca implies that abc=bca=cab=bac=cba=acb for all a,b,c∈S.
Therefore, S is a commutative ternary right almost semigroup.
Theorem4.3. If a regular ternary right almost semigroup S satisfies the identity abc=ba for all a,b,c∈S, then S is right regular.
Proof. LetS be a regular ternary right almost semigroup. Then for any a∈S, there exists x,y∈S such that a=axaya.
Given that Ssatisfies the identity abc=ba for all a,b,c∈S,
i.e., S is commutative, We have to prove that S is right regular. i.e., for any a∈S, there exists x,y∈S such that a=xya2.
Consider a=axaya=axyaa(aya=yaa)
=ayxa(xya=yx)=xyaa(ayx=xay)=xya2
a=xya2 for all a∈S. Therefore, S is right regular.
Theorem4.4. If a regular ternary right almost semigroup S satisfies the identity abc=ba for all a,b,c∈S, then S is completely regular.
Proof. Let Sbe a regular ternary right almost semigroup. Then for any a∈S, there exist x,y∈S such that a=axaya. Given that S satisfies the identity abc=ba for all a,b,c∈S,
from theorem 4.2, S is commutative, and we have to prove that S is completely regular,
i.e., if a∈S, then there exist x,y∈S such that a=axaya and
axa=aax=xaa=aya=aay=yaa=axy=yxa=xay=yax,
By the regularity of S we have axaya=a for all a∈S. To prove that
Corollary4.6. If a regular ternary right almost semigroup S satisfies the identity abc=ba for all a,b,c∈S, then S is quasi commutative.
Proof. Let S be a regular ternary right almost semigroup satisfying the identity abc=ba for all a,b,c∈S. Then from theorem 4.2, S is commutative. From result 3.31, every commutative ternary right almost semigroup is a quasi-commutative ternary right almost semigroup. Hence, S is quasi commutative.
Corollary4.7. If a regular ternary right almost semigroup S satisfies the identity abc=ba for all a,b,c∈S, then S is normal.
Proof. Let S be a regular ternary right almost semigroup satisfying the identity abc=ba for all a,b,c∈S. Then from corollary 4.6, S is quasi commutative. From result 3.29, every quasi-commutative ternary right almost semigroup is normal. Hence, S is normal.
Corollary4.8. If a regular ternary right almost semigroup S satisfies the identity abc=ba for all a,b,c∈S, then S is pseudo commutative.
Proof. Let S be a regular ternary right almost semigroup satisfying the identity abc=ba for all a,b,c∈S. Then from theorem 4.2, S is commutative. Again, from result 3.31, every commutative ternary right almost semigroup is a pseudo commutative ternary right almost semigroup. Hence, S is pseudo commutative.
Theorem4.9. If a right pseudo commutative ternary right almost semigroup S satisfies the identity abc=ba for all a,b,c∈S, then Sis commutative.
Proof. Let S be a right pseudo commutative ternary right almost semigroup. Then for all a,b,c∈S such that
abcde=abdec=abecd=abdce=abedc=abced
we have to prove that S is commutative. Since Ssatisfies the identity abc=ba for all
we have that abc=cab=acb=bca=cba=bacimplies that abc=bca=cab=bac=cba=acb for all a,b,c∈S.
Therefore, S is commutative.
Theorem4.10. If a pseudo commutative ternary right almost semigroup S satisfies the identity abc=ba for all a,b,c∈S, then S is commutative.
Proof. The theorem follows from the above three theorems.
Corollary4.11. If a pseudo commutative ternary right almost semigroup S satisfies the identity
abc=ba for all a,b,c∈S, then S is quasi commutative.
Proof. Let S be a pseudo commutative ternary right almost semigroup and S satisfies the identity abc=ba for all a,b,c∈S. Then from theorem 4.10, Sis commutative. From result 3.31, every commutative ternary right almost semigroup is quasi commutative. Hence S is quasi commutative.
Corollary4.12. If a pseudo commutative ternary right almost semigroup S satisfies the identity
abc=ba for all a,b,c∈S, then S is normal.
Proof. Let Sbe a pseudo commutative ternary right almost semigroup satisfying the identity abc=ba for all a,b,c∈S. Then from corollary 4.11, S is a quasi-commutative ternary right almost semigroup. From result 3.29, every quasi-commutative ternary right almost semigroup is normal. Hence, S is normal.
<bold id="s-07d64655ff53">Conclusion</bold>
The properties of ternary semigroups, ternary right almost semigroups, commutative ternary right almost semigroups, regular and normal ternary right almost semigroups were discussed. We also proved that a regular ternary right almost semigroup satisfying the identity abc=ba for all a,b,c∈S, is a commutative ternary right almost semigroup, completely regular, right cancellative, quasi-commutative and pseudo commutative ternary right almost semigroup satisfying the identity abc=ba for all a,b,c∈S, is commutative.
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