This work is continuation of the paper “Regular Elements of Defined by the Class” whose sections are labelled 1-2-3. In the introduction and the second section, some definitions and well known results are stated with references. In Section 3 we give a full description of regular elements of the semigroup when an empty set is not included in D and

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

The method used in this part does not differ from the method given in [1] .

We denoted the following semilattices by symbols:

1), where (see diagram 1 of the Figure 1);

2) where (see diagram 2 of the Figure 1);

3) where and (see diagram 3 of the Figure 1);

4) where and (see diagram 4 of the Figure 1);

5) where, , , (see diagram 5 of the Figure 1);

6) where, , , (see diagram 6 of the Figure 1);

7), where, , , , , (see diagram 7 of the Figure 1);

8), where, , , , , , (see diagram 8 of the Figure 1);

Note that the semilattices 1)-8), which are given by diagram 1-8 of the Figure 1 always are XI-semilattices (see [2] , Lemma 1.2.3).

Remark that

Lemma 1. Let be an isomorphism between and semilattices, , and. If X is a finite set and and, then the following equalities are true:

1)

2)

3)

4)

5)

6)

7)

8)

Proof. Let. Then given Lemma immediately follows from ( [1] , Lemma 3). □

Theorem 1. Let and. Then a binary relation

of the semigroup whose quasinormal representation has a form will be a

regular element of this semigroup iff there exist a complete a-isomorphism of the semilattice on some subsemilattice of the semilattice D which satisfies at least one of the following conditions:

・

・ , for some and which satisfies the condition;

・ , for some, , and which satis- fies the conditions:, ,;

・ , for some, and which satisfies the conditions:, , , ,;

・ , where, , , and satisfies the conditions:, , ,;

・ , where, , , , and satisfies the conditions:, , , , ,.

・ , where,

, , , and satisfies the conditions:,

, , ,;

・ , where, , , and satisfies the conditions:, , , , ,.

Proof. Let. Then given Theorem immediately follows from ( [1] , Theorem 2). □

Lemma 2. Let and. Let be set of all

regular elements of the semigroup such that each element satisfies the condition a) of Theorem 1. Then.

Now let a binary relation of the semigroup satisfy the condition b) of Theorem 1 (see diagram 2 of the Figure 1). In this case we have, where and. By definition of the semi- lattice D it follows that

It is easy to see and. If

then

(see remark page 5 in [1] ).

Lemma 3. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition b) of Theorem 1. Then

Proof. Let, and. Then quasinormal representation of a binary relation has a form for some and by statement b) of Theorem 1 satisfiesthe conditions and. By definition of the semilattice D we have, i.e., and. It follows that. Therefore the inclusion holds. By the Equality(1) we have

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

□Let binary relation of the semigroup satisfy the condition c) of Theorem 1 (see diagram 3 of the Figure 1). In this case we have, where and. By definition of the

semilattice D it follows that

It is easy to see and. If-1

then

(see remark page 5 in [1] and Theorem 1).

Lemma 4. Let X be a finite set, and. Let

be set of all regular elements of the semigroup such that each element satisfies the condition c) of Theorem 1. Then

where

Proof. Let be arbitrary element of the set and. Then

quasinormal representation of a binary relation has a form for some

, and by statement c) of Theorem 1 satisfies the conditions

, and. By definition of the semilattice D we have. From

this and by the condition, , we have

i.e., where. It follows that, from the last inclusion and by

definition of the semilattice D we have for all, where

Therefore the following equality holds

Now, let, and. Then for the binary relation we have

From the last condition it follows that.

1). Then we have, that. But the inequality

contradicts the condition that representation of binary relation is quasinormal. So,

the equality is true. From last equality and by definition of the semilattice D we have

for all, where

2), , , ,

and are true. Then we have

and

respectively, i.e., or if and only if

Therefore, the equality is true. From last equality and by defi-

nition of the semilattice D we have: for all, where

3), ,

, , and are true. Then we have

and

respectively, i.e., and if and only if

Therefore, the equality is true. From last equality and by definition of the semilattice D we have: for all, where

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

where

□

Lemma 5. Let, , where and. If quasinormal repre- sentation of binary relation of the semigroup has a form for some, and, then iff

Proof. If, then by statement c) of theorem 1 we have

From the last condition we have

since by assumption. On the other hand, if the conditions of (6) holds, then the conditions of (5) follow, i.e.. □

Lemma 6. Let, and X be a finite set. Then the following equality holds

Proof. Let, where. Assume that

and a quasinormal representation of a regular binary relation has a form

for some, and. Then according to Lemma 5, we have

Further, let be a mapping from X to the semilattice D satisfying the conditions for all., and are the restrictions of the mapping on the sets, , respec-

tively. It is clear that the intersection of elements of the set is an empty set, and

. We are going to find properties of the maps, ,.

1). Then by the properties of D we have, i.e., and by

definition of the sets and. Therefore for all. By suppose we have that

, i.e. for some. Therefore for some.

2). Then by properties of D we have, i.e.,

and by definition of the sets, and. Therefore for all. By suppose we have, that, i.e. for some. If. Then

. Therefore by definition of the set and. We have contradiction to

the equality. Therefore for some.

3). Then by definition quasinormal representation binary relation a and by property of D we have

, i.e. by definition of the sets and. Therefore

for all. Therefore for every binary relation there exists

ordered system. It is obvious that for disjoint binary relations there exists disjoint ordered

systems. Further, let

be such mappings, which satisfy the conditions: for all and for some

; for all and for some; for all

. Now we define a map f from X to the semilattice D, which satisfies the condition:

Further, let, , and. Then bi-

nary relation may be represented by

and satisfy the conditions:

(By suppose for some and for some), i.e., by lemma 5 we have

that. Therefore for every binary relation and ordered system

there exists one to one mapping. By Lemma 1 and by Theorem 1 in [1] the number of the mappings are respectively:

Note that the number does not depend on choice of chains

of the semilattice D. Since the number of such different chains of the semilattice D is equal to 15, for arbitrary where, the number of regular elements of the set is equal to

□

Therefore, we obtain:

Lemma 7. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition c) of Theorem 1. Then

Proof. Let. Then the given Lemma immediately follows from Lemma 4 and from the Equalities (3).

□

Now let binary relation of the semigroup satisfy the condition d) of Theorem 1 (see diagram 4 of the Figure 1). In this case we have where and. By de- finition of the semilattice D it follows that

It is easy to see and. If

then

(see Definition [1] , Definition 4 and [1] , Theorem 2).

Lemma 8. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition d) of Theorem 1. Then

Proof. Let Then the given Lemma immediately follows from ( [1] , Lemma 10). □

Now let binary relation of the semigroup satisfy the condition e) of Theorem 1 (see diagram 5 of the Figure 1). In this case we have where and and . By definition of the semilattice D it follows that

It is easy to see and. If

then

(see [1] , Definition 4 and [1] , Theorem 1).

Lemma 9. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition e) of Theorem 1. Then

where

Proof. Let. Then the given Lemma immediately follows from ( [1] , Lemma 13). □

Lemma 10. Let and be arbitrary elements of the set, where, and. Then the following equality holds

Proof. Let. Then the given Lemma immediately follows from definition semilattice D and by ( [1] , Lemma 13). □

Lemma 11. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition

e) of Theorem 1. Then, where

and

Proof. Let. Then the given Lemma immediately follows from Lemma 9 and 10. □

Let f be a binary relation of the semigroup satisfy the condition g) of Theorem 1 (see diagram 7

of the Figure 1). In this case we have where, and

. By definition of the semilattice D it follows that

It is easy to see and. If

Then

(see Definition [1] , Definition 4 and [1] , Theorem 2).

Lemma 12. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition f) of Theorem 1. Then

Proof. Let. Then the given Lemma immediately follows from ( [1] , Lemma 15). □

Now let g be a binary relation of the semigroup satisfy the condition f) of Theorem 1 (see

diagram 6 of the Figure 1). In this case we have, where,

and. By definition of the semilattice D it follows that

It is easy to see and. If

then

(see [1] , Definition 4 and [1] , Theorem 2).

Lemma 13. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition g) of Theorem 1. Then

Proof. Let. Then the given Lemma immediately follows from ( [1] , Lemma 16). □

Let h be a binary relation of the semigroup satisfy the condition h) of Theorem 1 (see diagram 8 of the Figure 1). In this case we have, Where, . By definition of the semilattice D it follows that

It is easy to see and. If

Then

(see [1] , Definition 4 and [1] , Theorem 2).

Lemma 14. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition h) of Theorem 1. Then

Proof. Let. Then the given Lemma immediately follows from ( [1] , Lemma 17). □

Let us assume that

Theorem 2. Let,. If X is a finite set and is a set of all regular elements of the semigroup, then.

Proof. This Theorem immediately follows from ( [1] , Theorem 2) and Theorem 1. □

Example 1. Let,

Then, , , , , , , , and.

We have, , , , , , , , ,.

Theorem 3. Let. Then the set of all regular elements of the semigroup is a subsemigroup of this semigroup.

Proof. From ( [1] , Lemma 2), and by definition of the semilattice D it follows that the diagrams of XI- semilattices have the form of one of the diagrams given ( [1] , Figure 2). Now the given Theorem immediately follows from ( [3] , Theorem 2). □

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