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Concepts of Bi-Ternary Semi Groups 
 

1N. Sandhya Rani, *2G. Srinivasa Rao, *3Ch. Ramprasad 
1Research Scholar, Department of Mathematics & Statistics, School of Applied Sciences & Humanities, VFSTR Deemed 

to be University, Vadlamudi, Guntur, A.P., India. Email: n.sandhayarani@rguktrkv.ac.in 

1Mathematics Mentor, RGUKT, R.K.Valley, Idupulapaya, A.P., India. 

2Associate Professor, Department of Mathematics & Statistics, School of Applied Sciences & Humanities, VFSTR 

Deemed to be University, Vadlamudi, Guntur, A.P., India.Email:gsrinulakshmi77@gmail.com 

3Associate Professor, Department of Mathematics, VVIT, Guntur, A.P., India.Email: ramprasadchegu1984@gmail.com 

 

Article History: 

Received: 01-06-2024 

Revised: 03-07-2024 

Accepted: 29-07-2024 

Abstract:  

In this paper we made an attempt to study the algebraic structure of bi -groups, bi- ternary 

semi group ,bi-ternary sub semi group , ideals in bi ternary semi group and discussed some 

of its properties with counter examples. 

Keywords: Group, Bi-groups, Bi-Ternary Semi Groups. 

 

Introduction 

Bi-groups are a particularly useful tool since they provide solutions to a significant difficulty that all 

groups encounter, namely that the union of two subgroups does not create any algebraic framework, 

but they do make a good bi-algebraic framework. The bi-group research was conducted from 1994 to 

1996. Maggu were the initial one to use a syntax for bi-groups. Vasantha Kandaswamy and Meiyappan 

expanded this theory in 1997. Several individuals propose adjustments to some of Maggu's already 

proven outcomes. These conclusions included supplementary bi-group characterisation results. 

However, Vasantha Kandaswamy has lately researched the idea of bi-algebraic framework.Agboola 

and Akinola investigated bi-cosets in a bivector space..I expanded upon it the ternary operation is one 

of the operations in algebraic frame work that I used, and then used the same idea in every way that I 

could. G. Srinivasa Rao et.al[1-2, 13-18] discussed about ternary semirings, ordered ternary semirings 

and gamma semirings and their properties. 

1. Preliminaries: 

Definition 1.1: A non-empty set𝐺and′ ∗ ′is a binary operation on 𝐺 if it satisfies the following 

conditions, then algebraic structure(𝐺,∗) is called a group. 

𝑖) 𝑎, 𝑏 ∈ 𝐺 ⟹ 𝑎 ∗ 𝑏 ∈ 𝐺 

𝑖𝑖) 𝑎 ∗ (𝑏 ∗ 𝑐) = (𝑎 ∗ 𝑏) ∗ 𝑐∀𝑎, 𝑏, 𝑐 ∈ 𝐺 

𝑖𝑖𝑖) If 𝑒 ∈ 𝐺Such that 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎∀𝑎 ∈ 𝐺 

𝑖𝑣) meant for each 𝑎 ∈ 𝐺 present exist part 𝑏 ∈ 𝐺 such with the aim of 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 = 𝑒     

      where𝑏 = 𝑎−1 

mailto:ramprasadchegu1984@gmail.com


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Definition 1.2: A non-empty group 𝐺and ′ ∗ ′ is a dual procedure on 𝐺 if it satisfies the following 

conditions, then algebraic structure(𝐺,∗) be call a semi group. 

𝑖)𝑎, 𝑏 ∈ 𝐺 ⟹ 𝑎 ∗ 𝑏 ∈ 𝐺 

𝑖𝑖)𝑎 ∗ (𝑏 ∗ 𝑐) = (𝑎 ∗ 𝑏) ∗ 𝑐∀𝑎, 𝑏, 𝑐 ∈ 𝐺 

Definition 1.3: A non-empty group be a T. Then 𝑇 be understood to exist a ternary semi group 

from 𝑇 × 𝑇 × 𝑇 → 𝑇 which map [𝑥1𝑥2𝑥3] → 𝑥1𝑥2𝑥3  satisfy the condition:[(𝑥1𝑥2𝑥3)𝑥4𝑥5] =

[𝑥1(𝑥2𝑥3𝑥4)𝑥5] = [𝑥1𝑥2(𝑥3𝑥4𝑥5)] for all 𝑥𝑖 ∈ 𝑇, 𝑖 = 1𝑡𝑜5. 

Definition 1.4: A non-empty set (𝐺, +,∙) with dual action two are ′ + ′ & ′ ∙ ′ be call a bi-group if 

here be real two suitable sub sets 𝐺1 & 𝐺2 of 𝐺 follow  

𝑖) 𝐺 = 𝐺1 ∪ 𝐺2 

𝑖𝑖)  (𝐺1+)is a group. 

𝑖𝑖𝑖)(𝐺1,⋅) is group. 

Example 1.5: Let 𝐺 = {set of integers } ∪ {𝑖, −𝑖} be a bi group. 

Definition 1.6:A non-empty set (𝐺, +,∙) with dual action two are ′ + ′ & ′ ∙ ′ is call a bi semi group 

if here be real two suitable sub sets 𝐺1 & 𝐺2 of 𝐺 as follow 

𝑖)𝐺 = 𝐺1 ∪ 𝐺2 

𝑖𝑖)(𝐺1, +) be semi group. 

𝑖𝑖𝑖)(𝐺1,⋅) be semi group. 

Example 1.7: let 𝐺 = (𝑍+, +) ∪ ({1, −1, 𝑖, −𝑖},∙) is a bi semi group. 

2. Main Results: 

Definition 2.1:  A non-empty set 𝑇 = 𝑇1 ∪ 𝑇2 be a bi ternary semi group. Then both 𝑇1 and 𝑇2 are 

satisfy the conditions of ternary semi group. i.e.  [(𝑥1𝑥2𝑥3)𝑥4𝑥5] = [𝑥1(𝑥2𝑥3𝑥4)𝑥5] =

[𝑥1𝑥2(𝑥3𝑥4𝑥5)] ∀𝑥𝑖 ∈ 𝑇1 

Similarly [(𝑎𝑏𝑐)𝑑𝑒] = [𝑎(𝑏𝑐𝑑)𝑒] = [𝑎𝑏(𝑐𝑑𝑒)]∀𝑎, 𝑏, 𝑐, 𝑑, 𝑒 ∈ 𝑇2 

set of non-empty is 𝑇 with ternary growth be a mark of  [  ] is said to be a Bi-Ternary semi group if 

𝑇 = 𝑇1 ∪ 𝑇2 someplace 𝑇1 & 𝑇2 be proper subsets of 𝑇 such that  

𝑖)𝑇1 is ternary semi group. 

𝑖𝑖)𝑇2 is ternary semi group. 

Example 2.2: Let 𝑇 = {𝑖, −𝑖} ∪ {[
0 0
0 0

] , [
1 0
0 0

] , [
1 0
0 1

] , [
0 1
0 0

] , [
0 0
1 0

] , [
0 0
0 1

]}  then 𝑇  is bi 

ternary semi group under complex multiplication and matrix multiplication. 

Example 2.3: Let 𝑇 = ∁ ∪ 𝑍 is  of bi ternary semi group. 

Example 2.4: Let 𝑇 = 𝐶0
− ∪ 𝑍 is of bi ternary semi group. 



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Example 2.5: Let 𝑇 = 𝑇1 ∪ 𝑇2 where if   𝑇1 = (𝐶,∙) and   𝑇2 = ( 3𝒁 , [⬚]) are twoternary semi group 

so that 𝑇 is Bi ternary semi group. 

Definition 2.6: A nonempty sub set 𝑆 = 𝑆1 ∪ 𝑆2 of a bi ternary semi group  𝑇 be understood to be a bi 

ternary sub semi group, when if both 𝑆1&𝑆2  satisfy the condition: 𝑎1𝑏1𝑐1∀𝑎1, 𝑏1, 𝑐1 ∈ 𝑆1 

𝑎2𝑏2𝑐2∀𝑎2, 𝑏2, 𝑐2 ∈ 𝑆2 

Note 2.7: A non-blank sub group 𝑆 = 𝑆1 ∪ 𝑆2 of a bi-digit group 𝑇 be tell to be a bi ternary sub semi 

group if and only if𝑆1𝑆1𝑆1 ⊆ 𝑆1 

𝑆2𝑆2𝑆2 ⊆ 𝑆2 

Example 2.8: Let 𝑇 = 𝑇1 ∪ 𝑇2  where   𝑇1 = (𝐶,∙) and   𝑇2 = (𝒁 , [⬚]) are two ternary semi groups. 

Let 𝑆 = 𝑆1 ∪ 𝑆2  where 𝑆1 = (𝑪0
−, [⬚])  and   𝑆2 = ( 3𝒁 , [⬚])  are two ternary sub semi-groups of 

𝑇.So,𝑆 ⊂ 𝑇 is ternary sub semi group of 𝑇. 

Theorem 2.9: The non-empty intersection of two Bi-ternary sub semi groups of a Biternary semi 

group 𝑇 is a Bi-ternary sub semi-group of  𝑇. 

Proof. : Let 𝑆1 ,   𝑆2 be two Bi-ternary sub semi groups of  𝑇. let 𝑎, 𝑏, 𝑐 ∈ 𝑆1 ∩ 𝑆2 

𝑎, 𝑏, 𝑐 ∈ 𝑆1 ∩ 𝑆2 Then  𝑎, 𝑏, 𝑐 ∈ 𝑆1 𝑎𝑛𝑑 𝑎, 𝑏, 𝑐 ∈ 𝑆2 

𝑎, 𝑏, 𝑐 ∈ 𝑆1 , 𝑆1 is a Bi-ternary sub semi group of  𝑇  , 𝑎𝑏𝑐 ∈ 𝑆1 

𝑎, 𝑏, 𝑐 ∈ 𝑆2 , 𝑆2 is a Bi-ternary sub semi group of  𝑇, 𝑎𝑏𝑐 ∈ 𝑆2 

So, that𝑎𝑏𝑐 ∈ 𝑆1, 𝑎𝑏𝑐 ∈ 𝑆2 then 𝑎𝑏𝑐 ∈ 𝑆1 ∩ 𝑆2 

Therefore 𝑆1 ∩ 𝑆2 is a Bi-ternary sub semi group of  𝑇. 

Theorem 2.10: The meeting point of some relations of Bi-ternary sub semi groups of  𝑇 is the Bi 

ternary sub semi group of  𝑇. 

Proof : Let {𝑆𝛼}𝛼∈∆ be a family of bi ternary sub semi groups of T, And  𝑆 = ⋂ 𝑆𝛼𝛼∈∆ . 

Let 𝑎, 𝑏, 𝑐 ∈ 𝑆 ⇒ 𝑎, 𝑏, 𝑐 ∈ ⋂ 𝑆𝛼𝛼∈∆ ⇒ 𝑎, 𝑏, 𝑐 ∈ 𝑆𝛼 ∀ 𝛼 ∈ ∆ 

𝑎, 𝑏, 𝑐 ∈ 𝑆𝛼 , 𝑆𝛼 is a bi ternary sub semi group T then 𝑎𝑏𝑐 ∈ 𝑆𝛼 

Now, 𝑎𝑏𝑐 ∈ 𝑆𝛼  , ∀𝛼 ∈ ∆ ⇒ 𝑎𝑏𝑐 ∈ ⋂ 𝑆𝛼𝛼∈∆  

Then   𝑎𝑏𝑐 ∈ 𝑆. Therefore S is a bi ternary sub semi group of  𝑇. 

Theorem 2.11: Let 𝑇 be a bi ternary semi grouping & 𝐴 be a non-empty sub set of 𝑇. Then < 𝐴 >= 

the intersection of all bi ternary sub semi groups of 𝑇containing𝐴. 

Proof : Letting  be a group comprising all bi-ternary primary semi-groups of T containing A. 

bi ternary be T, sub semi group of `𝑇containing𝐴. 𝑇 ∈ ∆ , accordingly ∆≠ ∅ 

Agree 𝑆∗ = ⋂ 𝑆𝑠∈∆  

Giving that 𝐴 ⊆ 𝑆 ∀ 𝑆 ∈ ∆ and 𝐴 ⊆ 𝑆∗ 



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Since 𝑆∗ is the intersection of bi ternary sub semi groups of  𝑇. 

So,𝑆∗ is a bi ternary sub semi group of𝑇. 

Since 𝑆∗ ⊆ 𝑆 for all 𝑆 ∈ ∆, 𝑆∗ is the smallest bi ternary sub semi group of 𝑇 containing 𝐴. 

 Hence 𝑆∗ =< 𝐴 > 

Definition 2.12: A bi-ternary semi group 𝑇 be thought concurrent if for every 𝑎, 𝑏, 𝑐 ∈ 𝑇, then wehave  

𝑎𝑏𝑐 = 𝑏𝑐𝑎 = 𝑐𝑎𝑏 = 𝑎𝑐𝑏 = 𝑏𝑎𝑐 = 𝑐𝑏𝑎 

3.Ideals In Biternary Semi Groups: 

Left Ideals in Bi-ternary semi groups 

Definition 3.1: A nonempty sub set 𝐴 = 𝐴1 ∪ 𝐴2 of a bi-ternary semi group 𝑇 be thought concurrent 

left ideal of 𝑇 if both 𝐴1  and 𝐴2  are Left ideals of 𝑇1  and 𝑇2  such that if 𝑏, 𝑐 ∈ 𝑇1, 𝑎 ∈ 𝐴1  implies 

𝑏𝑐𝑎 ∈ 𝐴1 

Similarly, 𝑏, 𝑐 ∈ 𝑇2, 𝑎 ∈ 𝐴2implies 𝑏𝑐𝑎 ∈ 𝐴2. 

Note 3.2: A unfilled auxiliary set called A of a biternary semigroup T is considered that it's left 

idealistic if just if    𝑇𝑇𝐴 ⊆ 𝐴. 

Example 3.3:𝑇 = 𝑇1 ∪ 𝑇2 Be a Bi ternary semi group Where 𝑇1 = {[
𝑎 0
𝑏 𝑐

] , 𝑎, 𝑏, 𝑐  ∈ 𝑍}  ,  

𝑇2 = {[
𝑥 𝑦
0 𝑧

] , 𝑥  , 𝑦, 𝑧 ∈ 𝑍}are ternary semi groups of 𝑇. 𝐼 = 𝐼1 ∪ 𝐼2 where 

𝐼1 = {[
𝑎 0
𝑏 0

] , 𝑎, 𝑏 ∈ 𝑍}, 𝐼2 = {[
𝑎 0
0 0

] , 𝑎  ∈   𝑍}  are left ideals of 𝑇1 ∩ 𝑇2respectively. Thus 𝐼 is 

the left idealistic of Bi ternary semi group 𝑇. 

Theorem 3.4: The intersection of any two left ideals be a bi-ternary semi group 𝑇 be left T is idealistic. 

Proof. : Let 𝐴, 𝐵 left idealistic of two be𝑇. Let 𝑎 ∈ 𝐴 ∩ 𝐵 and𝑏, 𝑐 ∈ 𝑇 . 

If 𝑎  ∈   𝐴  ∩ 𝐵 then 𝑎 ∈ 𝐴 and 𝑎  ∈   𝐵 

𝑎  ∈   𝐴  ;   𝑏, 𝑐 ∈   𝑇 , 𝐴 be left idealistic 𝑇 then 𝑏𝑐𝑎 ∈ 𝐴. 

𝑎 ∈ 𝐵; 𝑏, 𝑐 ∈ 𝑇 , 𝐵 be left idealistic of 𝑇 then 𝑏𝑐𝑎 ∈ 𝐵 

therefore, 𝑏𝑐𝑎 ∈ 𝐴, 𝑏𝑐𝑎 ∈ 𝐵 ⟹ 𝑏𝑐𝑎 ∈ 𝐴 ∩ 𝐵. 

Therefore 𝐴 ∩ 𝐵 is a left ideals of 𝑇. 

Theorem 3.5: T's left ideally suited represents the non-empty intersection of any set of left ideals bi-

ternary semi groups. 

Proof: Let 𝐴𝛼 , 𝛼 ∈ ∆ be group to be left idealistic of 𝑇 & let 𝐴 = ⋂ 𝐴𝛼𝛼∈∆  

Let𝑎 ∈ 𝐴; 𝑏, 𝑐 ∈ 𝑇. Now 𝑎 ∈ 𝐴, 𝑎 ∈ ⋂ 𝐴𝛼𝛼∈∆ ⟹ 𝑎 ∈ 𝐴𝛼 for every𝛼 ∈ ∆. 

𝑎 ∈ 𝐴𝛼; 𝑏, 𝑐 ∈ 𝑇  , 𝐴𝛼 be left idealistic of left of 𝑇 ⟹ 𝑏𝑐𝑎 ∈ 𝐴𝛼 

𝑏𝑐𝑎 ∈ 𝐴𝛼 for all 𝛼 ∈ ∆⟹ 𝑏𝑐𝑎 ∈ ⋂ 𝐴𝛼𝛼∈∆  ⟹ 𝑏𝑐𝑎 ∈ 𝐴. 



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Therefore 𝐴 be left idealistic of bi ternary semi group 𝑇. 

Theorem 3.6: The unification of other two left ideals of bi-ternary semi group 𝑇 is a left ideal of 𝑇. 

Proof: Let 𝐼1, 𝐼2 idealistic left two of bi ternary semi group𝑇. 

Let 𝑎 ∈ 𝐼1 ∪ 𝐼2 ⟹ 𝑎 ∈ 𝐼1 or 𝑎 ∈ 𝐼2 or both and  𝛼, 𝛽 ∈ 𝑇 

𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼1 ⟹ 𝛼𝛽𝑎 ∈ 𝐼1 

𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼2 ⟹ 𝛼𝛽𝑎 ∈ 𝐼2 

𝛼𝛽𝑎 ∈ 𝐼1, 𝛼𝛽𝑎 ∈ 𝐼2 ⟹ 𝛼𝛽𝑎 ∈ 𝐼1 ∪ 𝐼2 

𝛼𝛽𝑎 ∈ 𝐼1 ∪ 𝐼2 

Then𝐼1 ∪ 𝐼2 is left ideal of 𝑇.  

Theorem 3.7: Any family combined of left idealistic of bi-ternary semi group 𝑇 be left ideal of 𝑇. 

Proof.: Agree to 𝐴𝛼 , 𝛼 ∈ ∆ be a related of left ideals of 𝑇 and let 𝐴 = ⋃ 𝐴𝛼𝛼∈∆  

Clearly 𝐴  is a non empty sub set of𝑇. 

Let𝑎 ∈ 𝐴; 𝑏, 𝑐 ∈ 𝑇. Now 𝑎 ∈ 𝐴, 𝑎 ∈ ⋃ 𝐴𝛼𝛼∈∆  then 𝑎 ∈ 𝐴𝛼 for some  𝛼 ∈ ∆ 

𝑎 ∈ 𝐴𝛼; 𝑏, 𝑐 ∈ 𝑇  , 𝐴𝛼 be left idealistic of 𝑇 ⟹ 𝑏𝑐𝑎 ∈ 𝐴𝛼 

𝑏𝑐𝑎 ∈ 𝐴𝛼 for all 𝛼 ∈ ∆⟹ 𝑏𝑐𝑎 ∈ ⋃ 𝐴𝛼𝛼∈∆  ⟹ 𝑏𝑐𝑎 ∈ 𝐴. 

Therefore 𝐴 be left idealistic of bi ternary semi group 𝑇. 

Lateral Ideals in Bi-ternary semi groups: 

Definition 3.8: A non − blank auxiliary set 𝐴 = 𝐴1 ∪ 𝐴2  of a bi-ternary semi group𝑇 be lateral 

ideal of 𝑇  if both 𝐴1  and 𝐴2  are Lateral ideals of 𝑇1  and 𝑇2  such that if 𝑏, 𝑐 ∈ 𝑇1, 𝑎 ∈ 𝐴1  implies 

𝑏𝑎𝑐 ∈ 𝐴1  

Similarly, 𝑏, 𝑐 ∈ 𝑇2, 𝑎 ∈ 𝐴2 implies 𝑏𝑎𝑐 ∈ 𝐴2. 

Note 3.9: A non-empty auxiliary set 𝐴 of a bi-ternary semi group𝑇 be to lateral ideal if and only if  

𝑇𝐴𝑇 ⊆ 𝐴. 

Example 3.10: 𝑇 = 𝑇1 ∪ 𝑇2 Be a Bi-ternary semi group Where𝑇1 = {[
𝑎 0
𝑏 𝑐

] , 𝑎, 𝑏, 𝑐 ∈   𝑍} , 

𝑇2 = {[
𝑥 𝑦
0 𝑧

] , 𝑥, 𝑦, 𝑧  ∈   𝑍} are ternary semi groups of T. 𝐼 = 𝐼1 ∪ 𝐼2 where 

𝐼1 = {[
0 0
𝑏 0

] , 𝑏 ∈ 𝑍}, 𝐼2 = {[
0 𝑎
0 0

] , 𝑎 ∈ 𝑍}  are lateral ideals of 𝑇1&𝑇2respectively. Thus I is the 

lateral Ideal of Bi ternary semi group T. 

Theorem 3.11: The connection by two lateral ideals of a bi-ternary semi group 𝑇 be a lateral idealistic 

of  𝑇. 

Proof: Agree 𝐴, 𝐵 be two lateral perfects of 𝑇.  Be in agreement 𝑎  ∈   𝐴  ∩   𝐵 and  𝑏, 𝑐 ∈ 𝑇 . 

If 𝑎  ∈   𝐴  ∩   𝐵 then 𝑎 ∈ 𝐴 and 𝑎 ∈ 𝐵 



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𝑎 ∈ 𝐴; 𝑏, 𝑐 ∈ 𝑇 , 𝐴 is a lateral perfect of 𝑇 then𝑏𝑎𝑐 ∈ 𝐴. 

𝑎 ∈ 𝐵; 𝑏, 𝑐 ∈ 𝑇 , 𝐵 is a lateral perfect of 𝑇 then 𝑏𝑎𝑐 ∈ 𝐵 

Hence, 𝑏𝑎𝑐 ∈ 𝐴, 𝑏𝑎𝑐 ∈ 𝐵 ⟹ 𝑏𝑎𝑐 ∈ 𝐴 ∩ 𝐵. 

Therefore 𝐴 ∩ 𝐵 is a on the side perfect of 𝑇. 

Theorem 3.12: The non-empty joint of any family of lateral ideals bi ternary semi group 𝑇 is a lateral 

perfect of 𝑇. 

Proof: Agree 𝐴𝛼 , 𝛼 ∈ ∆ be a combined of on the side perfect of 𝑇 and let 𝐴 = ⋂ 𝐴𝛼𝛼∈∆  

Let 𝑎 ∈ 𝐴; 𝑏, 𝑐 ∈ 𝑇. Now 𝑎 ∈ 𝐴, 𝑎 ∈ ⋂ 𝐴𝛼𝛼∈∆ ⟹ 𝑎 ∈ 𝐴𝛼 for every𝛼 ∈ ∆. 

𝑎 ∈ 𝐴𝛼; 𝑏, 𝑐 ∈ 𝑇  , 𝐴𝛼 were perfect on side perfect & 𝑇 ⟹ 𝑏𝑎𝑐 ∈ 𝐴𝛼 

𝑏𝑐𝑎 ∈ 𝐴𝛼 for all 𝛼 ∈ ∆⟹ 𝑏𝑎𝑐 ∈ ⋂ 𝐴𝛼𝛼∈∆  ⟹ 𝑏𝑎𝑐 ∈ 𝐴. 

thus 𝐴 be a lateral idealistic of bi ternary semi group  𝑇. 

Theorem 3.13: The amalgamation by two lateral ideals of bi-ternary semi grouping 𝑇 is a on the side 

perfect of 𝑇. 

Proof : Agree to 𝐼1, 𝐼2 any two on the side perfect of bi-ternary semi-grouping 𝑇. 

Let 𝑎 ∈ 𝐼1 ∪ 𝐼2 ⟹ 𝑎 ∈ 𝐼1 or 𝑎 ∈ 𝐼2 or both and  𝛼, 𝛽 ∈ 𝑇 

𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼1 ⟹ 𝛼𝑎𝛽 ∈ 𝐼1 

𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼2 ⟹ 𝛼𝑎𝛽 ∈ 𝐼2 

𝛼𝑎𝛽 ∈ 𝐼1, 𝛼𝑎𝛽 ∈ 𝐼2 ⟹ 𝛼𝑎𝛽 ∈ 𝐼1 ∪ 𝐼2 

𝛼𝑎𝛽 ∈ 𝐼1 ∪ 𝐼2 

Then𝐼1 ∪ 𝐼2 is lateral ideal of 𝑇. 

Theorem 3.14: The combined be all related of lateral ideals be bi-ternary semi grouping 𝑇 is a lateral 

ideal of 𝑇. 

Proof: Agree to 𝐴𝛼 , 𝛼 ∈ ∆ become a group of transverse ideals. of 𝑇 and let 𝐴 = ⋃ 𝐴𝛼𝛼∈∆  

Clearly 𝐴  is a non-blank sub set of 𝑇. 

 Let 𝑎 ∈ 𝐴; 𝑏, 𝑐 ∈ 𝑇. Now 𝑎 ∈ 𝐴, 𝑎 ∈ ⋃ 𝐴𝛼𝛼∈∆  then 𝑎 ∈ 𝐴𝛼 for some  𝛼 ∈ ∆ 

𝑎 ∈ 𝐴𝛼; 𝑏, 𝑐 ∈ 𝑇  , 𝐴𝛼 is a transverse ideals of 𝑇 ⟹ 𝑏𝑎𝑐 ∈ 𝐴𝛼  

𝑏𝑐𝑎 ∈ 𝐴𝛼 for all 𝛼 ∈ ∆⟹ 𝑏𝑎𝑐 ∈ ⋃ 𝐴𝛼𝛼∈∆  ⟹ 𝑏𝑎𝑐 ∈ 𝐴. 

thus 𝐴 is a lateral ideals of bi ternary semi group 𝑇. 

Right Ideals in Bi-ternary semi groups: 

Definition 3.15: A non-empty sub set 𝐴 = 𝐴1 ∪ 𝐴2  of become a family of lateral ideas.  

A bi-ternary semi-group T is considered the proper optimum of T if equally 𝐴1 and 𝐴2 are right ideals 

of 𝑇1 and 𝑇2 such that if 𝑏, 𝑐 ∈ 𝑇1, 𝑎 ∈ 𝐴1 implies 𝑎𝑏𝑐 ∈ 𝐴1 



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Note 3.16: A non-empty auxiliary set. A of a bi-ternary semi-group T is considered to be correct if 

just if 𝐴𝑇𝑇 ⊆ 𝐴. 

Example 3.17:𝑇 = 𝑇1 ∪ 𝑇2 Be a Bi ternary semi group Where 𝑇1 = {[
𝑎 0
𝑏 𝑐

] , 𝑎, 𝑏, 𝑐 ∈ 𝑍} , 𝑇2 =

{[
𝑥 𝑦
0 𝑧

] , 𝑥, 𝑦, 𝑧  ∈   𝑍} are ternary semi groups of 𝑇. 𝐼 = 𝐼1 ∪ 𝐼2 where 

𝐼1 = {[
𝑎 𝑏
0 0

] , 𝑎, 𝑏  ∈   𝑍  }, 𝐼2 = {[
0 𝑎
0 0

] , 𝑎 ∈ 𝑍}  are right ideals of 𝑇1& 𝑇2respectively. Thus 𝐼is 

the right Ideal of Bi ternary semi group 𝑇. 

Theorem 3.18: T's right ideal is the non-empty confluence of multiple right conceptions of a bi-ternary 

semi group. 

Proof: Agree to 𝐼1, 𝐼2 be two ideals of bi-ternary semi group 𝑇. 

Let 𝑎, 𝑏 ∈ 𝐼1 ∩ 𝐼2&𝛼, 𝛽 ∈ 𝑇 

Also 𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼1 ⟹ 𝛼𝛽𝑎 ∈ 𝐼1 

𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼2 ⟹ 𝛼𝛽𝑎 ∈ 𝐼2 

𝛼𝛽𝑎 ∈ 𝐼1, 𝛼𝛽𝑎 ∈ 𝐼2 ⟹ 𝛼𝛽𝑎 ∈ 𝐼1 ∩ 𝐼2 

Hence   𝐼1 ∩ 𝐼2 is the right ideal of Bi ternary semi group𝑇. 

Theorem 3.19: The non-empty intersections of any group of right conceptions be Bi-ternary semi 

grouping T is a right ideal of T.. 

Theorem 3.20: The amalgamation of neither of the right ideals be bi-digit semi grouping T were right 

picture of T. 

Proof : Agree 𝐼1, 𝐼2 be multiple (two) right ideals of bi ternary semi group𝑇. 

Let 𝑎 ∈ 𝐼1 ∪ 𝐼2 ⟹ 𝑎 ∈ 𝐼1 or 𝑎 ∈ 𝐼2 or both and  𝛼, 𝛽 ∈ 𝑇 

Also𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼1 ⟹ 𝛼𝛽𝑎 ∈ 𝐼1 

𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼2 ⟹ 𝛼𝛽𝑎 ∈ 𝐼2 

𝛼𝛽𝑎 ∈ 𝐼1, 𝛼𝛽𝑎 ∈ 𝐼2 ⟹ 𝛼𝛽𝑎 ∈ 𝐼1 ∪ 𝐼2 

𝛼, 𝛽 ∈ 𝑇, 𝑎 ∈ 𝐼1 ∪ 𝐼2, 𝛼𝛽𝑎 ∈ 𝐼1 ∪ 𝐼2 

Then𝐼1 ∪ 𝐼2 is rightideal of𝑇. 

Definition 3.21: A nonblank set 𝐴 be bi ternary semi group𝑇 is said to be ternary ideal or just an 

perfect of 𝑇 if  

𝑏, 𝑐 ∈ 𝑇, 𝑎 ∈ 𝐴Then , 𝑏𝑐𝑎 ∈ 𝐴 , 𝑏𝑎𝑐 ∈ 𝐴 , 𝑎𝑏𝑐 ∈ 𝐴. 

Definition 3.22: An Ideal 𝐴 = 𝐴1 ∪ 𝐴2 be a Bi-ternary semi group 𝑇  be maximal perfect. Both 

𝐴1&𝐴2  are maximal ideals of 𝑇1&𝑇2  provided that 𝐴1&𝐴2  any good ideals of 𝑇1&𝑇2 . Hence 𝐴 is 

maximal ideal of Bi ternary semi group𝑇. 



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Definition3.23: An ideal 𝐴 be  bi-ternary semi group 𝑇, the Principal Bi ternary ideal generated 

by a if 𝐴1&𝐴2 are ideals of 𝑇1&𝑇2 generated by {𝑎} for some 𝑎 ∈ 𝑇. is denoted by 〈𝑎〉. 

Definition 3.24: An ideal 𝐴 of a bi ternary semi group𝑇 is said to be globally idempotent.If 𝑨𝟑 =

𝑨. 

Definition 3.25:A bi ternary semi group𝑇 is said to be globally idempotent.𝑻𝟑 = 𝑻. 

Theorem 3.26: If 𝑇 is a bi ternary semi group with unity 1 then the union of all proper ideals of 𝑇 is 

the unique maximal ideal of 𝑇. 

Proof: Let 𝑀 be the union of all proper ideals of 𝑇. Since 1 is not an element of any proper ideal of 𝑇. 

1 ∉ 𝑀. Therefore  𝑀 is aproper sub set of 𝑇. By theorem union of all ideals of  𝑇 is an ideal of 𝑇.𝑀 is 

an ideal of 𝑇. Thus 𝑀 is a proper ideal of 𝑇. Since 𝑀 contains all proper ideals of  , 𝑀 is a maximal 

ideal of 𝑇. If 𝑊 is any maximal ideal of  , then 

𝑊 ⊆ 𝑀 ⊆ 𝑇 and hence 𝑊 = 𝑀. Therefore 𝑀is the unique maximal ideal of 𝑇. 

Definition 3.27: Let 𝑇 be a bi ternary semi group and 𝐴 be non empty subset of 𝑇. The smallest left 

ideal of 𝑇containing 𝐴 is call left perfect of 𝑇generated by 𝑨. 

Theorem 3.28: The intersection of all left ideals of T that include A is the left perfect of a bi modal 

semigroup T created by a non-empty auxiliary set A. 

Proof: Agree of ∆ a put the left ideals of 𝑇 contaiing 𝐴. Given that T is a left perfect and includes A, 

𝑇 ∈ ∆. So ∆≠ ∅. Agree 𝑆∗ = ⋂ 𝑆𝑠∈∆ . Seeing as 𝐴 is a resulting put of 𝑆 for all 𝑆 ∈ ∆, 𝐴 ⊆ 𝑆∗. 𝑆∗ is the 

left ideal of 𝑆 . Let 𝑃  be the left ideal of 𝑇  containing𝐴 .𝑃  be the left perfect of 𝑇 . Clearly 𝐴 ⊆

𝑃.Therefore𝑃 ∈ ∆⟹ 𝑆∗ ⊆ 𝑃 & hence 𝑆∗ be the left perfect of 𝑇containing 𝐴. 

Definition 3.34: Preservative auxiliary semi group 𝑄 of a bi-ternary semiring 𝑇 is called quasi ideal 

of bi ternary semiring T then 𝑄 =  𝑄1  ∪   𝑄2 mutually 𝑄1 & 𝑄2 are quasi standards of  𝑇1&  𝑇2  if 

  𝑄1𝑇1 𝑇1 ∩ (  𝑇1𝑄1𝑇 1 +  𝑇1  𝑇1𝑄1𝑇1  𝑇1) ∩   𝑇1  𝑇1𝑄1 ⊆ 𝑄1 

  𝑄2𝑇2 𝑇2 ∩ (  𝑇 2𝑄2𝑇 2 +  𝑇2  𝑇2𝑄2𝑇2  𝑇2) ∩   𝑇2  𝑇2𝑄2 ⊆ 𝑄2. Again 𝑄 is a quasi-ideal of 𝑇. 

Definition 3.35: A apt perfect of 𝑃 = 𝑃1 ∪ 𝑃2 of a commutative bi ternary semi group𝑇 is prime 

ideal if, together 𝑃1  & 𝑃2  were main ideals of 𝑇1&𝑇2  correspondingly if  𝑋1 , 𝑌1, 𝑍1  are ideals 

of 𝑇1and𝑋2, 𝑌2, 𝑍2 are ideals of𝑇2 such that 

𝑋1𝑌1𝑍1 ⊆ 𝑃1 ⟹ 𝑋1 ⊆ 𝑃1𝑜𝑟𝑌1 ⊆ 𝑃1𝑜𝑟 𝑍1 ⊆ 𝑃1 

𝑋2𝑌2𝑍2 ⊆ 𝑃2 ⟹ 𝑋2 ⊆ 𝑃2𝑜𝑟𝑌2 ⊆ 𝑃2𝑜𝑟 𝑍2 ⊆ 𝑃2 

Definition 3.36: A put perfect by 𝑃 = 𝑃1 ∪ 𝑃2  of a commutative bi ternary semi group 𝑇  is 

completely prime ideal if, together 𝑃1  & 𝑃2  are completely prime ideals of 𝑇1&𝑇2  separately 

if 𝑥1 , 𝑦1, 𝑧1 are of 𝑇1and𝑥2, 𝑦2, 𝑧2 are of𝑇2 such that 

𝑥1𝑦1𝑧1 ∈ 𝑃1 ⟹ 𝑥1 ∈ 𝑃1𝑜𝑟𝑦1 ∈ 𝑃1𝑜𝑟 𝑧1 ∈ 𝑃1 

𝑥2𝑦2𝑧2 ∈ 𝑃2 ⟹ 𝑥2 ∈ 𝑃2𝑜𝑟𝑦2 ∈ 𝑃2𝑜𝑟 𝑧2 ∈ 𝑃2 



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Theorem 3.37: Let 𝑇 be a bi ternary semi group with identity. Then T's maximum notions are all 

fundamental concepts. 

Proof: Agree 𝑃 be a ultimate perfect of 𝑇. Let 𝐴, 𝐵, 𝐶 be ideals of 𝑇 such that𝐴𝐵𝐶 ⊆ 𝑃.Suppose that 

𝐴, 𝐵 not contained in 𝑃 . Then 𝐴 ∪ 𝑃 = 𝑇  and 𝐵 ∪ 𝑃 = 𝑇 . As 𝑒 ∈ 𝑇  so 𝑒 ∈ 𝐴 ∪ 𝑃  and  𝑒 ∈ 𝐵 ∪ 𝑃 

implies 𝑒 ∈ 𝐴 or 𝑒 ∈ 𝑃 and 𝑒 ∈ 𝐵 or 𝑒 ∈ 𝑃. Since 𝑒 ∉ 𝑃 so 𝑒 ∈ 𝐴 and  𝑒 ∈ 𝐵 ⇒ 𝐴 = 𝑇 and 𝐵 = 𝑇. 

Now since 𝑒 ∈ 𝑇, So 𝑇𝑇𝐶 = 𝐶  and 𝐶 = 𝑇𝑇𝐶 = 𝐴𝐵𝐶 ⊆ 𝑃 suggests 𝐶 ⊆ 𝑃. Henceforth 𝑃 is a main 

idealistic of 𝑇. 

4.HOMOMARPHISM ON BI TERNARY SEMI GROUP: 

Definition 4.1: Let (𝑆,∙,∗)  𝑤ℎ𝑒𝑟𝑒 𝑆 = (𝑆1,∙) ∪ (𝑆2,∗)  and ( 𝑇,∘,⊗ ) = (𝑇1,∘ ) ∪ (𝑇2,⊗)  be two bi 

ternary semi groups. A plan ∅ = ∅1 ∪ ∅2  on or after 𝑆 to 𝑇  is bi ternary semi group homo-

morphism. If ∅1preserves homomorphism from 𝑆1to𝑇1 , ∅2 preserves homomorphism from 𝑆2to 𝑇2 

such that  

1) ∅(𝑎 ∙  𝑏 ∙  𝑐) = ∅(𝑎)  ∘  ∅( 𝑏 )   ∘ ∅( 𝑐 ) 

2) ∅(𝑎 ∗  𝑏 ∗  𝑐) =  ∅( 𝑎 ) ⨂ ∅ ( 𝑏 )⨂ ∅( 𝑐 ) for all        𝑎, 𝑏, 𝑐 ∈ 𝑆 & ∅(𝑎), ∅(𝑏), ∅(𝑐) ∈ 𝑇. 

Definition 4.2: A bi ternary semi group homomorphism ∅: 𝑆 → 𝑇 is onto homomorphism is called 

Epimorphism  

∅: 𝑆 → 𝑇. 

Definition 4.3: A bi ternary semi group homomorphism ∅: 𝑆 → 𝑇 is one-to-one homomorphism is 

called Monomorphism ∅: 𝑆 → 𝑇. 

Definition 4.4: A bi ternary semi group homomorphism ∅: 𝑆 → 𝑇is both one-to-one and onto (i.e. 

Bijective) homomorphism is called Isomorphism∅: 𝑆 → 𝑇. it says that 𝑆 is isomorphic to 𝑇. 𝑆 ≅ 𝑇. 

Definition 4.5: A bi ternary isomorphism ∅ is defined from same bi ternary semiring onto itself is 

called an Automorphism. 

Definition 4.6: A mapping ∅: 𝑆 → 𝑇 is bi ternary semi group homomorphism the kernel of the 

homomorphism 𝑘𝑒𝑟∅ is defined as 𝑘𝑒𝑟∅ = {𝑒 ∈ 𝑆: ∅(𝑒) = 𝑒′}  Where  𝑒′ ∈ 𝑇 is the identity of 𝑇. 

Theorem 4.7: Let 𝑆 = 𝑆1 ∪ 𝑆2 and𝑇 = 𝑇1 ∪ 𝑇2 be two bi ternary semi groups with ∅ is a bi ternary 

semi group homomorphism from 𝑆 → 𝑇 then 𝑘𝑒𝑟∅ = 𝑘𝑒𝑟∅1 ∪ 𝑘𝑒𝑟∅2 is ideal of the set 𝑆. 

Proof: Given that 𝑆 is a bi ternary semi group. 

Let 𝑘𝑒𝑟∅ = {𝑎 ∈ ∅: ∅(𝑎) = 0′} 

We prove that 𝑘𝑒𝑟∅ is an ideal of  𝑆. 

We first prove 𝑘𝑒𝑟∅ is a non empty set. 

Let us assume 0 is the 0(zero) section of 𝑆, also 0′ is the 0(zero) section of 𝑇. 

Let 0 ∈ 𝑆 ⇒ 0 ∈ 𝑆1 ∪ 𝑆2 ⟹ 0 ∈ 𝑆1& 0 ∈ 𝑆2 ⇒ ∅1(0) = 0′&∅2(0) = 0′ 

 Thus  0 ∈ 𝑘𝑒𝑟∅ .Hence  𝑘𝑒𝑟∅ is non empty. 



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Let 𝑥, 𝑦 ∈ 𝑘𝑒𝑟∅ by the definition of 𝑘𝑒𝑟∅ 

If 𝑥 ∈ 𝑘𝑒𝑟∅ ⇒ ∅(𝑥) = 0′&∅(𝑦) =  0′ 

Let 𝑎, 𝑏 ∈ 𝑘𝑒𝑟∅ & 𝑠 ∈ 𝑆 

Since 𝑎, 𝑏 ∈ 𝑘𝑒𝑟∅ ⇒ ∅(a) = 0′& ∅(b) = 0′ 

Consider,∅(𝑎𝑏𝑠) = ∅(𝑎)∅(𝑏)∅(𝑠) = 0′. 0′. ∅(𝑠) = ∅(𝑠) = 0′ 

∴ 𝑎𝑏𝑠 ∈ 𝑘𝑒𝑟∅ 

Similarly 𝑠𝑎𝑏 ∈ 𝑘𝑒𝑟∅ , 𝑎𝑠𝑏 ∈ 𝑘𝑒𝑟∅  

Hence ∀𝑎, 𝑏 ∈ 𝑘𝑒𝑟∅ &𝑠 ∈ 𝑆 𝑡ℎ𝑒𝑛 𝑎𝑏𝑠 ∈ 𝑘𝑒𝑟∅ , 𝑠𝑎𝑏 ∈ 𝑘𝑒𝑟∅ , 𝑎𝑠𝑏 ∈ 𝑘𝑒𝑟∅ 

So 𝑘𝑒𝑟∅ is an ideal of S. 

Theorem 4.8: The homomorphic image of an ideal is an ideal.  If ∅: 𝑆 → 𝑇  is an bi ternary semi 

group onto homomorphism and 𝐴 is an ideal of 𝑆 then ∅(𝐴) is an ideal  of 𝑇. 

Proof: Perfect 0 & 0′ be the additive identity of 𝑆 & 𝑇 respectively. 

∅(𝐴) = {𝜇′ ∈ 𝑇: ∃𝜇 ∈ 𝑆 𝑠. 𝑡 ∅(𝜇) = 𝜇′} 

0 ∈ 𝐴 𝑠. 𝑡 ∅(0) ∈ ∅(𝐴) ⇒ 0′ ∈ ∅(𝐴) where  ∅(0) = 0′ ∈ 𝑇 

∴ ∅(𝐴) is a non-empty. 

∴  ∅(𝐴) ⊆ 𝑇 

Let 𝑎′ , 𝑏′ ∈ ∅(𝐴)∃𝑎, 𝑏 ∈ 𝐴 

Such that∅(𝑎) = 𝑎′ , ∅(𝑏) = 𝑏′ 

Let 𝑎′ , 𝑏′ ∈ ∅(𝐴)& 𝑡 ∈ 𝑇. 

We prove 𝑎′𝑏′𝑡 ∈ ∅(𝐴) 

Since ∅ is onto ∃𝑠 ∈ 𝑆  𝑠. 𝑡 ∅(𝑠) = 𝑡 

Consider, 𝑎′𝑏′𝑡 = ∅(𝑎)∅(𝑏)∅(𝑠) = ∅(𝑎𝑏𝑠) ∈ ∅(𝐴) 

Since 𝑎, 𝑏 ∈ 𝐴 &𝑠 ∈ 𝑆 ⇒ 𝑎𝑏𝑠 ∈ 𝐴 has a perfect of S. 

∴ 𝑎′𝑏′𝑡 ∈ ∅(𝐴) 

Similarly 𝑡𝑎′𝑏′ ∈ ∅(𝐴)&𝑎′𝑡𝑏′ ∈ ∅(𝐴) 

Hence ∅(𝐴) is the left, on the side, right perfect of 𝑇. 

Hereafter ∅(𝐴) be idealistic  of 𝑇. 

Definition 5.1:Bi ternary semi field: Let 𝑇 is a bi ternary semi ring, 𝑇 said of T is a bi-digit semi field 

condition 𝑇1 is a ternary semi field & 𝑇2 is a ternary semi field everyplace 𝑇 = 𝑇1 ∪ 𝑇2.i.e. equally 𝑇1 

& 𝑇2 stand shifting ternary semi rings thru non zero(0) element has multiplicative inverse. 

 



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