^{1}

^{1}

^{2}

^{3}

This part of the paper is the continuotion of paper “Regular Elements of <i>B<sub>X</sub></i> (D) Defined by the Class ∑<sub>1</sub><i>(X,10)-Ⅰ</i>”.

This work is continuation of the paper “Regular Elements of

In the present work our aim is to identify regular elements of thesemigroup

The method used in this part does not differ from the method given in [

We denoted the following semilattices by symbols:

1)

2)

3)

4)

5)

6)

7)

8)

Note that the semilattices 1)-8), which are given by diagram 1-8 of the

Remark that

Lemma 1. Let

1)

2)

3)

4)

5)

6)

7)

8)

Proof. Let

Theorem 1. Let

regular element of this semigroup iff there exist a complete a-isomorphism

・

・

・

・

・

・

・

・

Proof. Let

Lemma 2. Let

regular elements of the semigroup

Now let a binary relation

It is easy to see

then

(see remark page 5 in [

Lemma 3. Let X be a finite set,

Proof. Let

From this equality and by statement b) of Lemma 1 it immediately follows that

□Let binary relation

semilattice D it follows that

It is easy to see

then

(see remark page 5 in [

Lemma 4. Let X be a finite set,

where

Proof. Let

quasinormal representation of a binary relation

this and by the condition

i.e.

definition of the semilattice D we have

Therefore the following equality holds

Now, let

From the last condition it follows that

1)

the equality

2)

and

respectively, i.e.,

Therefore, the equality

nition of the semilattice D we have:

3)

and

respectively, i.e.,

Therefore, the equality

Now, by Equality (2) and by conditions 1), 2) and 3) it follows that the following equality is true

where

□

Lemma 5. Let

Proof. If

From the last condition we have

since

Lemma 6. Let

Proof. Let

Further, let

tively. It is clear that the intersection of elements of the set

1)

definition of the sets

2)

and

the equality

3)

ordered system

systems. Further, let

be such mappings, which satisfy the conditions:

Further, let

nary relation

and satisfy the conditions:

(By suppose

that

Note that the number

□

Therefore, we obtain:

Lemma 7. Let X be a finite set,

Proof. Let

□

Now let binary relation

It is easy to see

then

(see Definition [

Lemma 8. Let X be a finite set,

Proof. Let

Now let binary relation

It is easy to see

then

(see [

Lemma 9. Let X be a finite set,

where

Proof. Let

Lemma 10. Let

Proof. Let

Lemma 11. Let X be a finite set,

e) of Theorem 1. Then

and

Proof. Let

Let f be a binary relation

of the

It is easy to see

Then

(see Definition [

Lemma 12. Let X be a finite set,

Proof. Let

Now let g be a binary relation

diagram 6 of the

It is easy to see

then

(see [

Lemma 13. Let X be a finite set,

Proof. Let

Let h be a binary relation

It is easy to see

Then

(see [

Lemma 14. Let X be a finite set,

Proof. Let

Let us assume that

Theorem 2. Let

Proof. This Theorem immediately follows from ( [

Example 1. Let

Then

We have

Theorem 3. Let

Proof. From ( [

Yasha Diasamidze,Nino Tsinaridze,Neşet Aydn,Ali Erdoğan, (2016) Regular Elements of B_{X} (D) Defined by the Class ∑_{1}(X,10)-Ⅱ. Applied Mathematics,07,894-907. doi: 10.4236/am.2016.79079