Generated Sets of the Complete Semigroup Binary Relations Defined by Semilattices of the Class Σ8 (X, n + k + 1)

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http://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385

DOI: 10.4236/am.2018.94028 Apr. 27, 2018 369 Applied Mathematics

Generated Sets of the Complete Semigroup

Binary Relations Defined by Semilattices of the

Class

Σ

₈

(

X n

,

+ +

k

1

)

Yasha Diasamidze, Omari Givradze, Nino Tsinaridze, Giuli Tavdgiridze

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

Abstract

In this article, we study generated sets of the complete semigroups of binary relations defined by X-semilattices unions of the class Σ8

(

X n, + +k 1

)

, and

find uniquely irreducible generating set for the given semigroups.

Keywords

Semigroup, Semilattice, Binary Relation

1. Introduction

Let X be an arbitrary nonempty set, D isan X-semilattice of unions which is closed with respect to the set-theoretic union of elements from D, f be an arbi-trary mapping of the set X in the set D. To each mapping f we put into corres-pondence 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

( )

X

B D . It is easy to prove that B_X

( )

D is a semigroup with respect to the

op-eration of multiplication of binary relations, which is called a complete semi-group of binary relations defined by an X-semilattice of unions D.

We denote by ∅_{an empty binary relation or an empty subset of the set}_X_{. The}

condition

( )

x y, ∈

α

will be written in the form

x y

α

. Further, let x y X, ∈ ,

Y⊆X ,

α

∈B_X

( )

D ,

Y D

D Y

∈

=





and T∈D. We denote by the symbols y

α

,

Yα

, V D

(

,

α

)

, X∗ and V X

(

∗,

α

)

the following sets:

{

}

(

) {

}

{

}

(

)

{

}

| , , , | ,

| , , | ,

y Y

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

X Y Y X V X Y Y X

α α α α α α

α α

∈

∗ ∗

= ∈ = = ∈

= ∅ ≠ ⊆ = ∅ ≠ ⊆



How to cite this paper: Diasamidze, Y., Givradze, O., Tsinaridze, N. and Tavdgi-ridze, G. (2018) Generated Sets of the Complete Semigroup Binary Relations Defined by Semilattices of the Class

(

)

8 X n, k 1

Σ + + _.Applied Mathematics, 9, 369-382.

https://doi.org/10.4236/am.2018.94028

Received: March 22, 2018 Accepted: April 24, 2018 Published: April 27, 2018

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

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DOI: 10.4236/am.2018.94028 370 Applied Mathematics

{

|

}

,

{

|

}

.

T T

D = Z∈D T ⊆Z Yα = y∈X yα =T

It is well known the following statements: Theorem 1.1. Let D=

{

D Z Z, 1, 2, ,Zm−1

}



 be some finite X-semilattice of unions and C D

( ) {

= P P P0, ,1 2,,Pm−1

}

be the family of sets of pairwise

nonin-tersecting subsets of the set X (the set ∅_{can be repeated several times). If}ϕ_{is a}

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

( )

which satisfies the conditions

1 2 1

0 1 2 1

m

D Z Z Z

P P P P

ϕ

−

 

= _ _

 



 

and ˆ _\

Z Z

D =D D , 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

∈

= ∪ ∪ ∪ ∪ = ∪







(1.1)

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 (1.1), then among the parameters Pi

(

0< ≤ −i m 1

)

there exist such

pa-rameters that cannot be empty sets for D. Such sets Pi are called bases sources,

where sets Pj

(

0≤ ≤ −j m 1

)

, which can be empty sets too are called

com-pleteness sources.

It is proved that under the mapping ϕ_{the number of covering elements of the}

pre-image of a

bases source is always equal to one, while under the mapping ϕ_{the number of}

covering elements of the pre-image of a completeness source either does not ex-ist or is always greater than one (see [1][2] chapter 11).

Definition 1.1. We say that an element α_{of the semigroup}B_X

( )

D is ex-ternal if

α δ β

≠ _ _{for all}

δ β

, ∈B_X

( ) { }

D \

α

(see [1][2] Definition 1.15.1).

It is well known, that if B is all external elements of the semigroup BX

( )

D and

B′

_{is any generated set for the} B_X

_{( )}

D , then B⊆B′ (see [1][2] Lemma 1.15.1).

Definition 1.2. The representation

(

T

)

T D

Yα T

α

∈

=

_

× of binary relation α_is

called quasinormal, if T T D

Yα X

∈

=



and YT YT

α α

′

∩ = ∅_{for any}T T, ′∈D,

T≠T′ (see [1][2] chapter 1.11).

Definition 1.3. Let

α β

, ⊆ ×X X . Their product

δ α β

=  is defined as follows: x y

δ

(

x y, ∈X

)

if there exists an element z∈X such that x z y

α β

(see [1], chapter 1.3).

2. Result

Let Σ8

(

X n, + +k 1

)

(

3≤ ≤k n

)

be a class of all X-semilattices of unions

whose every element is isomorphic to an X-semilattice of unions

{

1, 2, , n k,

}

D= Z Z Z + D



 , which satisfies the condition:

(

)

(

)

(

)

, 1, 2, , ; , 1, 2, , ;

\ and \ 1 .

n i i j p q q p

Z Z D i k Z D j n k

Z Z Z Z p q n k

+ ⊂ ⊂ = ⊂ = +

≠ ∅ ≠ ∅ ≤ ≠ ≤ +

 

 

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[image:3.595.299.453.71.139.2]

Figure 1. Diagram of the semilattice D.

It is easy to see that D=

{

Z Z1, 2,,Zn k+

}

is irreducible generating set of the

semilattice D.

Let C D

( ) {

= P P P0, ,1 2,,Pn k+

}

be a family of sets, where P P P0, ,1 2,,Pn k+

are pairwise disjoint subsets of the set X and 1 2

0 1 2

n k

D Z Z Z

P P P P

ϕ

+

 

= _ _

 





 is a map

ping of the semilattice D onto the family of sets C D

( )

. Then the formal

equali-ties of the semilattice D have a form:

0 0, 0,

,

; , 1, 2, , ; , 1, 2, , .

n k n k n k

i j i n q i

i i i

i j i q n q

D P Z P j n Z P q k

+ + +

+

= = =

≠ ≠ +

= = = = =



 



(2.0)

Here the elements P ii

(

=1, 2,,n+k

)

are bases sources, the element P0

are sources of completeness of the semilattice D. Therefore X ≥ +n k (by

symbol X we denoted the power of a set X), since Pi ≥1

(

i=1, 2,,n+k

)

(see [1][2] chapter 11).

In this paper we are learning irreducible generating sets of the semigroup

( )

X

B D defined by semilattices of the class Σ8

(

X n, + +k 1

)

.

Note, that it is well known, when k=2, then generated sets of the complete

semigroup of binary relations defined by semilattices of the class

(

)

(

)

8 X, 2 2 1 8 X, 5

Σ + + = Σ _.

In this paper we suppose, that 3≤ ≤k n.

Remark, that in this case (i.e. k≥3), from the formal equalities of a semilattice D follows, that the intersections of any two elements of a semilattice D is not empty.

Lemma 2.0 If D∈Σ8

(

X n, + +k 1

)

, then the following statements are true:

a) 0

1 ;

n k

i i

Z P

+

=



b) Zj+1\Zj=Pj, j=1, 2,,n−1;

c) Z_q\Z_{n q}₊ =P_{n q}₊ , 1,q= , .k

Proof. From the formal equalities of the semilattise D immediately follows the following statements:

0 1

1 0, 0,

1

0, 0,

,

, \ \ , 1, 2, , 1;

\ \ , 1, , .

n k n k n k

i j j i i j

i i i

i j i j

n k n k

q n q i i n q

i i

i q i q n q

Z P Z Z P P P j n

Z Z P P P q k

+ + +

+

= = =

≠ + ≠

+ +

= =

≠ ≠ +

       

= =_ _{ } _= = −    

       

   

=_ _{ } _= =    

   







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Lemma 2.0 is proved.

We denoted the following sets by symbols D1, D2 and D3:

{

}

{

}

{

}

1 1, 2, , k , 2 k1, k 2, , n , 3 n1, n2, , n k .

D = Z Z  Z D = Z + Z +  Z D = Z + Z +  Z +

Lemma 2.1. Let D∈Σ8.0

(

X n, + +k 1

)

and

α

∈BX

( )

D . Then the following statements are true:

1) Let T T, ′∈D2∪D T3, ≠T′. If T T, ′∈V D

(

,

α

)

, then α is external element

of the semigroup BX

( )

D ;

2) Let T∈D1, T′∈D2∪D3. If T′ ⊄T and T T, ′∈V D

(

,

α

)

, then α is

ex-ternal element of the semigroup BX

( )

D .

3) Let T T, ′∈D1 and

T

≠

T′

. If T T, ′∈V D

(

,

α

)

and k≥3, then α is

( )

D ; Proof. Let Z0=D



and

α δ β

= _ _{for some}

δ β

, ∈BX

( ) { }

D \

α

. If

quasi-normal representation of binary relation δ_{has a form}

(

)

( , ) T T V D

Yδ T

δ

∈

=



× , then

(

)

( , ) T T V D

Yδ T

δ

α δ β

β

∈

=  =



× . (2.1) From the formal equalities (2.0) of the semilattice D we obtain that:

0

0 0,

0, ,

; , 1, 2, , ;

, 1, 2, , .

n k n k i j i

i i

i j n k

n q i i i q n q

Z P Z P j n

Z P q k

β β β β

β β

+ +

= =

≠ +

+ = ≠ +

= = =

= =





(2.2)

where Piβ ≠ ∅ for any Pi≠ ∅

(

i=0,1, 2,,n+k

)

and

β

∈BX

( )

D by

definition of a semilattice D from the class Σ8.0

(

X n, + +k 1

)

.

Now, let Zmβ =T and Zj

β

=T′ for some

T

≠

T′

, T T, ′∈D2∪D3, then

from the equalities (2.3) follows that T =P0β =T′ since T and T′ are

minim-al elements of the semilattice D and P0≠ ∅ by preposition. The equality

T=T′ contradicts the inequality T≠T′.

The statement a) of the Lemma 2.1 is proved.

Now, let Zmβ =T and Zj

β

=T′, for some T∈D1, T′∈D2∪D3 and

T′ ⊄T, then from the equalities 2.3 follows, that

0 0

n k

j i

i

T Z

β

Z

β

P

β

+

=

′ = = =

_

_{, if} j=0 , or

0,

n k

j i

i i j

T Z β Pβ

+

= ≠

′ = =

_

_, 1≤ ≤j n , or

0, ,

n k

n q i

i i q n q

T Z β Pβ

+ +

= ≠ +

′ = =



where j= +n q. For the Z_j

β

=T′ we consider the following cases:

1) If 0 0

n k

i i

T Z

β

P

β

+

=

′ = =



, then we have

0 1 n k

Pβ=Pβ==P+ β=T′,

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0, 0,

, ,

, if 1 ;

, if .

n k n k i i i i m i m m n k n k

i i i i q n q i q n q

P T T m n

T Z

P T T m n q

β β β + + = = ≠ ≠ + + = = ≠ + ≠ +  ₌ _′₌ _′ _{≤ ≤}   = _{= }  = ′= ′ = +  



But the equality

T

=

T

′

contradicts the inequality

T

≠

T

′

. Thus we have,

that j≠0.

2) Let 1≤ ≤j n, i.e.

0,

n k

j i

i i j

T Z β Pβ

+

= ≠

′ = =

_

_{, then we have, that}

0 1 j1 j1 n k

P

β

=P

β

==P₋

β

=P₊

β

==P₊

β

=T′,

since

T′

_{is a minimal element of a semilattice}_D_{. On the other hand:}

0 0,

0, 0,

,

0, ,

, if 0;

, if 1 , ;

, if ;

n k n k

i i j j i i

i j n k n k

i i j j i i

i m i m j m

n k i i i j n j

i i

P P P T P m

P P P T P m n m j

T Z

P T m n j

P

β β β β

β β β + + = = ≠ + + = = ≠ ≠ + = ≠ + =     ₌ _∪ ₌ _′_∪ ₌  _{ } _   _ _       ₌ _∪ ₌ _′_∪ _{≤ ≤} _≠             = =    ₌ _′ _{= +}      



0, 0,

, , ,

, if , .

n k n k

i j j i

i q n q i q n q j

Pβ Pβ T Pβ m n q q j

+ + = ≠ + ≠ +                        ₌ _∪ ₌ _′_∪ _{= +} _≠             



The equality

T

=

T

′

T

≠

T′

. Also, the equality

j

T = ∪T′ P

β

(

P_j

β

∈D

)

T

≠ ∪

T

′

Z

for any

Z∈D and T′ ⊄T (T′ ⊄T, by preposition) by definition of a semilattice D. 3) If j= +n q

(

1≤ ≤q k

)

, i.e.

0, ,

n k

n q i

i i q n q

T Z β Pβ

+ +

= ≠ +

′ = =



, then we have, that

0 1 q1 q1 n q1 n q1 n k

P

β

=P

β

==P₋

β

=P₊

β

==P_{+ −}

β

=P_{+ +}

β

==P₊

β

=T′,

since

T′

_{is a minimal element of a semilattice}_D_{. On the other hand:}

0, 0,

,

0, 0,

,

0,

, if 0;

, if 1 ;

n k n k

i i q n q q n q

i i

i q n q

n k n k

i i n q n q

i i

i q i q n q

m _{n k}

i i

i i m

P P P P T P P m

P P P T P m q n

T Z

P P

β β β β β β

β β β β

β β β + + + + = = ≠ + + + + + = = ≠ ≠ + + = ≠    _{ } _ ′ = ∪ ∪ = ∪ ∪ =  _{ } _   _ _       ₌ _∪ ₌ _′_∪ _{≤ = ≤}             = _{= }   _{ =}      



0, , ,

0, 0, 0,

, , , , , ,

, if 1 ;

,

n k

q n q q n q

i i m q n q

n k n k n k

i i i q n q q n q

i i i

i m i q n q i q n q q n q

P P T P P m q n

P P P P P T P P

β β β β

β β β β β

+ + + = ≠ + + + + + + = = = ′ ′ ′ ′ ≠ ≠ + ≠ + +    _∪ _∪ ₌ _′_∪ _∪ _{≤ ≠ ≤}               ₌  ₌ _∪ _∪ _{= ∪} _∪                  



if n 1 m n q n k q, q since j m.

(6)

The equality

T

=

T

′

T

≠

T′

. Also, the equality

q n q

T = ∪T′ P

β

∪P₊

β

, or T = ∪T′ P_{n q}₊

β

(

P_q

β

,P_{n q}₊

β

∈D

)

contradicts the

in-equality

T

≠ ∪

T

′

Z

for any Z∈D and T′ ⊄T by definition of a semilattice

D.

The statement 2) of the Lemma 2.1 is proved.

Let T T, ′∈D1 and

T

≠

T

′

. If k≥3 and Zj

β

=T′, Zmβ =T, then from

the formal equalities (2.0) of a semilattice D there exists such an element, that

q j

P ⊆Z and P_q⊆Z_m, where 0≤ ≤ +q m k. So, from the equalities (2.3)

fol-lows that Pq

β

⊆Zj

β

=T′ and Pq

β

⊆Zm

β

=T . Of from this and from the

equalities (2.3) we obtain that there exists such an element Z∈D, for which the

equalities

T

′

= ∪

Z

′

and

T

= ∪

Z

Z′′

, where Z Z′ ′′∈, D. But such elements

by definition of a semilattice D do not exist. The statement c) of the Lemma 2.1 is proved. Lemma 2.1 is proved.

Lemma 2.2. Let D∈Σ8

(

X n, + +k 1

)

and

α

∈BX

( )

D . Then the following statements are true:

1) Let V D

(

,

α

)

∩D1≠ ∅,V D

(

,

α

)

∩D2= ∅,V D

(

,

α

)

∩D3= ∅. If

(

,

)

1 2

V D

α

∩D ≥ , then α is external element of the semigroup BX

( )

D ;

2) Let V D

(

,

α

)

∩D1= ∅,V D

(

,

α

)

∩D2≠ ∅,V D

(

,

α

)

∩D3= ∅. If

(

,

)

2 2

V D

α

( )

D ;

3) Let V D

(

,

α

)

∩D1= ∅,V D

(

,

α

)

∩D2= ∅,V D

(

,

α

)

∩D3≠ ∅. If

(

,

)

3 2

V D

α

( )

D ;

4) Let V D

(

,

α

)

∩D1= ∅,V D

(

,

α

)

∩D2≠ ∅,V D

(

,

α

)

∩D3≠ ∅, then α is

( )

D ;

5) Let V D

(

,

α

)

∩D1≠ ∅,V D

(

,

α

)

∩D2= ∅,V D

(

,

α

)

∩D3≠ ∅. If

(

,

)

1 2

V D

α

∩D ≥ , V D

(

,

α

)

∩D3 =1, or V D

(

,

α

)

∩D1 =1,

(

,

)

3 2

V D

α

∩D ≥ then α is external element of the semigroup BX

( )

D ;

6) Let V D

(

,

α

)

∩D1≠ ∅,V D

(

,

α

)

∩D2≠ ∅,V D

(

,

α

)

∩D3= ∅, then α is

( )

D ;

7) Let V D

(

,

α

)

∩D1≠ ∅,V D

(

,

α

)

∩D2≠ ∅,V D

(

,

α

)

∩D3≠ ∅, then α is

( )

D .

Proof. Let α_{be any element of the semigroup}B_X

( )

D . It is easy that

(

,

)

V D

α

∈D. We consider the following cases:

Let V D

(

,

α

)

∩D1= ∅,V D

(

,

α

)

∩D2= ∅,V D

(

,

α

)

∩D3= ∅, then

(

,

)

{ }

V D

α

∈ D since V D

(

,

α

)

is subsemilattice of the semilattice D.

1) Let V D

(

,

α

)

∩D1≠ ∅,V D

(

,

α

)

∩D2= ∅,V D

(

,

α

)

∩D3= ∅.

If V D

(

,

α

)

∩D1 =1, then V D

(

,

α

)

∈

{ }

Zj , or V D

(

,

α

)

∈

{

Z Dj,

}



, where

1, 2, ,

j=  k, since V D

(

,

α

)

If V D

(

,

α

)

∩D1 ≥2, then by statement c) of the Lemma 2.1 follows that αis

external element of the semigroup BX

( )

D .

2) Let V D

(

,

α

)

∩D1= ∅,V D

(

,

α

)

∩D2≠ ∅,V D

(

,

α

)

∩D3= ∅.

If V D

(

,

α

)

∩D2 =1, then V D

(

,

α

)

∈

{ }

Zj , or V D

(

,

α

)

∈

{

Z Dj,

}



, where

1, 2, ,

(7)

If V D

(

,

α

)

∩D2 ≥2, then by statement a) of the Lemma 2.1 follows that αis

( )

D .

3) Let V D

(

,

α

)

∩D1= ∅,V D

(

,

α

)

∩D2= ∅,V D

(

,

α

)

∩D3≠ ∅.

If V D

(

,

α

)

∩D3 =1 , then V D

(

,

α

)

∈

{ }

Zj , or V D

(

,

α

)

∈

{

Z Dj,

}



,

1, 2, ,

j= +n n+  n+k, since V D

(

,

α

)

If V D

(

,

α

)

∩D3 ≥2, then by statement a) of the Lemma 2.1 follows that α is

( )

D .

4) Let V D

(

,

α

)

∩D1= ∅,V D

(

,

α

)

∩D2≠ ∅,V D

(

,

α

)

∩D3≠ ∅, then by the

statement a) of the Lemma 2.1 follows that α_{is external element of the}

semi-group BX

( )

D .

5) Let V D

(

,

α

)

∩D1≠ ∅,V D

(

,

α

)

∩D2= ∅,V D

(

,

α

)

∩D3≠ ∅.

If V D

(

,

α

)

∩D1 =1,V D

(

,

α

)

∩D3 =1 , then V D

(

,

α

)

=

{

Zn q+ ,Zq

}

, or

(

,

)

{

n q, Z ,q

}

V D

α

= Z + D , or V D

(

,

α

)

=

{

Zn q+ ,Z Dj,

}



where Z1≤Zj≤Zk and

1, 2, ,

q=  k.

If V D

(

,

α

)

=

{

Zn q+ ,Z Dj,

}



where j≠q q, =1, 2,,k, then by the statement

2) of the Lemma 2.1 follows that α_{is external element of the semigroup}

( )

X

B D ;

If V D

(

,

α

)

∩D1 =1,V D

(

,

α

)

∩D3 ≥2, or

(

,

)

1 2,

(

,

)

3 1

V D

α

∩D ≥ V D

α

∩D = , then from the statement 1) and 3) of the

Lemma 2.1 follows that α_{is external element of the semigroup} BX

( )

D

respec-tively.

6) Let V D

(

,

α

)

∩D1≠ ∅,V D

(

,

α

)

∩D2≠ ∅,V D

(

,

α

)

∩D3= ∅. Then from

the statement b) of the Lemma 2.1 follows that α_{is external element of the}

semi-group BX

( )

D .

7) Let V D

(

,

α

)

∩D1≠ ∅,V D

(

,

α

)

∩D2≠ ∅,V D

(

,

α

)

∩D3≠ ∅, then by the

statement a) of the Lemma 2.1 follows that α_{is external element of the}

semi-group BX

( )

D .

Now we learn the following subsemilattices of the semilattice D:

{

}

{

}

{

}

{

}

{

}

{

}

{ }

{

}

1

2

3

4

, , , where 1, 2, , ;

, , where 1, 2, , ;

, , where 1, 2, , .

n j j j n j j

j

Z Z D j k

Z D j n k

Z Z j k

Z D j n k

+

= =

= = +

= =

= = +



 

  



A

We denoted the following sets by symbols A0 and B

( )

A0 :

(

)

(

)

{

}

( )

{

( ) (

)

}

0 1 2 3 4

0 0

, | , ,

| , .

X

V D D V D

B B D V D

α α

= ⊆ ∉ ∪ ∪ ∪

= ∈ ∈

A A A A A

A A

By definition of a set B

( )

A0 follows that any element of the set is external

element of the semigroup BX

( )

D .

(

X n, + +k 1

)

. If quasinormal representation of a

(8)

(

Yn j Zn j

) (

Yj Zj

) (

Y0 D

)

,

α α α

α

= + × + ∪ × ∪ ×



where Yn j,Yj ,Y0

{ }

α α α

+ ∉ ∅ and j=1, 2,,k, then α is generated by elements of

the elements of set B

( )

A0 .

Proof. 1). Let quasinormal representation of binary relations δ_andβ_{have a}

form

(

) (

)

(

) (

(

)

(

0

)

(

)

,

\ \ \ ,

n j n j j j q q

n j n j j n j j j q

Y Z Y Z Y Z Y D

Z Z Z Z Z D Z Z X D D

δ δ δ δ

δ β

+ +

+ + +

= × ∪ × ∪ × ∪ ×



  

where Yn j,Yj ,Yq

{ }

,Z1 Zq Z qk, j j, 1, ,k

α α α

+ ∉ ∅ ≤ ≤ ≠ =  .

(

)

(

) (

)

(

)

(

)

0 1,

,

\ \ \

\ \

n j j n j j n k

i n j j i

i j n j

Z Z Z D Z X D

P P P P X D D X D X

+ +

+

+ =

≠ +

∪ ∪ ∪

 

 

=_ ∪ _∪ ∪ ∪ = ∪ =

 

 

 

  



,

since the representation of a binary relation β_{is quasinormal and by}

state-ment 3) of the Lemma 2.1 binary relations δ and β are external elements of the semigroup BX

( )

D . It is easy to see, that:

,

n j n j

Z ₊

β

=Z ₊

0 0

1, 1,

,

n k n k

j i i n j

i i

i j i j n j n j n j n j j j

Z P P P P P

Z P Z Z Z

β β β

β β

+ +

+

= =

≠ ≠ +

+ + +

 

   

 

   

=_ ∪ _ =__ ∪ _∪ _

    

    

= ∪ = ∪ =



0 1,

0

,

n k

q i n j j q i

i q n k

i n j j q i

Z P P Z Z Z D

D P Z Z Z D

β β

+

+ =

≠ +

+ =

 

 

=_ ∪ _ = ∪ ∪ =

 

 

= = ∪ ∪ =



 



since Z_q∩Z_{n j}₊ ≠ ∅,Z_q∩

(

Z_j\Z_{n j}₊

)

=P_{n j}₊ ≠ ∅,Z_q∩

(

D Z\ _j

)

=P_j ≠ ∅ (see

equality (2.0))

(

) (

)

(

) (

)

(

) (

(

)

0

0 ,

n j n j j j q q

n j n j j j q

Y Z Y Z Y Z Y D

Y Z Y Z Y D Y D

Y Z Y Z Y Y D

δ δ δ δ

α δ β β β β β

α

+ +

= = × ∪ × ∪ × ∪ × = × ∪ × ∪ × ∪ ×

= × ∪ × ∪ ∪ × =

 

 



if Yn j Yn j

δ α

+ = + , Yj Yj

δ ₌ α_and

0 0

q

Yδ ∪Yδ =Yα. Last equalities are possible since

0 1

q

Yδ∪Yδ ≥ (Y0 0

δ _≥ _{, by preposition).} Lemma 2.3 is proved.

(

X n, + +k 1

)

bi-nary relation α_{has a form}

α

=

(

Y_jα×Z_j

) (

∪ Y₀α×D

)

, where Y_jα,Y₀α∉ ∅

{ }

,

1, 2, ,

j=  n+k, then binary relation α is generated by elements of the elements of set B

( )

A0 .

(9)

form:

(

) (

)

(

)

(

)

(

)

0 ,

\ \ ,

j j q q j j j q

Y Z Y Z Y D

Z Z D Z Z X D D

δ δ δ

δ β

= × ∪ × ∪ ×



  

where Y Y_jδ, _qδ∉ ∅

{ }

and Z₁≤Z_j≠Z_q ≤Z_{n k}₊ . Then from the statements a), b)

and c) of the Lemma 2.1 follows, that δ_andβ_{are generated by elements of the}

set B

( )

A0 and

,

j j

Z

β

=Z

q

Z

β

=D , since Zq∩

(

D Z\ j

)

=Pj≠ ∅, ;

D

β

=D

(

) (

)

(

) (

)

(

) (

(

)

0

0 ,

j j q q

j j q

Y Z Y Z Y D

Y Z Y D Y D

Y Z Y Y D

δ δ δ

δ β β β β

α = × ∪ × ∪ × = × ∪ × ∪ × = × ∪ ∪ × =

 

 



if Yj Yj

δ ₌ α

, Yq Y0 Y0

δ _∪ δ ₌ α

and q=1, 2,,n+k. Last equalities are possible

since Yq Y0 1

δ_∪ δ _≥ ₍

0 0

Yδ ≥ by preposition).

(

X n, + +k 1

)

bi-nary relation α_{has a form}

(

Yn j Zn j

) (

Yj Zj

)

α α

α

= + × + ∪ × , where Yn j,Yj

{ }

α α + ∉ ∅ ,

1, 2, ,

j=  k, then binary relation α is generated by elements of the elements of set B

( )

A0 .

Proof. Let quasinormal representation of a binary relations δ_,β_{have a form}

(

) (

)

(

)

(

)

(

)

0 ,

\ \ ,

n j n j q q n j n j n j j

Y Z Y Z Y D

Z Z D Z Z X D D

δ δ δ

δ β

+ +

+ + +

= × ∪ × ∪ ×



  

where Y_{n j}δ₊ ,Y_qδ∉ ∅

{ }

, j≠q and j=1, 2,,k. Then from the Lemma 2.2

follows that β is generated by elements of the set B

( )

A0 ,

δ

∈B

( )

A0 and ,

n j n j

Z ₊

β

=Z ₊

q n j j j

Z

β

=Z ₊ ∪Z =Z , since Z_q∩Z_{n j}₊ ≠ ∅ ,

(

\

)

,

q n j n j

Z ∩ D Z+ =P+ ≠ ∅ ≠j q 

(see equality (2.0))

j

D

β

=Z since D∩

(

X D\ 

)

= ∅,

(

) (

)

(

) (

)

(

) (

(

)

0

0 ,

n j n j q q

n j n j q j j

n j n j q j

Y Z Y Z Y D

Y Z Y Z Y Z

Y Z Y Y Z

δ δ δ

δ β β β

α

+ +

= × ∪ × ∪ × = × ∪ × ∪ × = × ∪ ∪ × =

 

if Yn j Yn j

δ α

+ = + and Yq Y0 Yj

δ _∪ δ ₌ α

. Last equalities are possible since

0 1

q

Yδ∪Yδ ≥ (Y0 0

δ _≥ _{by preposition).} Lemma 2.5 is proved.

(

X n, + +k 1

)

. Then the following statements are true:

(10)

(

j=1, 2,,k

)

, then binary relation α is generated by elements of the set

( )

0

B A .

2) If quasinormal representation of a binary relation α_{has a form}α = ×X D,

then binary relation α_{is generated by elements of the set} B

( )

A0 .

Proof. 1)Let T∈D\

(

D2∪D3

)

. If quasinormal representation of a binary

re-lations δ_,β_{have a form}

(

) (

)

(

) (

(

)

(

)

0 ,

\ \ ,

j j

n j n j j n j j

Y Z Y D

Z Z Z Z Z X D D

δ δ

δ

β ₊ ₊ ₊

= × ∪ ×

= × ∪ × ∪ ×



 

where Y Yj, 0

{ }

δ δ_{∈ ∅} _,

1, 2, ,

j=  k

(

)

(

)

(

)

(

)

(

)

0, ,

\ \

n j j n j

n k

i j n j

i i j n j

Z Z Z X D

P P P X D D X D X

+ +

+

+ =

≠ +

∪ ∪

   

=_ _∪ ∪ ∪ = ∪ =  

 



  



(see equalities (2.0) and (2.1)), then from the Lemma 2.4 follows that δ_is

gener-ated by elements of the set B

( )

A0 and from the Lemma 2.3 element β is

gen-erated by elements of the set B

( )

A0 and

,

j n j j j

Z

β

=Z ₊ ∪Z =Z

j

D

β

=Z , since D∩

(

X D\ 

)

= ∅,

(

Yj Zj

) (

Y0 D

) (

Yj Zj

) (

Y0 Zj

)

X Zj ,

δ δ δ δ

δ β

 = ×

β

∪ × 

β

= × ∪ × = × =

α

since representation of a binary relation δ_{is quasinormal.}

The statement a) of the lemma 2.6 is proved.

2) Let quasinormal representation of a binary relation δ_{have a form}

(

Z_{n j} Z_q

) (

(

X Z\ _{n j}

)

D

)

,

δ = ₊ × ∪ ₊ × 

where j≠q, then from the Lemma 2.4 follows that δ is generated by elements

of the set B

( )

A0 and

0, 0,

,

n k n k

q i i q

i i

i q i q

Z δ P δ Pδ Z D D Dδ D

+ +

= =

≠ ≠

       

=_ _ =_ _= ∪ = =    

   

   



, since

1

, _q _n

j≠q Z

δ

∩Z ₊ ≠ ∅ and Zq

δ

∩

(

X Z\ n+₁

)

≠ ∅;

(

) (

(

)

(

)

(

)

\

n j q n j n j n j

Z Z X Z D

Z D X Z D X D

δ δ δ δ

α

+ +

= × ∪ ×

= × ∪ × = × =

 

   ,

since representation of a binary relation δ_{is quasinormal.}

The statement b) of the lemma 2.6 is proved. Lemma 2.6 is proved.

(

X n, + +k 1

)

. Then the following statements are true:

a) If X D\  ≥1 and T∈D2∪D3, then binary relation

α = ×

X T

is

(11)

b) If X =D and T∈D₂∪D₃, then binary relation

α = ×

X T

is external

element for the semigroup BX

( )

D .

Proof. 1) If quasinormal representation of a binary relation δ_{has a form}

(

0

)

(

)

1

n k

j j j k

Yδ D Yδ Z

δ +

= +

= ×  ∪



× _,

where Yj

δ _{≠ ∅}_{for all}

1, 2, ,

j= +k k+  n+k, then

δ

∈B

( ) { }

A0 \

α

. Let

qua-sinormal representation of a binary relations β have a form

(

)

(

{ } ( )

)

\

t X D

D T t f t

β

′∈

′ ′ = × ∪ ×





, where f is any mapping of the set X D\  in the

set

(

D2∪D3

) { }

\ T . It is easy to see, that

β α

≠ and two elements of the set

2 3

D ∪D belong to the semilattice V D

(

,

β

)

, i.e.

δ

∈B

( ) { }

A0 \

α

. In this case

we have that Zj

β

=T for all j= +k 1,k+2,,n+k.

(

)

(

)

(

)

(

)

0

1

0

1

0 1

,

n k

j j j k

n k j j k n k

j j k

Y D Y Z

Y T Y T

Y Y T X T

δ δ

δ β δ β β

α

+ = + + = + + = +

= = × ∪ ×

= × ∪ ×

  

=_ ∪ × _= × =

 







since the representation of a binary relation δ_{is quasinormal. Thus, the element} α_{is generated by elements of the set} B

( )

A0 .

The statement a) of the lemma 2.7 is proved.

2) Let X =D,

α = ×

X T

, for some T∈D₂∪D₃ and

α δ β

=  for some

( ) { }

, BX D \

δ β

∈

α

_{. Then we obtain that} Z_j

β

=T since T is a minimal element

of the semilattice D.

Now, let subquasinormal representations

β

of a binary relation β_{have a form}

{ }

( )

(

2

)

0 \

n k

i

i t X D

P T t t

β

+

β

′

= ∈

   _′ _′ =__ _× _∪ ×

 









 ,

where 0 1 2

1

n k

P P P P

T T T T

β

 + 

=  

 



 is normal mapping. But complement mapping

2

β

is empty, since X D\ = ∅, i.e. in the given case, subquasinormal

represen-tation β _{of a binary relation}β_{is defined uniquely. So, we have that}

X T

β β

= = × =

α

_{(see property 2) in the case 1.1), which contradict the}

condi-tion, that

β

∉BX

( ) { }

D \

α

.

Therefore, if X =D and

α = ×

X T

, for some T∈D₂∪D₃, then α is

( )

D .

The statement 2) of the Lemma 2.7 is proved. Lemma 2.7 is proved.

Theorem 2.1. Let D∈Σ8

(

X n, + +k 1

)

, k≥3, and

{

}

{

}

{

}

1 1, 2, , k , 2 k1, k 2, , n , 3 n1, n2, , n k ;

D = Z Z  Z D = Z + Z +  Z D = Z + Z +  Z+

{

}

{

}

1= Zn q+ ,Z Dq, , whereq=1, 2, , ;k





(12)

{

}

{

}

2= Z Dj, , where j=1, 2, ,n+k;





A

{

}

{

}

3= Zn j+ ,Zj , where j=1, 2,, ;k

A

{ }

{

}

4 = Zj , D , where j=1, 2, ,n+k;





A

(

)

(

)

{

}

0= V D,

α

⊂D V D| ,

α

∉ 1∪ 2∪ 3∪ 4 ,

A A A A A

( )

0

{

X

( ) (

| ,

)

0

}

,

B A =

α

∈B D V D

α

∈A

{

}

0 | 2 3

B = X T T× ∉D ∪D

Then the following statements are true:

1) If X D\  ≥1, then the S0=B

( )

A0 is irreducible generating set for the semigroup BX

( )

D ;

2) If X =D, then the S1=B0∪B

( )

A0 is irreducible generating set for the semigroup BX

( )

D .

Proof. Let D∈Σ8

(

X n, + +k 1

)

, k≥3 and X D\ ≥1



. First, we proved that every element of the semigroup BX

( )

D is generated by elements of the set S0.

Indeed, let α_{be an arbitrary element of the semigroup} BX

( )

D . Then

quasi-normal representation of a binary relation α_{has a form}

(

0

)

(

)

1

n k

i i

i

Yα D Yα Z

α

+

=

= ×  ∪



× ,

where

0

n k

i i

Yα X

+

=



and Yi Yj

α_∩ α _{= ∅}

₍

₎

0≤ ≠ ≤ +i j n k . For the V X

(

∗,

α

)

we consider the following cases: 1) If V X

(

,

α

)

1 2 3 4

∗ _∉_A _∪_A _∪_A _∪_A _{, then}

( )

0

B S

α

∈ A₀ ⊆ by definition of

a set S0.

Now, let V X

(

,

α

)

1 2 3 4

∗ _∈_A _∪_A _∪_A _∪_A _. 2) If V X

(

,

α

)

1

∗ _∈_A _{, then quasinormal representation of a binary relation}_α has a form

(

Yn j Zn j

) (

Yj Zj

) (

Y0 D

)

α α α

α

= + × + ∪ × ∪ ×



, where Yn j,Yj ,Y0

{ }

α α α

+ ∉ ∅

(

j=1, 2,,k

)

and from the Lemma 2.3 follows that α is generated by elements

of the elements of set B

( )

A0 ⊆S0 by definition of a set S0.

3) If V X

(

,

α

)

2

∗ _∈_A _{, then quasinormal representation of a binary relation}_α has a form

(

Yj Zj

) (

Y0 D

)

α α

α

= × ∪ ×  _{, where}Y_jα,Y₀α∉ ∅

{ }

, j=1, 2,,n+k

and from the Lemma 2.4 follows that α_{is generated by elements of the elements}

of set B

( )

4) If V X

(

,

α

)

3

∗ _∈_A _{, then quasinormal representation of a binary relation}_α

has a form

(

Yn j Zn j

) (

Yj Zj

)

α α

α

= + × + ∪ × , where Yn j,Yj

{ }

α α

+ ∉ ∅ , j=1, 2,,k and from the Lemma 2.5 follows that α_{is generated by elements of the elements}

of set B

( )

Now, let V X

(

,

α

)

4

∗ _∈

A , then quasinormal representation of a binary

rela-tion α has a form α = ×X D, or

α

= ×X Z_j, where j=1, 2,,n+k.

5) If α = ×X D, then from the statement b) of the Lemma 2.6 follows that

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6) If

α

= ×X Zj, where j=1, 2,,n+k, then from the statement a) of the

Lemma 2.6 and 2.7 follows that binary relation α_{is generated by elements of the}

set B

( )

A0 .

Thus, we have that S0 is a generating set for the semigroup BX

( )

D .

If X D\  ≥1, then the set S0 is an irreducible generating set for the

semi-group BX

( )

D since, S0 is a set external elements of the semigroup BX

( )

D .

The statement a) of the Theorem 2.1 is proved.

Now, let X =D. First, we proved that every element of the semigroup

( )

X

B D is generated by elements of the set S1. The cases 1), 2), 3), 4) and 5)

are proved analogously of the cases 1), 2), 3), 4) and 5 given above and consider case, when V X

(

,

α

)

1

∗ _∈

A .

If V X

(

,

α

)

Zj

∗ ₌ _{, where}

1, 2, ,

j=  k, then from the statement a) of the

Lemma 2.7 follows that binary relation α is generated by elements of the set

( )

0

B A .

If V X

(

,

α

)

Zj

∗ ₌ _{, where}

2 3

j

Z ∈D ∪D , then from the statement b) of the

Lemma 2.6 follows that binary relation

α

= ×X T is external element for the

semigroup BX

( )

D .

Thus, we have that S1 is a generating set for the semigroup BX

( )

D .

If X =D, then the set S1 is an irreducible generating set for the semigroup

( )

X

B D since S₁ is a set external elements of the semigroup BX

( )

D .

The statement b) of the Theorem 2.1 is proved. Theorem 2.1 is proved.

Corollary 2.1. Let D∈Σ8

(

X n, + +k 1

)

(

k≥3

)

and

{

}

{

}

{

}

1 1, 2, , k , 2 k1, k 2, , n , 3 n1, n2, , n k ;

D = Z Z  Z D = Z + Z +  Z D = Z + Z +  Z +

{

}

{

}

1= Zn q+ ,Z Dq, , whereq=1, 2, , ;k





A

{

}

{

}

2= Z Dj, , where j=1, 2, ,n+k;





A

{

}

{

}

3= Zn j+ ,Zj , where j=1, 2,, ;k

A

{ }

{

}

4 = Zj , D , where j=1, 2, ,n+k;





A

(

)

(

)

{

}

0= V D,

α

⊂D V D| ,

α

∉ 1∪ 2∪ 3∪ 4 ,

A A A A A

( )

0

{

X

( ) (

| ,

)

0

}

,

B A =

α

∈B D V D

α

∈A

{

}

0 | 2 3

B = X T T× ∉D ∪D

Then the following statements are true:

1) If X D\  ≥1, then S0=B

( )

A0 is the uniquely defined generating set for the semigroup BX

( )

D ;

2) If X =D, then S1=B0∪B

( )

A0 is the uniquely defined generating set for the semigroup BX

( )

D .

Proof. It is well known, that if B is all external elements of the semigroup

( )

X

B D and

B′

is any generated set for the B_X

( )

D , then B⊆B′ (see [1][2]

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( )

1 0

S =B ∪B A0 are defined uniquely, since they are sets external elements of

the semigroup BX

( )

D .

Corollary 2.1 is proved.

It is well-known, that if B is all external elements of the semigroup BX

( )

D

and

B′

is any generated set for the BX

( )

D , then B⊆B′ (Definition 1.1).

In this article, we find irredusible generating set for the complete semigroups of binary relations defined by X-semilattices of unions of the class

(

)

8 X n, k 1

Σ + +

(

k≥3

)

. This generating set is uniquely defined, since they are

defined by elements of the external elements of the semigroup BX

( )

D .

References

[1] Diasamidze, Y. and Makharadze, S. (2013) Complete Semigroups of Binary Rela-tions. Kriter, Turkey, 1-519.