Regular Elements of the Semigroup B X (D) Defined by Semilattices of the Class Σ2 (X, 8) and Their Calculation Formulas

(1)

http://dx.doi.org/10.4236/am.2015.614199

Regular Elements of the Semigroup B

X

(D)

Defined by Semilattices of the Class Σ

2 (X, 8)

and Their Calculation Formulas

Nino Tsinaridze, Shota Makharadze, Guladi Fartenadze

Department of Mathematics, Faculty of Physics, Mathematics and Computer Sciences, Shota Rustaveli Batumi State University, Batumi, Georgia

Received 13 February 2015; accepted 27 December 2015; published 30 December 2015

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

The paper gives description of regular elements of the semigroup BX(D) which are defined by se-milattices of the class Σ2(X, 8), for which intersection the minimal elements is not empty. When X is

a finite set, the formulas are derived, by means of which the number of regular elements of the semigroup is calculated. In this case the set of all regular elements is a subsemigroup of the semi-group BX(D) which is defined by semilattices of the class Σ2(X, 8).

Keywords

Semilattice, Semigroup, Binary Relation, Regular Element

1. Introduction

An element

α

taken from the semigroup B_X

( )

D is called a regular element of B_X

( )

D , if in B_X

( )

D

there exists an element β such that α β α α  = (see [1][2]).

Definition 1.1. We say that a complete X-semilattice of unions D is an XI-semilattice of unions if it satisfies the following two conditions:

1) ∧

(

D D, t

)

∈D for any t∈D



; 2)

(

, _t

)

t Z

Z D D

∈

=



∧ for any nonempty element Z of D (see ([1], Definition 1.14.2), ([2], Definition 1.14.2)).

Definition 1.2. The one-to-one mapping ϕ between the complete X-semilattices of unions

φ

(

Q Q,

)

and

D′′ is called a complete isomorphism if the condition

( )

1

T D

D T

ϕ ϕ

′∈

′

(2)

set D1 of the semilattice D' (see ([1], Definition 6.3.2), ([2], Definition 6.3.2) or [3]).

Definition 1.3. Let

α

be some binary relation of the semigroup BX

( )

D . We say that the complete iso-morphism ϕ between the complete semilattices of unions Q and D′ is a complete

α

-isomorphism if

1) Q V D=

(

,

α

)

;

2)

ϕ

( )

∅ = ∅ for ∅∈V D

(

,

α

)

and

ϕ

( )

T

α

=T for all T∈V D

(

,

α

)

(see ([1], Definition 6.3.3), ([2], Definition 6.3.3) or [3]).

Theorem 1.1.Let R be the set of all regular elements of the semigroup BX

( )

D . Then the following

state-ments are true:

1) R D

( )

′ ∩R D

( )

′′ = ∅ for any D D′ ′′∈Σ, XI

( )

D and D′≠D′′;

2)

( )

XI

D D

R R D

′∈Σ

′

=

_

;

3) If X is a finite set, then

( )

XI

D D

R R D

′∈Σ

′

=

∑

(see ([1], Theorem 6.3.6) or ([2], Theorem 6.3.6) or[3]).

2. Result

By the symbol Σ₂

(

X,8

)

we denote the class of all X-semilattices of unions whose every element is isomorphic to an X-semilattice of form D=

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁,

}

, where

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

{

}

6 3 1 6 4 1 6 4 2

7 4 1 7 4 2 7 5 1

, , ,

\ , , 7, 6 , 6, 7 , 5, 4 , 4, 5 , 5, 3 , 3, 5 , 4, 3 , 3, 4 , 2,1 , 1, 2

i j

Z Z Z D Z Z Z D Z Z Z D

Z Z i j

⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂

≠ ∅ ∈

 

  

(see [4]).

Now assume that D∈ Σ₂

(

X,8

)

. We introduce the following notation: 1) Q₁=

{ }

T , where

T

∈

D

(see diagram 1 inFigure 1);

2) Q₂=

{

T T, ′

}

, where T T, ′∈D and T⊂T′ (see diagram 2 inFigure 1);

3) Q3 =

{

T T T, ′ ′′,

}

, where T T T, ′ ′′∈, D and T ⊂T′⊂T′′ (see diagram 3 inFigure 1);

4) Q₄=

{

T T T D, ′ ′′, , 

}

, where T T T, ′ ′′∈, D and T ⊂T′⊂T′′⊂D (see diagram 4 in Figure 1); 5) Q₅=

{

T T T T, ′ ′′ ′, , ∪T′′

}

, where T T T, ′ ′′∈, D, T ⊂T′, T⊂T′′ and T′\T′′ ≠ ∅, T′′\T′ ≠ ∅

6) Q₆=

{

T Z Z Z D, ₄, , ′, 

}

, where T∈

{

Z Z7, 6

}

, Z Z, ′∈

{

Z Z2, 1

}

, Z ≠Z′ and Z Z\ ′ ≠ ∅, Z′\Z≠ ∅

7) Q₇ =

{

T T T T, ′ ′′ ′, , ∪T D′′, 

}

, where T T T, ′ ′′∈, D, T⊂T′, T⊂T′′ and T′\T′′ ≠ ∅, T′′\T′ ≠ ∅ (see diagram 7 inFigure 1);

8) Q₈=

{

T T Z Z, ′, ₄, ₄∪T Z D′, ,

}

, where T∈

{

Z Z₇, ₆

}

, T′∈

{

Z Z₅, ₃

}

, T⊂T′_, Z₄∪T′, ,Z∈

{

Z Z₂ ₁

}

,

4

Z ∪T′≠Z, T′\Z₄ ≠ ∅, Z₄\T′ ≠ ∅ and

(

Z4∪T′

)

\Z ≠ ∅, Z \

(

Z4∪T′

)

≠ ∅

9) Q₉=

{

T T T, ′, ∪T′

}

, where T T, ′∈D, T T\ ′ ≠ ∅, T′\T ≠ ∅ and T∩ = ∅T′ (see diagram 9 inFigure 1);

10) Q10=

{

T T T, ′, ∪T T′ ′′,

}

, where T T T, ′ ′′∈, D,

(

T∪T′

)

⊂T′′, T T\ ′ ≠ ∅, T′\T ≠ ∅ and

T∩ = ∅T′ (see diagram 10 inFigure 1);

(3)

12) Q₁₂ =

{

Z Z Z Z Z D₇, ₆, ₄, ₂, ₁, 

}

, where Z₇∩Z₆= ∅ (see diagram 12 inFigure 1);

13) Q₁₃=

{

T T T, ′, ∪T T Z′ ′′, ,

}

, where T T T, ′ ′′, ,Z∈D,

(

T∪T′

)

⊂Z, T′⊂T′′⊂Z,

(

T∪T′

)

\T′′≠ ∅,

(

)

\

T′′ T∪T′ ≠ ∅ and T∩T′′= ∅ (see diagram 13 inFigure 1);

14) Q₁₄=

{

T T Z Z Z D, ′, ₄, , ′, 

}

, where T T Z Z, ′, , ′∈D, T T, ′∈

{

Z Z₇, ₆

}

, T≠T′, Z₄⊂Z′⊂D,

T′⊂ ⊂Z Z′, Z₄ \Z ≠ ∅, Z Z\ ₄ ≠ ∅ and T∩ = ∅Z (see diagram 14 inFigure 1);

15) Q₁₅=

{

T T Z T Z T′, , ₄, ′′, , ′′∪Z D₄,

}

, where T T, ′∈

{

Z Z₇, ₆

}

, T≠T′, T⊂T′′, T′′∈

{

Z Z₅, ₃

}

, Z₄⊂Z,

(

T

′′ ∪

Z

4

)

∪ =

Z

D



, T′′\Z₄≠ ∅, Z₄\T′′ ≠ ∅,

(

T′′ ∪Z4

)

\Z ≠ ∅, Z\

(

T′′ ∪Z4

)

≠ ∅ and

T′∩T′′= ∅ (see diagram 15 inFigure 1);

16) Q₁₆=

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

, where Z₅∩Z₃= ∅ (see diagram 16 inFigure 1).

Denote by the symbol ∑

( ) (

Qi 1, 2,i= ,16

)

the set of all XI-subsemilattices of the semilattice D

isomor-phic to Q_i. Assume that D′∈ ∑

( )

Qi and denote by the symbol R D

( )

′ the set of all regular elements α of the semigroup B_X

( )

D′ , for which the semilattices V D

(

,

α

)

and Q_i are mutually α isomorphic and

(

,

)

i

V D

α

=Q.

Definition 1.4. Let the symbol ∑′XI

(

X D,

)

denote the set of all XI-subsemilattices of the semilattice D.

Let, further, D D, ′∈ ∑′

(

X D,

)

and

ϑ

XI ⊆ ∑′XI

(

X D,

)

× ∑′XI

(

X D,

)

. It is assumed that DϑXID′ if and only if there exists some complete isomorphism ϕ between the semilattices D and D′. One can easily verify that the binary relation ϑ_XI is an equivalence relation on the set ∑′XI

(

X D,

)

.

Let the symbol Q_iϑ_XI denote the ϑ_XI-class of equivalence of the set ∑′XI

(

X D,

)

, where every element is

isomorphic to the X-semilattice Q_i and

( )

i XI i

D Q

R Q R D

ϑ ∗

′∈

′

=



(see ([1], Definition 6.3.5), ([2], Definition 6.3.5) or [5]).

Lemma 1.1. If X be a finite set and Ω

( )

Q =m₀, then the following equalities are true: 1) R Q

( )

₁ =1;

2) R Q

( )

₂ =m₀⋅

(

2T T′\ − ⋅1 2

)

X T\ ′;

[image:3.595.93.478.435.725.2]

3) R Q

( )

₃ =m₀⋅

(

2T T′\ − ⋅1

) (

3T T′′ ′\ −2T T′′ ′\

)

⋅3X T\ ′′;

(4)

4) R Q

( )

₄ =m₀⋅

(

2T T′\ − ⋅1

) (

3T T′′ ′\ −2T T′′ ′\

)

⋅

(

4D T\′′ −3D T\′′

)

⋅4X D\

  

;

5) R Q

( )

₅ = ⋅2 m₀⋅

(

2T T′ ′′\ − ⋅1

) (

2T T′′ ′\ − ⋅1 4

)

X T\( ′∪T′′);

6)

( )

(

4\

)

( )\ 4

(

\ \

) (

\ \

)

\

6 2 0 2 1 2 3 2 3 2 5

X D Z Z Z

Z T Z Z Z Z Z Z Z Z

R Q = ⋅m ⋅ − ⋅ ∩ ′ ⋅ ′ − ′ ⋅ ′ − ′ ⋅



;

7) R Q

( )

₇ = ⋅2 m₀⋅

(

2T T′ ′′\ − ⋅1

) (

2T T′′ ′\ − ⋅1

)

(

5D T\( ′∪T′′) −4D T\( ′∪T′′)

)

⋅5X D\

  

;

8)

( )

(

\

) (

4\

)

(

\( 4 ) \( 4 )

)

\

8 2 0 2 1 2 1 3 2 6

X D

Z Z T Z Z T

T Z Z T

R Q = ⋅m ⋅ ′ − ⋅ ′ − ⋅ ∪ ′ − ∪ ′ ⋅



;

9) R Q

( )

₉ = ⋅2 m₀⋅3X\(T∪T′) ;

10) R Q

( )

₁₀ = ⋅2 m₀⋅

(

4T′′\(T∪T′) −3T′′\(T∪T′)

)

⋅4X T\ ′′ ;

11)

( )

(

\ 4 \ 4

)

(

\ \

)

\

11 2 0 4 3 5 4 5

D Z D Z X D Z Z Z Z

R Q = ⋅m ⋅ − ⋅ − ⋅

  

;

12)

( )

(

1\ 2 1\ 2

) (

2\1 2\1

)

\

12 4 0 4 3 4 3 6

X D

Z Z Z Z Z Z Z Z

R Q = ⋅m ⋅ − ⋅ − ⋅



;

13) R Q

( )

₁₃ =m₀⋅

(

2T′′\(T′∪T) − ⋅1 5

)

X Z\ ;

14)

( )

(

\ 4

)

(

\ \

)

\

14 0 2 1 6 5 6

D Z D Z X D Z Z

R Q =m ⋅ − ⋅ ′ − ′ ⋅

  

;

15)

( )

(

\

)

(

\( 4) \( 4)

)

\

15 0 2 1 4 3 7

X D

Z T Z Z T Z

T Z

R Q =m ⋅ ′′ − ⋅ ′′∪ − ′′∪ ⋅



;

16)

( )

(

5\ 1

) (

3\ 2

)

\

16 2 0 2 1 2 1 8

X D

Z Z Z Z

R Q = ⋅m ⋅ − ⋅ − ⋅



.

Proof. The statements 1)-4) immediately follows from the Theorem 13.1.2 in [1], Theorem 13.1.2 in [2]; the statements 5)-7) immediately follows from the Theorem 13.3.2 in [1], Theorem 13.3.2 in [2]; the statement 8) immediately follows from the Theorem 13.7.5 in [1], Theorem 13.7.5 in [2]; the statements 9)-11) immediately follows from the Theorem 13.2.2 in [1], Theorem 13.2.2 in [2]; the statement 12) immediately follows from the Theorem 13.5.2 in [1], Theorem 13.5.2 in [2]; the statements 13), 14) immediately follows from the Theorem 13.4.2 in [1], Theorem 13.4.2 in [2], the statement 15) immediately follows from the Corollary 13.10.2 in [1]

and the statement 16) immediately follows from the Theorem 2.2 in [4]. The lemma is proved.

Lemma 1.2. Let D=

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

∈ Σ₂

(

X, 8

)

and Z₇∩Z₆ ≠ ∅. Then the following sets exhibit all XI-subsemilattices of the given semilattice D:

1)

{ }

D ,

{ } { } { } { } { } { } { }

Z₁ , Z₂ , Z₃ , Z₄ , Z₅ , Z₆ , Z₇ , (see diagram 1 of the Figure 1);

2)

{

Z Z₇, ₅

} {

, Z Z₇, ₄

} {

, Z Z₇, ₂

} {

, Z Z₇, ₁

}

,

{

Z D₇, 

}

,

{

Z Z₆, ₄

} {

, Z Z₆, ₃

} {

, Z Z₆, ₂

} {

, Z Z₆, ₁

}

,

{

Z D₆,

}

,

{

Z Z5, 2

}

,

{

Z D5,

}

,

{

Z Z4, 2

} {

, Z Z4, 1

}

,

{

Z D4,

}

,

{

Z Z3, 1

}

,

{

Z D3,

} {

, Z D2,

} {

, Z D1,

}

    

, (see diagram 2 of the Figure 1);

3)

{

Z Z Z₇, ₅, ₂

}

,

{

Z Z D₇, ₅, 

}

,

{

Z Z Z₇, ₄, ₂

} {

, Z Z Z₇, ₄, ₁

}

,

{

Z Z D₇, ₄, 

} {

, Z Z D₇, ₂,

} {

, Z Z D₇, ₁, 

}

,

{

Z Z Z6, 4, 2

}

,

{

Z Z D6, 4,

}

,

{

Z Z Z6, 4, 1

}

,

{

Z Z D6, 2,

}

,

{

Z Z Z6, 3, 1

}

,

{

Z Z D6, 3,

} {

, Z Z D6, 1,

}

   

(5)

4)

{

Z Z Z D₇, ₅, ₂, 

} {

, Z Z Z D₇, ₄, ₂, 

} {

, Z Z Z D₇, ₄, ₁,

} {

, Z Z Z D₆, ₄, ₂,

} {

, Z Z Z D₆, ₄, ₁, 

} {

, Z Z Z D₆, ₃, ₁, 

}

5)

{

Z Z Z Z₇, ₅, ₄, ₂

}

,

{

Z Z Z D₇, ₅, ₁, 

} {

, Z Z Z D₇, ₂, ₁,

}

,

{

Z Z Z Z₆, ₄, ₃, ₁

}

,

{

Z Z Z D₆, ₃, ₂, 

} {

, Z Z Z D₆, ₂, ₁,

}

,

{

Z Z Z D4, 2, 1,

}



6)

{

Z Z Z Z D₇, ₄, ₂, ₁,

} {

, Z Z Z Z D₆, ₄, ₂, ₁,

}

, (see diagram 6 of the Figure 1); 7)

{

Z Z Z Z D₇, ₅, ₄, ₂,

} {

, Z Z Z Z D₆, ₄, ₃, ₁, 

}

, (see diagram 7 of the Figure 1); 8)

{

Z Z Z Z Z D7, 5, 4, 2, 1,

} {

, Z Z Z Z Z D6, 4, 3, 2, 1,

}

,

 

(see diagram 8 of the Figure 1);

Proof. The statements 1)-4) immediately follows from the Theorems 11.6.1 in [1], 11.6.1 in [2] or in [5], the statements 5)-7) immediately follows from the Theorems 11.6.3 in [1], 11.6.3 in [2] or in [5] and the statement 8) immediately follows from the Theorems 11.7.2 in [1].

The lemma is proved.

Theorem 2.1. Let D=

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

∈ Σ₂

(

X, 8

)

and Z₇∩Z₆≠ ∅. Then a binary relation

α

of the semigroup BX

( )

D that has a quasinormal representation of the form to be given below is a regular element of this semigroup iff there exist a complete

α

-isomorphism ϕ of the semilattice V D

(

,

α

)

on some subsemilattice D' of the semilattice D that satisfies at least one of the following conditions:

1) α= X×T, where

T

∈

D

;

2) α=

(

Y_Tα×T

) (

∪ Y_Tα′ ×T′

)

, where T T, ′∈D, T⊂T′,

Y

T

,

Y

T

{ }

α α

′

∉ ∅

and satisfies the conditions:

( )

T

Y

α

⊇

ϕ

T

,

Y

_Tα_′

∩

ϕ

( )

T

′

≠ ∅

;

3)

(

YT T

) (

YT T

) (

YT T

)

α α α

α = × ∪ ′× ′ ∪ ′′× ′′ , where ,T T T′ ′′∈, D, T ⊂T′⊂T′′,

Y

T

,

Y

T

,

Y

T

{ }

α α α

′ ′′

∉ ∅

and satis-fies the conditions:

Y

_Tα

⊇

ϕ

( )

T

,

Y

_Tα

∪

Y

_Tα_′

⊇

ϕ

( )

T

′

,

Y

_Tα_′

∩

ϕ

( )

T

′

≠ ∅

,

Y

_Tα_′′

∩

ϕ

( )

T

′′

≠ ∅

;

4) α=

(

Y_Tα×T

) (

∪ Y_Tα′ ×T′

) (

∪ Y_Tα′′×T′′

)

∪

(

Y₀α×D

)



, where T T T, ′ ′′∈, D, T ⊂T′⊂T′′⊂D_,

{ }

0

,

T T T

Y

α

Y

α_′

Y

α_′′

Y

α

∉ ∅

Y

_Tα

⊇

ϕ

( )

T

,

Y

_Tα

∪

Y

_Tα_′

⊇

ϕ

( )

T

′

,

Y

_Tα

∪

Y

_Tα_′

∪

Y

_Tα_′′

⊇

ϕ

( )

T

′′

,

( )

T

Y

α_′

∩

ϕ

T

′

≠ ∅

,

Y

_Tα_′′

∩

ϕ

( )

T

′′

≠ ∅

, Y₀α ∩ϕ

( )

D ≠ ∅;

5) α =

(

Y_Tα×T

) (

∪ Y_Tα′×T′

) (

∪ Y_Tα′′×T′′

)

∪

(

Y_Tα′∪_T′′×

(

T′∪T′′

)

, where T T T, ′ ′′∈, D, T⊂T′ , T⊂T′′ ,

\

T′ T′′ ≠ ∅, T′′\T′ ≠ ∅, Y_Tα,Y_Tα_′,Y_Tα_′′∉ ∅

{ }

Y

_Tα

∪

Y

_Tα_′

⊇

ϕ

( )

T

′

,

( )

T T

Yα∪Yα_′′ ⊇

ϕ

T′′ , Y_Tα_′ ∩

ϕ

( )

T′ ≠ ∅, Y_Tα_′′ ∩

ϕ

( )

T′′ ≠ ∅; 6) α=

(

Y_Tα×T

) (

∪ Y₄α×Z₄

) (

∪ Y_Zα′×Z′

) (

∪ Y_Zα×Z

) (

∪ Y₀α×D

)



, where T∈

{

Z Z₇, ₆

}

, Z Z, ′∈

{

Z Z₂, ₁

}

, Z≠Z′,

\

Z Z′ ≠ ∅, Z′\Z≠ ∅, Y_Tα,Y₄α,Y_Zα_′,Y_Zα∉ ∅

{ }

and satisfies the conditions Y_Tα ⊇

ϕ

( )

T , Y_Tα∪Y₄α ⊇

ϕ

( )

Z₄ ,

( )

4

T Z

Yα∪Yα∪Yα ⊇

ϕ

Z , Y_Tα∪Y₄α∪Y_Zα_′ ⊇

ϕ

( )

Z′ , Y₄α∩

ϕ

( )

Z₄ ≠ ∅, Y_Zα∩

ϕ

( )

Z ≠ ∅, Y_Zα_′∩

ϕ

( )

Z′ ≠ ∅; 7) α =

(

Y_Tα×T

) (

∪ Y_Tα′×T′

) (

∪ Y_Tα′′×T′′

)

∪

(

Y_Tα′∪_T′′×

(

T′∪T′′

)

∪

(

Y₀α×D

)



, where ,T T T′ ′′∈, D , T⊂T′ ,

T⊂T′′, T′\T′′ ≠ ∅ , T′′\T′ ≠ ∅ , Y_Tα,Y_Tα_′,Y_Tα_′′,Y₀α∉ ∅

{ }

and satisfies the conditions Y_Tα∪Y_Tα_′ ⊇

ϕ

( )

T′

( )

T T

Yα∪Yα_′′ ⊇

ϕ

T′′ , YT

( )

T , , YT

( )

T Y₀

( )

D

α _ϕ α _ϕ α _ϕ

′ ∩ ′ ≠ ∅ ′′ ∩ ′′ ≠ ∅ ∩  ≠ ∅;

8)

(

) (

)

(

)

) (

)

(

)

4

4 4 4 0

T T T Z Z

Yα T Yα T Yα Z Yα T Z Yα Z Yα D

α= × ∪ _′× ′ ∪ × ∪ _′∪ × ′∪ ∪ × ∪ ×  , where T∈

{

Z Z₇, ₆

}

,

{

5, 3

}

T′∈ Z Z , T⊂T′, Z₄∪T′, ,Z∈

{

Z Z₂ ₁

}

, Z₄∪T′≠Z, T′\Z₄≠ ∅, Z₄\T′ ≠ ∅,

(

Z₄∪T′

)

\Z≠ ∅,

(

4

)

\

Z Z ∪T′ ≠ ∅, Y_Tα,Y_Tα_′,Y₄α,Y_Zα∉ ∅

{ }

and satisfies the conditions Y_Tα∪Y_Tα_′ ⊇

ϕ

( )

T′ , Y_Tα∪Y₄α ⊇

ϕ

( )

Z₄ ,

( )

4 ,

T Z

(6)

Proof. In this case, when Z₇∩Z₆ ≠ ∅, from the Lemma 1.2 it follows that diagrams 1-8 given in Figure 1

exhibit all diagrams of XI-subsemilattices of the semilattices D, a quasinormal representation of regular elements of the semigroup BX

( )

D , which are defined by these XI-semilattices, may have one of the forms listed above. Then the validity of the statements 1)-4) immediately follows from the Theorem 13.1.1 in [1], Theorem 13.1.1 in [2], the statements 5)-7) immediately follows from the Theorem 13.3.1 in [1], Theorem 13.3.1 in [2] and the statement 8) immediately follows from the Theorem 13.7.1 in [1], Theorem 13.7.1 in [2].

The theorem is proved.

1) Lemm 2.1. Let D=

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

∈ Σ₂

(

X, 8

)

and Z₇∩Z₆≠ ∅. If by

R Q

∗

( )

₁ denoted all regular elements of the semigroup BX

( )

D satisfying the condition 1)of the Theorem 2.1, then

( )

1 8

R∗ Q = .

Proof. According to the definition of the semilattice D we have

{ } { } { } { } { } { } { }

{ }

{

}

1 XI 7 , 6 , 5 , 4 , 3 , 2 , 1 ,

Qϑ = Z Z Z Z Z Z Z D .

Assume that D₁′=

{ }

Z₇ ,D₂′=

{ }

Z₆ ,D₃′=

{ }

Z₅ ,D₄′=

{ }

Z₄ ,D₅′=

{ }

Z₃ ,D₆′=

{ }

Z₂ ,D₇′=

{ }

Z₁ ,D₈′=

{ }

D . Then from Theorem 1.1 we obtain

( )

1

( )

1

( )

2

( )

3

( )

4

( )

5

( )

6

( )

7

( )

8

R∗ Q = R D′ + R D′ + R D′ + R D′ + R D′ + R D′ + R D′ + R D′ . From this and by the statement 1)of Lemma 1.1 we obtain R∗

( )

Q₁ = + + + + + + + =1 1 1 1 1 1 1 1 8.

2) Now let binary relation

α

of the semigroup B_X

( )

D satisfying the condition 2) of the Theorem 2.1. In this case we have Q₂=

{

T T, ′

}

, where T T, ′∈D and T⊂T′. By definition of the semilattice D follows that

{

} {

}

{

}

{

} {

}

{

}

{

}

{

}

{

} {

}

{

}

{

}

{

} {

} { }

}

2 7 5 7 4 7 2 7 1 7 6 4 6 3 6 2 6 1

6 5 2 5 4 2 4 1 4 3 1 3 2 1

, , , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , , , , , , .

XI

Q Z Z Z Z Z Z Z Z Z D Z Z Z Z Z Z Z Z Z D Z Z Z D Z Z Z Z Z D Z Z Z D Z D Z D

θ = 

     

If the equalities

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

1 7 2 6 3 4 4 7 4 5 7 2 6 7 1 7 6 4

8 6 3 9 6 2 10 6 1 11 5 2 12 5 13 4 2

14 4 1 15 7 5 16 3 1 17 3 18 2 19 1

, , , , , , , , , , , , , ,

, , , , , , , , , , , ,

, , , , , , , , , , ,

D Z D D Z D D Z D D Z Z D Z Z D Z Z D Z Z

D Z Z D Z Z D Z Z D Z Z D Z D D Z Z

D Z Z D Z Z D Z Z D Z D D Z D D Z D

′= ′= ′= ′= ′= ′= ′=

′= ′= ′ = ′ = ′ = ′ =

′ = ′ = ′ = ′ = ′ = ′ =

  



  

Then from Theorem 1.1 we obtain:

( )

19

( )

2 1

i i

R∗ Q R D

= ′

=

_

. (2.1)

Lemma 2.2. Let D=

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

∈ Σ₂

(

X, 8

)

and Z₇∩Z₆ ≠ ∅. If

X

is a finite set, then

( )

2

( )

1

( )

2

( )

3

R∗ Q = R D′ + R D′ − R D′ .

Proof. Let D′=

{

Z Z, ′

}

∈Q₂

θ

XI, then Z Z, ′∈D and Z⊂Z′. If

α

∈R D

( )

′ then quasinormal representa-tion of a binary relarepresenta-tion

α

has form α =

(

Y_Tα×T

) (

∪ Y_Tα′ ×T′

)

for some T T, ′∈D, T⊂T′, YT ,YT

{ }

α α

′ ∉ ∅ and by statement 2) of the Theorem 2.1 satisfies the conditions Y_Tα ⊇Z and Y_Tα′ ∩Z′≠ ∅. Since Z7 and

6

Z are minimal elements of the semilattice D, we have Z⊇Z₇ or Z⊇Z₆.

On the other hand, D is maximal elements of the semilattice D, therefore

D



⊇

Z

′

. Hence, in the consi-dered case, only one of the following two conditions is fulfilled:

7

T

Yα ⊇Z and YT D α

′ ∩ ≠ ∅



or YT Z6 α _⊇

and YT D α

′ ∩ ≠ ∅



.

(7)

( )

2

( )

1

( )

2

.

R Q

∗

=

R D

′

∪

R D

′

(2.2) Now, let

α

∈R D

( )

1′ ∩R D

( )

2′ then

7

6

, ;

, .

T T

Y Z Y D Y Z Y D

α α

′

⊇ ∩ ≠ ∅



 (2.3)

Of this we have that YT Z₇ Z₆ Z₄, YT D

α α

′

⊇ ∪ = ∩ ≠ ∅ , i.e.

α

∈R D

( )

₃′ and R D

( )

₁′ ∩R D

( )

₂′ ⊆R D

( )

₃′ . Of the other hand if

α

∈R D

( )

3′ , then YT Z Y4, T D

α α

′

⊇ ∩ ≠ ∅ and the condition (2.3) is hold. Of this follows that

α

∈R D

( )

₁′ ∩R D

( )

₂′ , i.e. R D

( )

₃′ ⊆R D

( )

₁′ ∩R D

( )

₂′ . Therefore the equality

( )

1

( )

2

( )

3

R D′ ∩R D′ =R D′ (2.4) is fulfilled. Now of the equalities (2.2) and (2.4) follows the following equality

( )

2

( )

1

( )

2

( )

1

( )

2

( )

1

( )

2

( )

3 .

R∗ Q = R D′ + R D′ − R D′ ∩R D′ = R D′ + R D′ − R D′

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

∈ Σ₂

(

X, 8

)

and Z₇∩Z₆ ≠ ∅. If X is a finite set, then

( )

(

\ 7

)

\

(

\ 6

)

\

(

\ 4

)

\

2 19 2 1 2 19 2 1 2 19 2 1 2

D Z X D D Z X D D Z X D

R∗ Q = ⋅ − ⋅ + ⋅ − ⋅ − ⋅ − ⋅

     

.

Proof: It is easy to see Φ

(

Q Q₂, ₂

)

=1 and Ω

( )

Q₂ =19, then by statement 2) of the Lemma 1.1 and by Lemma 2.2 we obtain the validity of Lemma 2.3.

3) Let binary relation

α

of the semigroup B_X

( )

D satisfying the condition 3) of the Theorem 2.1. In this case we have Q₃=

{

T T T, ′ ′′,

}

, where T T T, ′ ′′∈, D and T ⊂T′⊂T′′. By definition of the semilattice D fol-lows that

{

}

{

}

{

} {

}

{

} {

}

{

} {

}

{

}

{

}

{

} {

}

{

} {

}}

3 7 5 2 7 5 7 4 2 7 4 1 7 4 7 2 7 1

6 4 2 6 4 1 6 4 6 3 1 6 3 6 2 6 1

5 2 4 2 4 1 3 1

, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .

XI

Q Z Z Z Z Z D Z Z Z Z Z Z Z Z D Z Z D Z Z D

Z Z Z Z Z Z Z Z D Z Z Z Z Z D Z Z D Z Z D

Z Z D Z Z D Z Z D Z Z D

θ =    

   

Now if

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

{

}

1 7 5 2 7 4 3 7 2 4 7 1 5 6 4

6 6 3 7 6 2 8 6 1 9 4 2 10 4 1

11 7 5 2 12 7 4 2 13 7 4 1 14 6 4 2

15 6 4 1

, , , , , , , , , , , , , , ,

, , , , , , , , , , , ,

, ,

D Z Z D D Z Z D D Z Z D D Z Z D D Z Z D D Z Z D D Z Z D D Z Z D D Z Z D D Z Z D D Z Z Z D Z Z Z D Z Z Z D Z Z Z

D Z Z Z

′= ′= ′= ′= ′=

′= ′= ′= ′= ′ =

′ = ′ = ′ = ′ =

′ =

    

{

}

{

}

{

}

16 6 3 1 17 5 2 18 3 1

,D′ = Z Z Z, , ,D′ = Z Z D, , ,D′ = Z Z D, ,  .

Then from Theorem 1.1 we obtain:

( )

18

( )

3 1

i i

R∗ Q R D

= ′

=



. (3.1)

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

∈ Σ₂

(

X, 8

)

and Z₇∩Z₆ ≠ ∅. If X is a finite set, then

( )

3 1 2 3 4 5 6 7 8

9 10 1 3 2 3 2 4

5 7 5 8 6 8

.

R Q R D R D R D R D R D R D R D R D R D R D R D R D R D R D R D R D R D R D R D R D R D R D

∗ ₌ _′ ₊ _′ ₊ _′ ₊ _′ ₊ _′ ₊ _′ ₊ _′ ₊ _′

′ ′ ′ ′ ′ ′ ′ ′

− − − ∩ − ∩ − ∩

′ ′ ′ ′ ′ ′

− ∩ − ∩ − ∩

(8)

quasinormal representation of a binary relation

α

has form α =

(

Y_Tα×T

) (

∪ Y_Tα′ ×T′

) (

∪ Y_Tα′′×T′′

)

for some

, , ,

T T T′ ′′∈D T⊂T′⊂T′′,

Y

_Tα

,

Y

_Tα′

,

Y

_Tα′′

∉ ∅

{ }

and by statement 3) of the Theorem 2.1 satisfies the conditions T

Yα ⊇Z, Y_Tα∪Y_Tα′ ⊇Z′, YT Z α

′ ∩ ′≠ ∅ and YT Z α

′′∩ ′′≠ ∅. By definition of the semilattice D we have 7

Z⊇Z or Z⊇Z₆ and

D



⊇

Z

′′

. Of this and by the conditions Y_Tα ⊇Z, Y_Tα∪Y_Tα′ ⊇Z′, YT Z α

′ ∩ ′≠ ∅, T

Yα′′∩Z′′≠ ∅ we have:

7, , ,

T T T T T

Yα ⊇Z Yα ∪Yα′ ⊇Z′ Yα′ ∩Z′≠ ∅ Yα′′∩ ≠ ∅D or

6, , ,

T T T T T

Yα ⊇Z Yα ∪Yα′ ⊇Z′ Yα′ ∩Z′≠ ∅ Yα′′∩ ≠ ∅D

i.e.

α

∈R D

( )

1′′ or

α

∈R D

( )

2′′ , where D1′′=

{

Z Z D7, ′,

}



and D₂′′=

{

Z Z D₆, ′, 

}

. Hence, using equality (3.1), we obtain

( )

3 8

( )

1

i i

R∗ Q R D

= ′

=

_

. (3.2)

Now we show that the following equalities are true:

( )

1 4 1 5 1 6 1 8

2 5 2 7 2 8 3 4

3 5 3 6 3 8 4 5

4 7 6 7 7 8

1 2 1 2

, , , ,

, , .

R D R D R D R D R D R D R D R D

R D R D R D R D R D R D

R D R D R D R D

′ ∩ ′ = ∅ ′ ∩ ′ = ∅ ′ ∩ ′ = ∅ ′ ∩ ′ = ∅

′ ∩ ′ = ∅ ′ ∩ ′ = ∅ ′ ∩ ′ = ∅

′ ∩ ′ = ′ ∩ ′ ∩

( )

3 1 7 1 3 7

2 6 2 4 6 2 6 8

3 7 9 4 6 4 6 8

5 6 5 6 8 4 8 10

, ,

,

, ,

, .

R D R D R D R D R D R D

R D R D R D R D R D R D R D R D

′ ′ ∩ ′ = ′ ∩ ′ ∩ ′

′ ∩ ′ = ′ ∩ ′ ∩ ′ = ′ ∩ ′ ∩ ′

′ ∩ ′ = ′ ′ ∩ ′ = ′ ∩ ′ ∩ ′

′ ∩ ′ = ′ ∩ ′ ∩ ′ ′ ∩ ′ = ′

(3.3)

For this we consider the following case. a) If

α

∈R D

( )

₁′ ∩R D

( )

₄′ , then

7 5 5

7 1 1

, , , ,

, , , .

T T T T T

Y Z Y Y Z Y Z Y D Y Z Y Y Z Y Z Y D

α α α α α

′ ′ ′′

⊇ ∪ ⊇ ∩ ≠ ∅ ∩ ≠ ∅

 

It follows that Y_Tα∪Y_Tα′ ⊇Z₁∪Z₅ =D and

(

YT YT

)

YT D YT

α α α α

′ ′′ ′′

∪ ∩ ⊇ ∩ ≠ ∅. But the inequality

(

YT YT

)

YT

α α α

′ ′′

∪ ∩ ≠ ∅ contradiction of the condition that representation of binary relation

α

is quazinormal. So, the equality R D

( )

₁′ ∩R D

( )

₄′ = ∅ is hold.

The similar way we can show that the following equalities are hold:

( )

1

( )

6

R D′ ∩R D′ = ∅, R D

( )

1′ ∩R D

( )

8′ = ∅, R D

( )

3′ ∩R D

( )

4′ = ∅, R D

( )

3′ ∩R D

( )

6′ = ∅,

( )

3

( )

8

R D′ ∩R D′ = ∅, R D

( )

₄′ ∩R D

( )

₇′ = ∅, R D

( )

₆′ ∩R D

( )

₇′ = ∅, R D

( )

₇′ ∩R D

( )

₈′ = ∅. b) If

α

∈R D

( )

₁′ ∩R D

( )

₅′ , then

7 5 5

6 4 4

, , , ,

, , ,

T T T T T

α α α α α

′ ′ ′′

⊇ ∪ ⊇ ∩ ≠ ∅ ∩ ≠ ∅

 

It follows that YT Z6 Z7 Z4 α _⊇ _∪ ₌

and YT YT Z4 YT

α α α

′ ′

∩ ⊇ ∩ ≠ ∅. But the inequality YT YT

α α

′

∩ ≠ ∅ contradiction of the condition that representation of binary relation

α

is quazinormal. So, the equality

( )

1

( )

5

R D′ ∩R D′ = ∅ is true.

( )

2

( )

5

R D′ ∩R D′ = ∅, R D

( )

₂′ ∩R D

( )

₇′ = ∅, R D

( )

₂′ ∩R D

( )

₈′ = ∅,

( )

3

( )

5

(9)

c) If

α

∈R D

( )

3′ ∩R D

( )

7′ , then

7 2 2

6 2 2

, , , ,

, , , .

T T T T T

α α α α α

′ ′ ′′

⊇ ∪ ⊇ ∩ ≠ ∅ ∩ ≠ ∅



 (3.4)

It follows that

6 7 4, ,2 2 , ,

T T T T T

Yα ⊇Z ∪Z =Z Yα∪Yα′ ⊇Z Yα′ ∩Z ≠ ∅ Yα′′∩ ≠ ∅D



(3.5)

i.e.,

α

∈R D

( )

₉′ . So, the inclusion R D

( )

₃′ ∩R D

( )

₇′ ⊆R D

( )

₉′ is hold.

Of the other hand, if

α

∈R D

( )

₉′ , then the conditions (3.4) and (3.5) are fulfilled, i.e.

α

∈R D

( )

₃′ ∩R D

( )

₇′ and R D

( )

₉′ ⊆R D

( )

₃′ ∩R D

( )

₇′ . Therefore, the equality R D

( )

₉′ =R D

( )

₃′ ∩R D

( )

₇′ is true.

The similar way we can show that the following equality is hold: R D

( )

₁₀′ =R D

( )

₄′ ∩R D

( )

₈′ . d) If

α

∈R D

( )

₁′ ∩R D

( )

₂′ ∩R D

( )

₃′ , then

7 5 5

7 4 4

7 2 2

, , , ,

, , , .

T T T T T

Y Z Y Y Z Y Z Y D Y Z Y Y Z Y Z Y D Y Z Y Y Z Y Z Y D

α α α α α

′ ′ ′′

⊇ ∪ ⊇ ∩ ≠ ∅ ∩ ≠ ∅

 

 (3.6)

It follows that

7, , ,2 5 4 ,

T T T T T T

Yα ⊇Z Yα ∪Yα′ ⊇Z Yα′ ∩Z ≠ ∅Yα′ ∩Z ≠ ∅ Yα′′∩ ≠ ∅D (3.7)

i.e.,

α

∈R D

( )

1′ ∩R D

( )

2′ . So, the inclusion R D

( )

1′ ∩R D

( )

2′ ∩R D

( )

3′ ⊆R D

( )

1′ ∩R D

( )

2′ is hold.

Of the other hand, if

α

∈R D

( )

1′ ∩R D

( )

2′ , then the conditions (3.6) and (3.7) are fulfilled, i.e.,

( )

1

( )

2

( )

3

R D R D R D

α

∈ ′ ∩ ′ ∩ ′ and R D

( )

1′ ∩R D

( )

′2 ⊆R D

( )

1′ ∩R D

( )

2′ ∩R D

( )

3′ . Therefore, the equality

( )

1

( )

2

( )

3

( )

1

( )

2

R D′ ∩R D′ ∩R D′ =R D′ ∩R D′ is true.

( )

1

( )

7

( )

1

( )

3

( )

7

R D′ ∩R D′ =R D′ ∩R D′ ∩R D′ ,

( )

2

( )

6

( )

2

( )

4

( )

6

( )

2

( )

6

( )

8

R D′ ∩R D′ =R D′ ∩R D′ ∩R D′ =R D′ ∩R D′ ∩R D′ ,

( )

4

( )

6

( )

4

( )

6

( )

8

R D′ ∩R D′ =R D′ ∩R D′ ∩R D′ ,

( )

5

( )

6

( )

5

( )

6

( )

8

R D′ ∩R D′ =R D′ ∩R D′ ∩R D′ .

We have that all equalities of (3.3) are true. Now, by the equalities of (3.2) and (3.3) we obtain the validity of Lemma 3.1.

Lemma 3.2. Let D′=

{

Y Y D, ′, 

}

, D′′=

{

Y Y D₁, ,₁′ 

}

, where Y Y Y Y, ′, ,₁ ₁′∈D Y1⊇Y and Y1′⊇Y′. If

quasi-normal representation of binary relation α of the semigroup BX

( )

D has a form

(

YT T

) (

YT T

)

α α

α= × ∪ ′× ′ ∪

(

Y0 D

)

α_× 

for some T T, ′∈D, T⊂T′⊂D and Y_Tα,Y_Tα_′,Y₀α∉ ∅

{ }

, then

α

∈R D

( )

′ ∩R D

( )

′′ iff

1, , , 1 0

T T T T

Yα ⊇Y Yα∪Yα′ ⊇Y Y′ α′ ∩ ≠ ∅Y′ Yα∩ ≠ ∅D .

Proof. If

α

∈R D

( )

′ ∩R D

( )

′′ , then by statement 3) of the Theorem 2.1 we have

0

1 1 1 0

, , , ;

, , , .

T T T T

Y Y Y Y Y Y Y Y D Y Y Y Y Y Y Y Y D

α α α α α

′ ′

⊇ ∪ ⊇ ∩ ≠ ∅ ∩ ≠ ∅

′ ′

⊇ ∪ ⊇ ∩ ≠ ∅ ∩ ≠ ∅



 (3.8)

Of the last condition we have

1, , , 1 0

T T T T

Yα ⊇Y Yα∪Yα′ ⊇Y Y′ α′ ∩ ≠ ∅Y′ Yα∩ ≠ ∅D



, (3.9) since Y₁⊇Y and Y₁′⊇Y′ by assumption.

Of the other hand, if the conditions of (3.9) are hold, then also hold the conditions of (3.8), i.e.

( )

R D R D

(10)

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

∈ Σ₂

(

X, 8

)

and Z₇∩Z₆ ≠ ∅. If X is a finite set, then the following equalities are hold:

( )

(

)

(

)

( )

(

)

(

)

( )

(

)

(

)

( )

(

)

(

)

( )

2 2

2 5 5 7

2 2

4 7 2 4

1 1

4 7 1 4

2 2

4 6 2 4

\ \ \

\ \

1 3

\ \ \

\ \

2 3

\ \ \

\ \

2 4

\ \ \

\ \

5 7

5

18 2 2 1 3 2 3 ,

D Z D Z X D Z Z Z Z

D Z D Z X D Z Z

Z Z

D Z D Z X D Z Z

Z Z

D Z D Z X D Z Z

Z Z

R D R D R D R D R D R D R D R D R D

′ ∩ ′ = ⋅ ⋅ − ⋅ − ⋅

′

  

( )

(

)

(

)

( )

(

)

(

)

1 1

4 6 1 4

1 1

1 3 3 6

\ \ \

\ \

8

\ \ \

\ \

6 8

18 2 2 1 3 2 3 ,

18 2 2 1 3 2 3 .

D Z D Z X D Z Z

Z Z

D Z D Z X D Z Z Z Z

R D R D R D

′

∩ = ⋅ ⋅ − ⋅ − ⋅

′ ∩ ′ = ⋅ ⋅ − ⋅ − ⋅

  

Proof. Let D′=

{

Y Y D, ′, 

}

,D′′=

{

Y Y D₁, ,₁′ 

}

∈

{

D D₁′ ′, ₂,,D₈′

}

, where Y₁⊇Y and Y₁′⊇Y′. Assume that

( )

R D R D

α

∈ ′ ∩ ′′ and a quasinormal representation of a regular binary relation α has the form

(

YT T

) (

YT T

)

(

Y0 D

)

α α α

α= × ∪ _′× ′ ∪ ×  for some T T, ′∈D, T⊂T′⊂D and

Y

_Tα

,

Y

_Tα_′

,

Y

₀α

∉ ∅

{ }

. Then by statement c) of the Theorem 3.1.1, we have

1, , , 1 0

T T T T

Yα ⊇Y Yα∪Yα′ ⊇Y Y′ α′ ∩ ≠ ∅Y′ Yα∩ ≠ ∅D (3.10)

Let f_α is a mapping of the set X in the semilattice D satisfying the conditions f_α

( )

t =t

α

for all t∈X.

0

f_α, f₁_α, f₂_α and f₃_α are the restrictions of the mapping f_α on the sets Y₁, Y Y₁′\ ₁, D Y\ ₁′, X D\  re-spectively. It is clear, that the intersection disjoint elements of the set

{

Y Y Y D Y X₁, \₁′ ₁, \ ₁′, \D

}

is empty set, and Y₁∪

(

Y Y₁′\ ₁

)

∪

(

D Y\ ₁′

) (

∪ X\D

)

=X .

We are going to find properties of the maps f₀_α, f₁_α, f₂_α, f₃_α.

1) t∈Y₁. Then by the properties (3.10) we have Y₁⊆Y_Tα, i.e., t∈Y_Tα and tα=T by definition of the set T

Yα. Therefore f₀_α

( )

t =T for all t∈Y₁.

2) t∈Y Y₁′\ ₁ . Then by the properties (3.10) we have Y Y₁\ ₁ Y₁ YT YT

α α

′

′ ⊆ ⊆′ ∪ , i.e., t∈Y_Tα∪Y_Tα′ and

{

,

}

t

α

∈ T T′ by definition of the sets Y_Tα and Y_Tα′. Therefore f1α

( ) {

t ∈ T T, ′

}

for all t∈Y Y1′\ 1.

Preposition we have that Y_Tα′ ∩Y′≠ ∅, i.e. t′

α

=T′ for some t′∈Y′. If t′∈Y1, then t Y1 YT α

′∈ ⊆ .

Therefore t′ =

α

T. That is contradict of the equality t′

α

=T′, while T≠T′ by definition of the semilattice D. Therefore f₁_α

( )

t′ =T′ for some t′∈Y′\Y₁.

3) t∈D Y\ ₁′. Then by properties (3.10) we have D Y\ ₁ D YT YT Y₀ X

α α α

′

′⊆ ⊆ ∪ ∪ =

 

, i.e., t∈Y_Tα∪Y_Tα′ ∪Y₀α

and tα∈

{

T T D, ′,

}

by definition of the sets Y_Tα , Y_Tα′ and Y0 α

. Therefore f₃_α

( )

t ∈

{

T T D, ′, 

}

for all

1

\

t∈D Y ′.

Preposition we have that Y₀α∩ ≠ ∅D , i.e. t′′ =α D for some t′′∈D. If t′′∈Y₁′. Then t′′∈ ⊆Y₁′ Y_Tα∪Y_Tα′. Therefore t′′

α

∈

{

T T, ′

}

by definition of the set Y_Tα and Y_Tα′ . We have contradict of the equality t′′

α

=T′′. Therefore f₃_α

( )

t′′ =D for some t′′∈D Y\ ₁′.

4) t∈X D\ . Then by definition quasinormal representation binary relation

α

and by property (3.10) we have t X D\ X YT YT Y₀

α α α

′

∈ ⊆ = ∪ ∪ , i.e. tα∈

{

T T D, ′, 

}

by definition of the sets YT ,YT α α

′ and Y0 α

. There-fore f₄_α

( )

t ∈

{

T T D, ′, 

}

for all t∈X D\ .

Therefore for every binary relation

α

∈R D

( )

′ ∩R D

( )

′′ exist ordered system

(

f₀_α,f₁_α,f₂_α,f₃_α

)

. It is ob-vious that for disjoint binary relations exist disjoint ordered systems.

Now, let f₀:Y₁→

{ }

T , f Y Y₁: ₁′\ ₁→

{

T T, ′

}

, f₂:D Y\ ₁′→

{

T T D, ′, 

}

, f₃:X \D→

{

T T D, ′, 

}

are such mappings, which satisfying the conditions:

5) f₀

( )

t =T for all t∈Y₁;

6) f t₁

( ) {

∈ T T, ′

}

for all t∈Y Y₁′\ ₁ and f t₁

( )

′ =T′ for some t′∈Y′\Y₁;

7) f₂

( )

t ∈

{

T T D, ′, 

}

for all t∈D Y\ ₁′ and f t₂

( )

′′ =D for some t′′∈D Y\ ₁′; 8) f₃

( )

t ∈

{

T T D, ′, 

}

for all t∈X D\ .