**4.** *M***-RTB Generator**

We present some results on the *M*-right-tri-ideal (RTI) generator based on an ordered Γ-semigroup.

**Definition 12.** *Let* S *be an ordered* Γ*- semigroup.* G+S *is said to be an M-RTI of* S *if it meets the criteria listed below:*


**Definition 13.** *Let* G *be a subset* S *called a M-right tri-basis (RTB) of* S *if it satisfies the following conditions:*


**Theorem 6.** *Let* G *be an M-RTB of* S *and f*1, *f*<sup>2</sup> ∈ G*. If f*<sup>1</sup> ∈ (N (*f*<sup>2</sup> · Γ · *f*2) ∪ *f*<sup>2</sup> · Γ · *f*<sup>2</sup> · Γ · (S · Γ · ... · Γ · S) · Γ · *f*2]*, then f*<sup>1</sup> = *f*2*.*

**Proof.** The proof is the same as in Theorem 2.

**Lemma 6.** *Let* G *be an M-RTB of* S *and f*1, *f*2, *f*3, *f*<sup>4</sup> ∈ G*. If f*<sup>1</sup> ∈ (N (*f*<sup>3</sup> · Γ · *f*2) ∪ *f*<sup>2</sup> · Γ · *f*<sup>4</sup> · (S · Γ · ... · Γ · S) · Γ · *f*3]*, then f*<sup>1</sup> = *f*<sup>2</sup> *or f*<sup>1</sup> = *f*<sup>3</sup> *or f*<sup>1</sup> = *f*4*.*

**Proof.** Theorem 2 leads to the proof.

**Definition 14.** *For any s*1,*s*<sup>2</sup> ∈ S*, s*<sup>1</sup> !*mrt s*<sup>2</sup> ⇐⇒&*s*1'*mrt* + &*s*2'*mrt is called a quasi-order on* S*.*

**Remark 6.** *The order* !*mrt is not a partial order of* S*.*

**Example 4.** *By Example 2,* &*k*4'*mrt* + &*k*6'*mrt and* &*k*6'*mrt* + &*k*4'*mrt but k*<sup>4</sup> = *k*6*. Hence, the relation* !*mrt is not a partial order on* S*.*

If F is an *M*-RTB of S, then &F '*mrt* = S. Let *s* ∈ S. Then, *s* ∈ &F '*mrt* and so *s* ∈ &*f*1'*mrt* for some *f*<sup>1</sup> ∈ F. This implies &*s*'*mrt* + &*f*1'*mrt*. Hence, *s* !*mrt f*1.

**Remark 7.** *If* G *is an M-RTB of* S*, then for any s* ∈ S*, there exists f*<sup>1</sup> ∈ G *such that s* !*mrt f*1*.*

**Lemma 7.** *Let* G *be an M-RTB of* S*. If f*1, *f*<sup>2</sup> ∈ G *such that f*<sup>1</sup> = *f*2*, then neither f*<sup>1</sup> !*mrt f*<sup>2</sup> *nor f*<sup>2</sup> !*mrt f*1*.*

**Proof.** The proof follows from Lemma 3.

**Lemma 8.** *Let* G *be the M-RTB of* S *and f*1, *f*2, *f*<sup>3</sup> ∈ G *and s* ∈ S*.*


**Proof.** The proof follows from Lemma 4.

**Lemma 9.** *Let* G *be the M-RTB of* S*,*


**Proof.** The proof follows from Lemma 5.

**Theorem 7.** *Let* G *be the M-RTB of* S*, if and only if the following conditions are met by* G*.*


**Proof.** Theorem 3 leads to the proof.

**Theorem 8.** *Let* G *be an M-RTB of* S*. Then,* G *is an ordered* Γ*-subsemigroup of* S*, if and only if g*<sup>1</sup> · *π* · *g*<sup>2</sup> = *g*<sup>1</sup> *or g*<sup>1</sup> · *π* · *g*<sup>2</sup> = *g*2*, for any g*1, *g*<sup>2</sup> ∈ G *and π* ∈ Γ*.*

**Proof.** The proof is the same as Theorem 4.
