Regular Elements of the Complete Semigroups BX(D) of Binary Relations of the Class ∑2(X,8)

(1)

http://dx.doi.org/10.4236/am.2015.63042

Regular Elements of the Complete

Semigroups

B

_X

( )

D

of Binary Relations of

the Class

Σ

₂

(

X

, 8

)

Nino Tsinaridze, Shota Makharadze

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

Email: [email protected], [email protected]

Received 10 February 2015; accepted 28 February 2015; published 3 March 2015

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

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

Abstract

As we know if D is a complete X -semilattice of unions then semigroup BX

( )

D possesses a

right unit iff D is an XI-semilattice of unions. The investigation of those α-idempotent and

regular elements of semigroups BX

( )

D requires an investigation of XI-subsemilattices of

se-milattice D for which V D

(

,

α

)

= ∈ ΣQ 2

(

X, 8

)

. Because the semilattice Q of the class Σ2

(

X, 8

)

are not always XI-semilattices, there is a need of full description for those idempotent and

regu-lar elements when V D

(

,

α

)

=Q. For the case where X is a finite set we derive formulas by

cal-culating the numbers of such regular elements and right units for which V D

(

,

α

)

=Q.

Keywords

Semilattice, Semigroup, Binary Relation

1. Introduction

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Let X be an arbitrary nonempty set, recall that the set of all binary relations on X is denoted B_X. The binary operation " " on B_X defined by for α β ∈, BX

( )

x z, ∈

α β

 ⇔

( )

x y, ∈

α

and

( )

y z, ∈

β

, for some y∈X is associative and hence B_X is a semigroup with respect to the operation " " . This semigroup is called the semigroup of all binary relations on the set X. By ∅ we denote an empty binary relation or empty subset of the set X.

Let D be a X-semilattice of unions, i.e. a nonempty set of subsets of the set X that is closed with re-spect to the set-theoretic operations of unification of elements from D, f be an arbitrary mapping from X into D. To each such a mapping f there corresponds a binary relation α_f on the set X that satisfies the condition _f

(

{ } ( )

)

x X

x f x

α

∈

=

_

× . The set of all such α_f

(

f :X →D

)

is denoted by B_X

( )

D . It is easy to prove that B_X

( )

D is a semigroup with respect to the operation of multiplication of binary relations, which is called a complete semigroup of binary relations defined by a X-semilattice of unions D (see ([1], Item 2.1), ([2], Item 2.1)).

Let x y, ∈X , Y⊆X ,

α

∈B_X

( )

D , T∈D, ∅ ≠D′⊆D and

Y D

t D Y

∈

∈ =



. We use the notations:

{

}

(

)

{

}

{

}

(

)

(

)

{

}

{

}

{

}

{

}

, , , , ,

, \ , ,

, .

y Y

T T t

T T

y x X y x Y y V D Y Y D X T T X

l D T D D Y x X x T D Z D t Z

D Z D T Z D Z D Z T

α

α α α α α α

α

∗ ∈

= ∈ = = ∈ = ∅ ≠ ⊆

′ = ∪ ′ ′ = ∈ = ′= ′∈ ′ ∈ ′

′ = ′∈ ′ ⊆ ′ ′ = ′∈ ′ ′⊆



Let

α

∈B_X

( )

D , Y_Tα =

{

x∈X xα=T

}

and

[ ]

(

)

(

)

(

)

(

)

{ }

(

)

, , if ;

, , if , ;

, , if , and .

V X D

V V X V X

V X V X D

α

α α α

α α

∗

∗ ∗

 _{∅ ∉}

 

= ∅ ∈



∪ ∅ ∅ ∉ ∅ ∈



In general, a representation of a binary relation α_{of the form}

(

)

[ ] T

T V

Yα T

α

∈

=

_

× is called quasinormal.

Note that for a quasinormal representation of a binary relation α, not all sets Y_Tα

(

T∈V

[ ]

α

)

can be dif-ferent from an empty set. But for this representation the following conditions are always fulfilled:

a) Y_Tα∩Y_Tα′ = ∅, for any T T, ′∈D and T≠T′;

b)

[ ] T

T V

X Yα

α ∈

=



(see ([1], Definition 1.11.1), ([2], Definition 1.11.1)).

Let

ε

∈B_X

( )

D . ε is called right unit of the semigroup B_X

( )

D . If α ε α = for any

α

∈B_X

( )

D . An element α_{taken from the semigroup}B_X

( )

D called a regular element of the semigroup B_X

( )

D if in

( )

X

B D there exists an element β such that α β α α  = (see [1]-[3]).

In [1] [2] they show that β is regular element of BX

( )

D iff V

[ ]

β

=V D

(

,

β

)

is a complete XI-semilattice of unions.

A complete X-emilattice of unions D is an XI-emilattice of unions if it satisfies the following two con-ditions:

(a) ∧

(

D D, _t

)

∈D for any t∈D; (b)

(

, _t

)

t Z

Z D D

∈

=

_

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

1.14.2) or [4]). Under the symbol ∧

(

D D, _t

)

we mean an exact lower bound of the set D_t in the semilattice D.

Let D′ be an arbitrary nonempty subset of the complete X-semilattice of unions D. A nonempty element T is a nonlimiting element of the set D′ if T l D T\

(

′,

)

≠ ∅ and a nonempty element T is a limiting ele-ment of the set D′ if T l D T\

(

′,

)

= ∅ (see ([1], Definition 1.13.1 and Definition 1.13.2), ([2], Definition 1.13.1 and Definition 1.13.2)).

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the family of sets of pairwise nonintersecting subsets of the set X. If

ϕ

is a mapping of the semilattice D on the family of sets C D

( )

which satisfies the condition ϕ

( )

D =P₀ and

ϕ

( )

Z_i =P_i for any i=1, 2,,m−1

and Dˆ_Z =D\

{

T∈D Z⊆T

}

, then the following equalities are valid:

( )

0 1 2 1 0

ˆ

,

Zi

m i

T D

D P P P P₋ Z P ϕ T

∈

= ∪ ∪ ∪ ∪ = ∪







( )

•

In the sequel these equalities will be called formal.

It is proved that if the elements of the semilattice D are represented in the form

( )

• , then among the parameters P_i

(

i=0,1, 2,,m−1

)

there exist such parameters that cannot be empty sets for D. Such sets P_i

(

0< ≤ −i m 1

)

are called basis sources, whereas sets P_j

(

0≤ ≤ −j m 1

)

which can be empty sets too are called completeness sources.

It is proved that under the mapping

ϕ

the number of covering elements of the pre-image of a basis source is always equal to one, while under the mapping

ϕ

the number of covering elements of the pre-image of a com-pleteness source either does not exist or is always greater than one (see ([1], Item 11.4), ([2], Item 11.4) or [5]).

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

∈

′

∪ =

_

Is fulfilled for each nonempty subset D₁ of the semilattice D′ (see ([1], definition 6.3.2), ([2], definition 6.3.2) or [6]) and the complete isomorphism

ϕ

between the complete semilattices of unions Q and D′ is a complete α-isomorphism if (b)

(a) Q=V D

(

,

α

)

;

(b)

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)).

Lemma 1.1. Let D by a complete X-semilattice of unions. If a binary relation ε of the form

{ } (

)

(

, _t

)

(

\

)

t D

t D D X D D

ε

∈

= × ∧ ∪ ×



 



is right unit of the semigroup B_X

( )

D , then ε is the greatest right unit of that semigroup (see ([1], Lemma 12.1.2), ([2], Lemma 12.1.2)).

Theorem 1.1. Let D_j=

{

T T₁, ₂,,T_j

}

, X and Y—be three such sets, that ∅ ≠ ⊆Y X. If f is such mapping of the set X, in the set D_j, for which f y

( )

=T_j for some y∈Y, then the numbers of all those mappings f of the set X in the set D_j is equal to s= jX Y\ ⋅

(

jY − −

(

j 1

)

Y

)

(see ([1], Theorem 1.18.2), ([2], Theorem 1.18.2)).

Theorem 1.2. Let D be a finite X-semilattice of unions and α σ α α  = for some α and σ of the semigroup B_X

( )

D ; D

( )

α be the set of those elements T of the semilattice Q=BX

( ) { }

D \ ∅ which are

nonlimiting elements of the set Q_T. Then a binary relation α having a quasinormal representation of the form ( , )

T T V D

Yα T

α

∈

=

_

× is a regular element of the semigroup B_X

( )

D iff the set V D

(

,

α

)

is a XI-semilattice of unions and for α-isomorphism

ϕ

of the semilattice V D

(

,α

)

on some X-subsemilattice D′of the semilattice D the following conditions are fulfilled:

(a)

ϕ

( )

T =T

σ

for any T∈V D

(

,

α

)

;

(b)

( )

( )_T T

T D

Yα T

α

ϕ

∈

⊇





for any T∈D

( )

α

;

(c) Y_Tα∩

ϕ

( )

T ≠ ∅ for any element Tof the set

( )

T

D

α

(see ([1], Theorem 6.3.3), ([2], Theorem 6.3.3) or [6]).

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2. Results

Let D is any X-semilattice of unions and Q=

{

T T T T T T T T₇, ₆, ₅, ₄, ₃, ₂, ,₁ ₀

}

⊆D, which satisfies the following conditions:

7 5 2 0 7 4 2 0 7 4 1 0

6 4 2 0 6 4 1 0 6 3 1 0

7 6 4 5 4 2 4 3 1 2 1 0

1 2 2 1 3 4 4 3

3 5 5 3 4 5 5 4

, , , , , , , , , , \ , \ , \ , \ , \ , \ , \ , \

T T T T T T T T T T T T

T T T T T T T T

T T T T T T T T

⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂ ⊂

∪ = ∪ = ∪ = ∪ =

≠ ∅ ≠ ∅ ≠ ∅ ≠ ∅

≠ ∅ ≠ ∅ ≠ ∅ ≠

6 7 7 6

, \ , \ .

T T T T

∅

≠ ∅ ≠ ∅

(1)

The semilattice Q, which satisfying the conditions (1) is shown in Figure 1. By the symbol Σ₂

(

X,8

)

we denote the set of all X-semilattices of unions whose every element is isomorphic to Q.

Let C Q

( )

=

{

P P P P P P P P₇, ₆, ₅, ₄, ₃, ₂, ₁, ₀

}

is a family sets, where P P P P P P P P₇, , , , , , , ₆ ₅ ₄ ₃ ₂ ₁ ₀ are pairwise disjoint subsets of the set X and

7 6 5 4 3 2 1 0

T T T T T T T T

P P P P P P P P

ψ _{= } _

 

is a mapping of the semilattice Q into the family sets C Q

( )

. Then for the formal equalities of the semilattice Q we have a form:

0 0 1 2 3 4 5 6 7

1 0 2 3 4 5 6 7

2 0 1 3 4 5 6 7

3 0 2 4 5 6 7

4 0 3 5 6 7

5 0 1 3 4 6 7

6 0 5 7

7 0 3 6

, , , , , , , .

T P P P P P P P P

T P P P P P P P

T P P P P P P

T P P P P P

T P P P P P P

T P P P

= ∪ ∪ ∪ ∪ ∪ ∪ ∪ = ∪ ∪ ∪ ∪ ∪ ∪ = ∪ ∪ ∪ ∪ ∪ ∪ = ∪ ∪ ∪ ∪ ∪ = ∪ ∪ ∪ ∪ = ∪ ∪ ∪ ∪ ∪ = ∪ ∪ = ∪ ∪ (2)

here the elements P P₁, , , ₂ P P₃ ₅ are basis sources, the element P P P P₀, , , ₄ ₆ ₇ are sources of completenes of the semilattice Q. Therefore X ≥4 and

δ

=4.

Theorem 2.1. Let Q=

{

T T T T T T T T₇, ₆, ₅, ₄, ₃, ₂, ,₁ ₀

}

∈ ∑₂

(

X,8

)

. Then Q is XI-semilattice, when T₅∩T₃= ∅. Proof. Let t∈T₀, Q_t =

{

T∈Q t∈T

}

and ∧

(

Q Q, t

)

is the exact lower bound of the set Qt in Q. Then of the formal equalities

( )

2 follows, that

{

}

{

}

{

}

{

}

{

}

{

}

{

}

0

5 2 0 1

3 1 0 2

7 5 4 2 1 0 3

5 3 2 1 0 4

6 4 3 2 1 0 5

7 5 4 3 2 1 0 6

6 5 4 3 2 1 0 7

, if , , , , if , , , , if , , , , , , , if , , , , , , if , , , , , , , if , , , , , , , , if , , , , , , , , if ,

t

Q t P

T T T t P

T T T T T T t P

Q

T T T T T t P

T T T T T T t P

T T T T T T T t P

∈   _∈   ∈  _∈  =  _∈   ∈  ∈  ∈ 

(

)

5 1 3 2 7 3 6 5

, if , , if , ,

, if , , if ,

t

T t P

Q Q

T t P

(5)

Figure 1. Diagram of Q.

We have Q

{

(

Q Q, t

)

t T₀

}

{

T T T T₇, ₆, ₅, ₃

}

∧ _{= ∧} _∈ ₌

and ∧

(

Q Q, _t

)

∉Q if t∈ ∪P₀ P₄∪ ∪P₆ P₇. So, from the definition XI-semilattice follows that Q is not XI-semilattice.

If P₀=P₄=P₆=P₇= ∅ (since they are completeness sources), then ∧

(

Q Q, t

)

∈Q for all t∈T0 and

4 7 6

T =T ∪T , T₁=T₇∪T₃, T₂=T₆∪T₅. Of the last conditions and from the Definition XI-semilattice follows that Q is XI-semilattice. Of the equality P₀=P₄ =P₆=P₇= ∅ follows that

(

) (

)

5 3 0 1 3 4 6 7 0 2 4 5 6 7 0 4 6 7

T ∩ =T P ∪ ∪ ∪ ∪ ∪P P P P P ∩ P ∪ ∪ ∪ ∪ ∪P P P P P =P ∪ ∪ ∪P P P = ∅

Of the other hand, if T₅∩T₃= ∅ then by formal equalities follows that P₀=P₄=P₆ =P₇= ∅. Therefore, semilattice Q is XI-semilattice.

The Theorem is proved.

Lemma 2.1. Let Q=

{

T T T T T T T T₇, ₆, ₅, ₄, ₃, ₂, ,₁ ₀

}

∈ ∑₂

(

X,8

)

and T₅∩T₃= ∅. Then following equalities are true:

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

P =T P =T P =T T P =T T

Proof. The given Lemma immediately follows from the formal equalities (2) of the semilattice Q. The lemma is proved.

{

T T T T T T T T₇, ₆, ₅, ₄, ₃, ₂, ,₁ ₀

}

∈ ∑₂

(

X,8

)

and T₅∩T₃= ∅. Then the binary relation

(

T7 T7

) (

T6 T6

) (

(

T5\T1

)

T5

)

(

T3\T2

)

T3

)

(

X T\ 0

)

T0

)

ε= × ∪ × ∪ × ∪ × ∪ ×

is the largest right unit of the semigroup B_X

( )

D .

Proof. By preposition and from Theorem 2.1 follows that Q is XI-semilattice. Of this, from Lemma 1.1, from Lemma 2.1 and from Theorem 1.3 we have that the binary relation

{ } (

)

(

)

(

)

(

) (

(

)

(

0

) (

(

)

(

)

(

)

0 0 3 7 5 6 1 5 2 3 0 0

7 7 6 6 5 1 5 3 2 3 0 0

, \ \

\ \ \ .

t t T

t Q Q X T T P T P T P T P T X T T

T T T T T T T T T T X T T

ε

∈

= × ∧ ∪ × = × ∪ × ∪ × ∪ × ∪ ×

= × ∪ × ∪ × ∪ × ∪ ×



is the largest right unit of the semigroup B_X

( )

D . The lemma is proved.

{

T T T T T T T T₇, ₆, ₅, ₄, ₃, ₂, ,₁ ₀

}

∈ ∑₂

(

X,8

)

and T₅∩T₃= ∅. Binary relation α having quazi-normal representation of the form

(

Y7 T7

) (

Y6 T6

) (

Y5 T5

) (

Y4 T4

) (

Y3 T3

) (

Y2 T2

) (

Y1 T1

) (

Y0 T0

)

α α α α α α α α

α = × ∪ × ∪ × ∪ × ∪ × ∪ × ∪ × ∪ ×

where Y₇α, , , Y₆α Y₅α Y₃α∉ ∅

{ }

and V D

(

,

α

)

= ∈ ∑Q ₂

(

X,8

)

is a regular element of the semigroup B_X

( )

D iff for some complete α isomorphism 7 6 5 4 3 2 1 0

7 6 5 4 3 2 1 0

T T T T T T T T

P T T T T T T T

ϕ_{= } _

  of the semilattice Q on some

X-subsemilattice Q′ =

{

T T T T T T T T₇, ₆, ₅, ₄, ,₃ ₂, ,₁ ₀

}

(see Figure 2) of the semilattice Q satisfies the following conditions:

7 7, 6 6, 7 5 5, 6 3 3, 5 5 , 3 3 Yα ⊇T Yα ⊇T Yα∪Yα ⊇T Yα∪Yα ⊇T Yα∩ ≠ ∅T Yα∩ ≠ ∅T

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[image:6.595.258.340.85.170.2]

Figure 2.Diagram of Q’.

( )

{ }

( )

{ }

( )

{

}

( )

{

}

( )

7

{

}

( )

6

{

5

}

( )

{

4

}

3 2 1

7 6 7 5 7 6 4

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

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

T T T T

T T T

Q T Q T Q T T Q T T T

Q T T Q T T T T T Q T T T T T

α α α α

α α α

= = = =

= = =

   

  

By Statement b) of the Theorem 1.2 follows that the following conditions are true:

(

) (

)

7 7 6 6 7 5 5 7 6 4 4 6 3 3

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

7 6 4 7 6 4 4 4 4

7 6 5 4 2 7 5 7 6 4 2 5 4 2 2 2

, , , , ,

, ;

,

Y T Y T Y Y T Y Y Y T Y Y T

Y Y Y Y Y T Y Y Y Y Y T

Y Y Y T T Y T Y T

Y Y Y Y Y Y Y Y Y Y Y T T Y T Y

α α α α α α α α α

α α α α α α α α α α

α α α α α

α α α α α α α α α α α α α

⊇ ⊇ ∪ ⊇ ∪ ∪ ⊇ ∪ ⊇ ∪ ∪ ∪ ∪ ⊇ ∪ ∪ ∪ ∪ ⊇ ∪ ∪ ⊇ ∪ ∪ = ∪ ⊇ ∪ ∪ ∪ ∪ = ∪ ∪ ∪ ∪ ∪ ⊇ ∪ ∪ = ∪

(

) (

)

2

7 6 4 3 1 7 6 4 6 3 1 4 3 1 1 1 1

, ,

T

Yα Yα Yα Yα Yα Yα Yα Yα Yα Yα Yα T T Yα T Yα T

⊇

∪ ∪ ∪ ∪ = ∪ ∪ ∪ ∪ ∪ ⊇ ∪ ∪ = ∪ ⊇

i.e., the inclusions Y₇α∪Y₆α∪Y₄α ⊇T₄, , Y₇α∪Y₆α∪Y₅α∪Y₄α∪Y₂α ⊇T₂ Y₇α∪Y₆α∪Y₄α∪Y₃α∪Y₁α ⊇T₁ are always hold. Further, it is to see, that the following conditions are true:

(

)

(

{ }

)

(

)

(

)

(

{ }

)

(

)

(

)

(

{ }

)

(

)

(

)

(

{ }

)

(

)

(

)

(

{ }

)

7 7 7

6 6 6

5 5 5

3 3 3

4 4

7 7 7 7 7

6 6 6 6 6

5 5 7 5 5 5 7

3 3 6 3 3 3 6

4 4 , \ , \ , \ ; , \ , \ , \ ; , \ , \ , \ ; , \ , \ , \ ; , \

T T T

T T

l Q T Q T T l Q T T

l Q T Q T T T l Q T T T

l Q T Q T

= ∪ = ∅ = ∅ ≠ ∅ = ∪ = ∅ = ∅ ≠ ∅ = ∪ = = ≠ ∅ = ∪ = = ≠ ∅ = ∪ =              

(

)

(

)

(

{ }

)

(

)

(

)

(

{ }

)

(

)

4

2 2 2

1 1 1

4 4 4 4 4

2 2 2 2 2 2 2

1 1 1 1 1 1 1

, \ , \ ; , \ , \ , \ ; , \ , \ , \ ,

T

T T T

T T l Q T T T

l Q T Q T T T l Q T T T

l Q T Q T T T l Q T T T

= = ∅ = ∪ = = = ∅ = ∪ = = = ∅       

i.e., T₇, , , T₆ T₅ T₃ are nonlimiting elements of the sets

( )

7

T

Q α ,

( )

6

T

Q α ,

( )

5

T

Q α and

( )

3

T

Q α respec-tively. By Statement c) of the Theorem 1.2 it follows, that the conditions Y₇α∩T₇≠ ∅, Y₆α ∩T₆≠ ∅,

5 5

Yα∩ ≠ ∅T and Y₃α∩ ≠ ∅T₃ are hold. Since Z₇⊂Z₅, Z₆⊂Z₃ we have Y₅α∩ ≠ ∅T₅ and Y₃α∩ ≠ ∅T₃ . Therefore the following conditions are hold:

7 7, 6 6, 7 5 5, 6 3 3, 5 5 , 3 3 Yα ⊇T Yα ⊇T Yα∪Yα ⊇T Yα∪Yα ⊇T Yα∩ ≠ ∅T Yα∩ ≠ ∅T The lemma is proved.

Definition 2.1. Assume that Q′∈ Σ₂

(

X,8

)

. Denote by the symbol R Q

( )

′ the set of all regular elements

α of the semigroup B_X

( )

D , for which the semilattices Q′ and Q are mutually α-isomorphic and

(

,

)

V D

α

=Q′.

It is easy to see the number q of automorphism of the semilattice Q is equal to 2.

Theorem 2.2. Let Q=

{

T T T T T T T T7, 6, 5, 4, 3, 2, ,1 0

}

∈ ∑2

(

X,8

)

, T5∩T3= ∅ and Σ2

(

X,8

)

=m0. If X be

(7)

( )

(

5\1

) (

3\ 2

)

\0

0

2 2T T 1 2T T 1 8X T

R Q′ = ⋅m ⋅ − ⋅ − ⋅

Proof. Assume that

α

∈R Q

( )

′ . Then a quasinormal representation of a regular binary relation α has the form

(

Y7 T7

) (

Y6 T6

) (

Y5 T5

) (

Y4 T4

) (

Y3 T3

) (

Y2 T2

) (

Y1 T1

) (

Y0 T0

)

α α α α α α α α

α = × ∪ × ∪ × ∪ × ∪ × ∪ × ∪ × ∪ ×

where Y₇α, , , Y₆α Y₅α Y₃α∉ ∅

{ }

and by Lemma 2.3 satisfies the conditions: X

7 7, 6 6, 7 5 5, 6 3 3, 5 5 , 3 3

Yα ⊇T Yα ⊇T Yα∪Yα ⊇T Yα∪Yα ⊇T Yα∩ ≠ ∅T Yα∩ ≠ ∅T (3) Let f_α is a mapping the set X in the semilattice Q satisfying the conditions fα

( )

t =t

α

for all t∈X. 1

f_α, f₂_α, f₃_α, f₄_α, f₅_α are the restrictions of the mapping f_α on the sets T₇, , \T₆ T₃ T₂, \T₅ T X T₁, \ ₀ respectively. It is clear, that the intersection disjoint elements of the set

{

T T T₇, ₆, \₃ T T₂, \₅ T X T₁, \ ₀

}

are empty set and T₇∪ ∪T₆

(

T₃\T₂

) (

∪ T₅\T₁

) (

∪ X T\ ₀

)

=X .

We are going to find properties of the maps f₁_α, f₂_α, f₃_α, f₄_α, f₅_α.

1) t∈T₇. Then by Property (3) we have t∈T₇⊆Y₇α, i.e., t∈Y₇α and tα=T₇ by definition of the set Y₇α. Therefore f1α

( )

t =T7 for all t∈T7.

2) t∈T₆. Then by Property (3) we have t∈ ⊆T₆ Y₆α, i.e., t∈Y₆α and tα =T₆ by definition of the set Y₆α. Therefore f2α

( )

t =T6 for all t∈T6.

3) t∈T₃\T₂. Then by Property (3) we have t∈T₃\T₂ ⊆T₃ ⊆Y₆α∪Y₃α, i.e., t∈Y₆α∪Y₃α and t

α

∈

{

T T₆, ₃

}

by definition of the sets Y₆α and Y₃α. Therefore f₃_α

( )

t ∈

{

T T₆, ₃

}

for all t∈T₃\T₂. Preposition we have that Y₃α∩ ≠ ∅T₃ , i.e. t₃α =T₃ for some t₃∈T₃. If t₃∈T₂, then

3 7 6 5 4 2

t ∈Yα∪Yα∪Yα∪Yα∪Yα. So t₃

α

=

{

T T T T T₇, ₆, ₅, ₄, ₂

}

by definition of the sets Y₇α, , , , Y₆α Y₅α Y₄α Y₂α. The condition t₃

α

=

{

T T T T T₇, ₆, ₅, ₄, ₂

}

contradict of the equality t₃α =T₃, while T₃∉

{

T T T T T₇, ₆, ₅, ₄, ₂

}

. Therefore,

( )

3 3 3

fα t =T for some t∈T3\T2.

4) t∈T₅\T₁. Then by Property (3) we have t∈T₅\T₁⊆T₅⊆Y₇α∪Y₅α, i.e., t∈Y₇α∪Y₅α and t

α

∈

{

T T₇, ₅

}

by definition of the sets Y₇α and Y₅α. Therefore f₄_α

( )

t ∈

{

T T₇, ₅

}

for all t∈T₅\T₁. Preposition we have that Y₅α∩T₅≠ ∅, i.e. t₄α =T₅ for some t₄∈T₅. If t₄∈T₁, then

4 7 6 4 3 1

t ∈Yα∪Yα∪Yα∪Yα∪Yα. So t₄

α

=

{

T T T T T₇, ₆, ₄, ₃, ₁

}

by definition of the sets Y₇α, , , , Y₆α Y₄α Y₃α Y₁α. The condition t₄

α

=

{

T T T T T₇, ₆, ₄, ₃, ₁

}

contradict of the equality, t₄α =T₅, while T₅∉

{

T T T T T₇, ₆, ₄, ₃, ₁

}

. Therefore,

( )

4 4 5

fα t =T for some t∈T5\T1.

5) t∈X T\ ₀. Then by definition quasinormal representation binary relation α and by Property (3) we have

0 7 6 5 4 3 2 1 0

\

t∈X T ⊆X =Yα∪Yα∪Yα∪Yα∪Yα∪Yα∪Yα∪Yα, i.e. t

α

∈

{

T T T T T T T T₇, ₆, ₅, ₄, ₃, ₂, ,₁ ₀

}

by definition of the sets Y₇α, , , , , , , Y₆α Y₅α Y₄α Y₃α Y₂α Y₁α Y₀α. Therefore f₅α

( )

t ∈

{

T T T T T T T T₇, ₆, ₅, ₄, ₃, ₂, ,₁ ₀

}

for all t∈X T\ 0.

Therefore for every binary relation

α

∈R Q

( )

′ exist ordered system

(

f₁_α,f₂_α,f₃_α,f₄_α,f₅_α

)

. It is obvious that for different binary relations exist different ordered systems.

Let f₁:T₇ →

{ }

T₇ , f₂:T₆ →

{ }

T₆ , f₃:T₃\T₂→

{

T T₆, ₃

}

, f₄:T₅\T₁→

{

T T₇, ₅

}

, f₅:X T\ ₀→Q are such mappings, which satisfying the conditions:

)

6 f t₁

( )

∈

{ }

T₇ for all t∈T₇;

)

(8)

)

8 f3

( )

t ∈

{

T T6, 3

}

for all t∈T3\T2 and f t3

( )

3 =T3 for some t3∈T3\T2;

)

9 f4

( )

t ∈

{

T T7, 5

}

for all t∈T5\T1 and f4

( )

t4 =Z4 for some t4∈T5\T1;

)

10 f5

( )

t ∈

{

T T T T T T T T7, 6, 5, 4, 3, 2, ,1 0

}

for all t∈X T\ 0.

Now we define a map f of a set X in the semilattice Q, which satisfies the following condition:

( )

1 7

2 6

3 3 2

4 5 1

5 0

, if , , if , , if \ , , if \ , , if \ .

f t t T

f t t T

f t f t t T T

f t t T T

f t t X T

 ∈

 _∈



= ∈

 _∈



∈ 

Now let

(

{ } ( )

)

x X

x f x

β

∈

=

_

× , Y_iβ =

{

t tβ =T_i

}

(

i=1, 2,, 5

)

. Then binary relation β is written in the form

(

Y7 T7

) (

Y6 T6

) (

Y5 T5

) (

Y4 T4

) (

Y3 T3

) (

Y2 T2

) (

Y1 T1

) (

Y0 T0

)

β β β β β β β β

β = × ∪ × ∪ × ∪ × ∪ × ∪ × ∪ × ∪ ×

and satisfying the conditions:

7 7, 6 6, 7 5 5, 6 3 3, 5 5 , 3 3 Yβ ⊇T Yβ ⊇T Yβ ∪Yβ ⊇T Yβ∪Yβ ⊇T Yβ∩ ≠ ∅T Yβ ∩ ≠ ∅T From this and by Lemma 2.3 we have that

β

∈R Q

( )

′ .

Therefore for every binary relation

α

∈R Q

( )

′ and ordered system

(

f1α,f2α,f3α,f4α,f5α

)

exist one to one mapping.

By Theorem 1.1 the number of the mappings f₁α, , , , f₂α f₃α f₄α f₅α are respectively:

3\2 5\1 \0

1, 1, 2T T −1, 2T T −1, 8X T (see ([1], Corollary 1.18.1), ([2], Corollary 1.18.1)).

The number of ordered system

(

f₁_α,f₂_α,f₃_α,f₄_α,f₅_α

)

or number regular elements can be calculated by the formula

( )

(

5\1

) (

3\ 2

)

\0

0

2 2T T 1 2T T 1 8X T

R Q′ = ⋅m ⋅ − ⋅ − ⋅

(see ([1], Theorem 6.3.5), ([2], Theorem 6.3.5)). The theorem is proved.

Corollary 2.1. Let Q=

{

T T T T T T T T₇, ₆, ₅, ₄, ₃, ₂, ,₁ ₀

}

∈ ∑₂

(

X,8

)

, T₅∩T₃= ∅. If X be a finite set and ( )r

_{( )}

X

E Q be the set of all right units of the semigroup B_X

( )

Q , then the following formula is true ( )r

_{( )}

(

₂T T5\1 ₁

) (

₂T T3\2 _{1 8}

)

X T\0

X

E Q = − ⋅ − ⋅

Proof: This corollary immediately follows from Theorem 2.2 and from the ([1], Theorem 6.3.7) or ([2], Theorem 6.3.7).

The corollary is proved.

References

[1] Diasamidze, Ya. and Makharadze, Sh. (2013) Complete Semigroups of Binary Relations. Monograph. Kriter, Turkey, 1-520.

[2] Diasamidze, Ya. and Makharadze, Sh. (2010) Complete Semigroups of Binary Relations. Monograph. M., Sputnik+, 657 p. (In Russian)

[3] Lyapin, E.S. (1960) Semigroups. Fizmatgiz, Moscow. (In Russian)

[4] Diasamidze, Ya., Makharadze, Sh. and Rokva, N. (2008) On XI-Semilattices of Union. Bull. Georg. Nation. Acad. Sci.,

2, 16-24.

(9)

4319.

[6] Diasamidze, Ya., Makharadze, Sh. and Diasamidze, Il. (2008) Idempotents and Regular Elements of Complete Semi-groups of Binary Relations. Journal of Mathematical Sciences, 153, 481-499.