Complete Semigroups of Binary Relations Defined by Semilattices of the Class ∑1(X,10)

(1)

http://dx.doi.org/10.4236/am.2015.62026

Complete Semigroups of Binary Relations

Defined by Semilattices of the Class

(

X

)

1 Σ

,10

Shota Makharadze

1

, Neşet Aydın

2

, Ali Erdoğan

3

1_{Shota Rustavelli University, Batum, Georgia}

2_{Çanakkale Onsekiz Mart University, Çanakkale, Turkey} 3_{Hacettepe University, Ankara, Turkey}

Email: [email protected], [email protected], [email protected]

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

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

Abstract

In this paper we give a full description of idempotent elements of the semigroup BX (D), which are defined by semilattices of the class Σ1 (X, 10). For the case where X is a finite set we derive

formu-las by means of which we can calculate the numbers of idempotent elements of the respective se-migroup.

Keywords

Semilattice, Semigroup, Binary Relation

1. Introduction

Let X be an arbitrary nonempty set, D be an X-semilattice of unions, i.e. such a nonempty set of subsets of the

set X that is closed with respect to the set-theoretic operations of unification of elements from D, f be an

arbi-trary mapping of the set X in the set D. To each such a mapping f we put into correspondence a binary relation

f

α 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

semi-group with respect to the operation of multiplication of binary relations, which is called a complete semisemi-group of

(2)

Recall that we denote by ∅ an empty binary relation or empty subset of the set X. The condition

( )

x y, ∈α will be written in the form xαy. Further let , x y∈X , Y ⊆X , α ∈B_X

( )

D , T∈D, ∅ ≠D′⊆D,

D= ∪D and t∈D . Then by symbols we denoted the following sets:

{

}

{

}

{ }

(

)

{

}

{

}

{

}

{

}

(

)

(

)

, , 2 , 2 \ ,

, , , ,

, , \ .

X X

y Y

T T

t T

y x X y x Y y Y Y X X

V D Y Y D D T D T T D T D T T

D Z D t Z l D T D D

α α α α

α α

∗ ∈

= ∈ = = ⊆ = ∅

′ ′ ′ ′ ′ ′ ′ ′

= ∈ = ∈ ⊆ = ∈ ⊆

′= ′∈ ′ ∈ ′ ′ = ∪ ′ ′





By symbol Λ

(

D D, ′

)

is denoted an exact lower bound of the set D' in the semilattice D.

Definition 1. We say that the complete X-semilattice of unions D is an XI-semilattice of unions if it satisfies the following two conditions:

a) Λ

(

D D, t

)

∈D for any t∈D 

;

b)

(

, _t

)

t Z

Z=



_∈ Λ D D for any nonempty element Z of the semilattice D.

Definition 2. We say that 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 element of the set D' if T l D T\

(

′,

)

= ∅.

Definition 3. Let α ∈B_X

( )

D , T∈V X

(

∗,

α

)

, Y_Tα =

{

y∈X y

α

=T

}

. A representation of a binary relation

α

of the form

₍

₎

(

)

, T

T V X Y T

α α

α=



_∈ ∗ × is called quasinormal.

Note that, if

₍

₎

(

)

, T

T V X Y T

α α

α=



_∈ ∗ × is a quasinormal representation of the binary relation

α

, then the

fol-lowing conditions are true:

1)

₍

₎

, T T V X X =



_∈ ∗_α Yα;

2) Y_Tα∩Y_Tα′ = ∅ for T T, ,′∈V X

(

∗

α

)

and T≠T′.

Let

∑

_n

(

X m,

)

denote the class of all complete X-semilattices of unions where every element is isomorphic

to a fixed semilattice D.

The following Theorems are well know (see [1] and [3]).

Theorem 4. Let X be a finite set;δand q be respectively the number of basic sources and the number of all automorphisms of the semilattice D. If X = ≥n δ and Σ_n

(

X m,

)

=s, then

( )

( ) (

(

)

( ) (

)

1 1

1

1 ! !

1

1 ! 1 !

p i p n

p m

m p

p i

C C p i

s

q i p i

δ δ

δ

δ δ

+ + −

+

−

= =

  ₋ _⋅ _⋅ _⋅ _⋅ ₋ _⋅ 

  

= ⋅

 

 _ − ⋅ − + _

 

∑ ∑

where

( ) (

! !

)

!

k j

j C

k j k

=

⋅ − (see Theorem 11.5.1 [1]).

Theorem 5. Let D be a complete X-semilattice of unions. The semigroup B_X

( )

D possesses right unit iff D is an XI-semilattice of unions (see Theorem 6.1.3 [1]).

Theorem 6. Let X be a finite set and D

( )

α be the set of all those elements T of the semilattice

(

,

) { }

\

Q=V Dα ∅ which are nonlimiting elements of the set Q_T. A binary relation

α

having a quasinormal

representation ₍ ₎

(

)

, T

T V D Y T

α α

α=

_

_∈ × is an idempotent element of this semigroup iff a) V D

(

,α

)

is complete XI-semilattice of unions;

b) _{( )}

T T

T D Y T

α

α ′

′∈ ⊇



for any T∈D

( )

α ;

c) Y_Tα∩ ≠ ∅T for any nonlimiting element of the set

( )

T

D α (see Theorem 6.3.9 [1]).

Theorem 7. Let D, Σ

( )

D , E( )Xr

( )

D′ and I denote respectively the complete X-semilattice of unions, the set of all XI-subsemilatices of the semilattice D, the set of all right units of the semigroup B_X

( )

D′ and the set of all idempotents of the semigroup B_X

( )

D . Then for the sets E( )Xr

( )

D′ and I the following statements are true:

1) if ∅ ∈D and Σ_∅

( )

D =

{

D′∈ Σ

( )

D ∅ ∈D′

}

then

a) EX( )r

( )

D′ ∩E( )Xr

( )

D′′ = ∅ for any elements D′ and D′′ of the set Σ∅

( )

D that satisfy the condition D′≠D′′;

b) _{( )} ( )_Xr

( )

D D

I E D

∅

′∈Σ ′

=

_

c) the equality _{( )} ( )r

( )

X

D D

I E D

∅

′∈Σ ′

(3)

2) if ∅ ∉D, then

a) EX( )r

( )

D′ ∩E( )Xr

( )

D′′ = ∅ for any elements D′ and D′′ of the set Σ

( )

D that satisfy the condition

D′≠D′′;

b) _{( )} ( )_Xr

( )

D D

I=



_′∈Σ E D′

c) the equality _{( )} ( )_Xr

( )

D D

I =

∑

_′∈Σ E D′ is fulfilled for the finite set X (see Theorem 6.2.3 [1]).

Corollary 1. Let Y=

{

y y₁, ₂,,yk

}

and Dj =

{

T T1, 2,,Tj

}

be some sets, where k≥1 and j≥1. Then the number s k j

( )

, of all possible mappings of the set Y into any such subset D_j′ of the set D that _j Tj∈D′j can be calculated by the formula

( )

, k

(

1

)

k

s k j = j − −j (see Corollary 1.18.1 [1]).

2 . Idempotent Elements of the Semigroups

B D

_X

( )

Defined by Semilattices of the

Class

Σ

₁

(

X

,10

)

Let X and Σ₁

(

X,10

)

be respectively an arbitrary nonempty set and a class X-semilattices of unions, where

each element is isomorphic to some X-semilattice of unions D=

{

Z Z Z Z Z Z Z Z Z D₉, ₈, ₇, ₆, ₅, ₄, ₃, ₂, ₁,

}

that satis-fies the conditions:

9 4 1 9 5 1

9 6 1 9 6 2

9 6 3 9 7 3

9 8 3 1 2 2 1

1 3 3 1 2 3

3 2 4 5 5 4

4 6 6 4 4 7

7 4 4 8

, , , , , ,

, \ , \ ,

\ , \ , \ ,

\ , \ ,

Z Z Z D Z Z Z D

Z Z Z D Z Z Z Z

Z Z Z Z Z Z

Z Z Z Z

⊂ ⊂ ⊂ ⊂ ⊂ ⊂

⊂ ⊂ ⊂ ≠ ∅ ≠ ∅

≠ ∅ ≠ ∅ ≠ ∅

≠ ∅ ≠ ∅

 



8 4

5 6 6 5 5 7

7 5 5 8 8 5

6 7 7 6 6 8

8 6 7 8 8 7

1 2 1 3 2 3 4 2

4 3 4 7 4 8 5 2

5 3 5 7 5 8 7 1

7 2 8

\ ,

\ , \ , \ ,

Z Z

Z Z Z Z Z Z

Z Z Z Z Z Z Z Z

Z Z Z

≠ ∅

≠ ∅ ≠ ∅ ≠ ∅

∪ = ∪ = ∪ = ∪

= ∪ = ∪ = ∪ = ∪

= ∪ = ∪ ₁ ₈ ₂

4 5 4 6 5 6 1

6 7 6 8 7 8 3

, ,

.

Z Z Z D

Z Z Z Z Z Z Z

= ∪ =

∪ = ∪ = ∪ =



(1)

An X-semilattice that satisfies conditions (1) is shown inFigure 1.

[image:3.595.245.383.596.703.2]

Let C D

( ) {

= P P P P P P P P P P₀, ,₁ ₂, ₃, ₄, ₅, ₆, ₇, ₈, ₉

}

be a family of sets, where P0, P1, P2, P3, P4, P5, P6, P7, P8, P9

(4)

are pairwise disjoint subsets of the set X and 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

D Z Z Z Z Z Z Z Z Z P P P P P P P P P P

ϕ

=  

 



be a map-

ping of the semilattice D onto the family sets C D

( )

. Then for the formal equalities of the semilattice D we

have a form:

0 1 2 3 4 5 6 7 8 9

1 0 2 3 4 5 6 7 8 9

2 0 1 3 4 5 6 7 8 9

3 0 1 2 4 5 6 7 8 9

4 0 2 3 5 6 7 8 9

5 0 2 3 4 6 7 8 9

6 0 4 5 7 8 9

7

, , , , , , ,

D P P P P P P P P P P Z P P P P P P P P P Z P P P P P P P P P Z P P P P P P P P P Z P P P P P P P P Z P P P P P P P P Z P P P P P P

Z

= ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪

= ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪

= ∪ ∪ ∪ ∪ ∪ ∪ ∪

= ∪ ∪ ∪ ∪ ∪



0 1 2 4 5 6 8 9

8 0 1 2 4 5 6 7 9

9 0

, , .

P P P P P P P P Z P P P P P P P P Z P

= ∪ ∪ ∪ ∪ ∪ ∪ ∪

=

(2)

Here the elements P1, P2, P3, P4, P5, P6, P7, P8 are basis sources, the elements P0, P6, P9 are sources of

com-pleteness of the semilattice D. Therefore X ≥7 and

δ

=7 (see [2]).

Lemma 1. Let D∈ Σ₁

(

X,10

)

, Σ1

(

X,10

)

=s and X ≥ ≥δ 7. If X is a finite set, then

( )

(

)

1

1 4 7 5 21 6 35 7 35 8 21 9 11

8

n n n n n n n

s= − × + × − × + × − × + × + .

Proof. In this case we have: m = 10, δ = 7. Notice that an X-semilattice given inFigure 1 has eight automor-phims. By Theorem 1.1 it follows that

( )

( ) (

(

)

(

) (

)

1 7 7

1 10

3

7 1

1 7! 7 !

1

8 1 ! 1 !

p i p n

p

p i

C C p i

s

i p i

+ + −

+

= =

  ₋ _⋅ _⋅ _⋅ _⋅ ₋ _⋅ 

  

= ⋅

 

 _ − ⋅ − + _

 

∑ ∑

,

where

(

!

)

! !

k j

j C

k j k

=

⋅ − and that

( )

(

)

1

1 4 7 5 21 6 35 7 35 8 21 9 11

8

n n n n n n n

s= − × + × − × + × − × + × + .

Example 8. Let n=7, 8, 9, 10 Then:

( )

7 8 9 10

10 , 10 , 10 , 10

X

B D = .

(

X,10

)

. Then the following sets are all proper subsemilattices of the semilattice

{

9, 8, 7, 6, 5, 4, 3, 2, 1,

}

D= Z Z Z Z Z Z Z Z Z D :

1)

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

Z9 , , , , , , , , , Z8 Z7 Z6 Z5 Z4 Z3 Z2 Z1

{ }

D 

(see diagram 1 of the Figure 2);

2)

{

Z Z₉, ₈

} {

, ,Z Z₉ ₇

} {

, ,Z Z₉ ₆

} {

, ,Z Z₉ ₅

} {

, ,Z Z₉ ₄

} {

, ,Z Z₉ ₃

} {

, ,Z Z₉ ₂

} {

, ,Z Z₉ ₁

}

, ,

{

Z D₉ 

}

, ,

{

Z Z₈ ₃

}

,

{

Z D8,

}

, ,

{

Z Z7 3

}

, ,

{

Z D7

}

, ,

{

Z Z6 3

} {

, ,Z Z6 2

} {

, ,Z Z6 1

}

, ,

{

Z D6

}

, ,

{

Z Z5 1

}

, ,

{

Z D5

}

, ,

{

Z Z4 1

}

,

   

{

Z D4,

} {

, ,Z D3

} {

, ,Z D2

} { }

, ,Z D1

   

3)

{

Z Z Z₉, ₈, ₃

}

, ,

{

Z Z D₉ ₈, 

}

, ,

{

Z Z Z₉ ₇, ₃

}

, ,

{

Z Z D₉ ₇,

}

, ,

{

Z Z Z₉ ₆, ₃

} {

, ,Z Z Z₉ ₆, ₂

} {

, ,Z Z Z₉ ₆, ₁

}

,

{

Z Z D9, 6,

}

, ,

{

Z Z Z9 5, 1

}

, ,

{

Z Z D9 5,

}

, ,

{

Z Z Z9 4, 1

}

, ,

{

Z Z D9 4,

} {

, ,Z Z D9 3,

} {

, ,Z Z D9 2,

}

,

(5)

{

Z Z D9, 1,

} {

, ,Z Z D8 3,

} {

, ,Z Z D7 3,

} {

, ,Z Z D6 3,

} {

, ,Z Z D6 2,

} {

, ,Z Z D6 1,

} {

, ,Z Z D5 1,

} {

, ,Z Z D4 1,

}

       

(see diagram 3 of the Figure 2);

4)

{

Z Z Z D9, 4, 1,

} {

, ,Z Z Z D9 5, 1,

} {

, ,Z Z Z D9 6, 1,

} {

, ,Z Z Z D9 6, 2,

} {

, ,Z Z Z D9 6, 3,

}

,

    

{

Z Z Z D9, 7, 3,

} {

, Z Z Z D9, 8, 3,

}

 

5)

{

Z Z Z Z₉, ₅, ₄, ₁

} {

, ,Z Z Z Z₉ ₆, ₄, ₁

} {

, ,Z Z Z Z₉ ₆, ₅, ₁

} {

, ,Z Z Z Z₉ ₇, ₆, ₃

} {

, ,Z Z Z Z₉ ₈, ₆, ₃

} {

, ,Z Z Z Z₉ ₈, ₇, ₃

}

,

{

Z Z Z D9, 8, 4,

} {

, Z Z Z D9, 8, 5,

} {

, Z Z Z D9, 7, 2,

} {

, Z Z Z D9, 7, 4,

} {

, Z Z Z D9, 7, 5,

} {

, Z Z Z D9, 8, 1,

}

,

     

{

Z Z Z D9, 8, 2,

} {

, Z Z Z D9, 4, 2,

} {

, Z Z Z D9, 4, 3,

} {

, Z Z Z D9, 5, 2,

} {

, Z Z Z D9, 5, 3,

} {

, Z Z Z D9, 7, 1,

}

,

     

{

Z Z Z D9, 2, 1,

} {

, ,Z Z Z D9 3, 1,

} {

, ,Z Z Z D9 3, 2,

} {

, ,Z Z Z D6 2, 1,

} {

, ,Z Z Z D6 3, 1,

} {

, ,Z Z Z D6 3, 2,

}

;

     

6)

{

Z Z Z Z D₉, ₅, ₄, ₁, 

} {

, Z Z Z Z D₉, ₆, ₄, ₁,

} {

, Z Z Z Z D₉, ₆, ₅, ₁, 

} {

, Z Z Z Z D₉, ₇, ₆, ₃,

}

,

{

Z Z Z Z D9, 8, 6, 3,

} {

, Z Z Z Z D9, 8, 7, 3,

}

 

7)

{

Z Z Z Z D9, 6, 2, 1,

} {

, Z Z Z Z D9, 6, 3, 1,

} {

, Z Z Z Z D9, 6, 3, 2,

}

  

8)

{

Z Z Z Z Z D₉, ₈, ₆, ₃, ₂, 

} {

, Z Z Z Z Z D₉, ₈, ₆, ₃, ₁,

} {

, Z Z Z Z Z D₉, ₇, ₆, ₃, ₂,

} {

, Z Z Z Z Z D₉, ₇, ₆, ₃, ₁, 

}

,

{

Z Z Z Z Z D9, 6, 5, 3, 1,

} {

, Z Z Z Z Z D9, 6, 5, 2, 1,

} {

, Z Z Z Z Z D9, 6, 4, 3, 1,

} {

, Z Z Z Z Z D9, 6, 4, 2, 1,

}

   

9)

{

Z Z Z8, 7, 3

} {

, ,Z Z Z8 6, 3

}

, ,

{

Z Z D8 6,

} {

, ,Z Z D8 5,

} {

, ,Z Z D8 4,

} {

, ,Z Z D8 2,

} {

, ,Z Z D8 1,

}

, ,

{

Z Z Z7 6, 3

}

,

    

{

Z Z D7, 5,

} {

, ,Z Z D7 4,

} {

, ,Z Z D7 2,

} {

, ,Z Z D7 1,

}

, ,

{

Z Z Z6 5, 1

} {

, ,Z Z Z6 4, 1

} {

, ,Z Z Z5 4, 1

}

, ,

{

Z Z D5 3,

}

,

    

{

Z Z D5, 2,

} {

, ,Z Z D4 3,

} {

, ,Z Z D4 2,

} {

, ,Z Z D3 2,

} {

, ,Z Z D3 1,

} {

, ,Z Z D2 1,

}

     

10)

{

Z Z Z D₈, ₆, ₃,

} {

, ,Z Z Z D₈ ₇, ₃,

} {

, ,Z Z Z D₇ ₆, ₃,

} {

, ,Z Z Z D₅ ₄, ₁,

} {

, ,Z Z Z D₆ ₄, ₁,

} {

, ,Z Z Z D₆ ₅, ₁,

}

11)

{

Z Z Z Z D₆, ₅, ₃, ₁,

} {

, Z Z Z Z D₆, ₅, ₂, ₁,

} {

, Z Z Z Z D₆, ₄, ₃, ₁,

} {

, Z Z Z Z D₆, ₄, ₂, ₁,

}

,

{

Z Z Z Z D7, 6, 3, 2,

} {

, Z Z Z Z D7, 6, 3, 1,

} {

, Z Z Z Z D8, 6, 3, 2,

} {

, Z Z Z Z D8, 6, 3, 1,

}

   

12)

{

Z Z Z Z6, 5, 4, 1

} {

, ,Z Z Z Z8 7, 6, 3

}

, ,

{

Z Z Z D8 2, 1,

} {

, ,Z Z Z D3 2, 1,

} {

, ,Z Z Z D4 3, 2,

} {

, ,Z Z Z D5 3, 2,

}

,

   

{

Z Z Z D7, 2, 1,

} {

, Z Z Z D7, 4, 2,

} {

, Z Z Z D7, 5, 2,

} {

, Z Z Z D8, 4, 2,

} {

, Z Z Z D8, 5, 2,

}

    

13)

{

Z Z Z D7, 5, 3,

} {

, Z Z Z D4, 2, 1,

} {

, Z Z Z D8, 3, 1,

} {

, Z Z Z D8, 3, 2,

} {

, Z Z Z D8, 4, 1,

} {

, Z Z Z D4, 3, 1,

}

,

     

{

Z Z Z D5, 2, 1,

} {

, Z Z Z D5, 3, 1,

} {

, Z Z Z D7, 3, 1,

} {

, Z Z Z D7, 3, 2,

} {

, Z Z Z D7, 4, 1,

} {

, Z Z Z D7, 4, 3,

}

,

     

{

Z Z Z D7, 5, 1,

} {

, ,Z Z Z D8 4, 3,

} {

, ,Z Z Z D8 5, 1,

} {

, ,Z Z Z D8 5, 3,

}

   

(6)

{

Z Z Z Z D9, 5, 3, 2,

} {

, Z Z Z Z D9, 7, 2, 1,

} {

, Z Z Z Z D9, 7, 4, 2,

} {

, Z Z Z Z D9, 7, 4, 3,

} {

, Z Z Z Z D9, 7, 5, 2,

}

,

    

{

Z Z Z Z D9, 8, 2, 1,

} {

, Z Z Z Z D9, 8, 4, 2,

} {

, Z Z Z Z D9, 8, 5, 2,

}

  

15)

{

Z Z Z Z D₉, ₄, ₂, ₁, 

} {

, Z Z Z Z D₉, ₄, ₃, ₁, 

} {

, Z Z Z Z D₉, ₅, ₂, ₁,

} {

, Z Z Z Z D₉, ₅, ₃, ₁, 

} {

, Z Z Z Z D₉, ₇, ₃, ₁, 

}

,

{

Z Z Z Z D9, 7, 3, 2,

} {

, Z Z Z Z D9, 7, 4, 1,

} {

, Z Z Z Z D9, 7, 5, 1,

} {

, Z Z Z Z D9, 7, 5, 3,

} {

, Z Z Z Z D9, 8, 3, 1,

}

,

    

{

Z Z Z Z D9, 8, 3, 2,

} {

, Z Z Z Z D9, 8, 4, 1,

} {

, Z Z Z Z D9, 8, 4, 3,

} {

, Z Z Z Z D9, 8, 5, 1,

} {

, Z Z Z Z D9, 8, 5, 3,

}

    

16)

{

Z Z Z Z D₅, ₄, ₃, ₁, 

} {

, Z Z Z Z D₅, ₄, ₂, ₁,

} {

, Z Z Z Z D₇, ₅, ₄, ₁, 

} {

, Z Z Z Z D₈, ₅, ₄, ₁,

} {

, Z Z Z Z D₈, ₇, ₃, ₂, 

}

,

{

Z Z Z Z D8, 7, 3, 1,

} {

, Z Z Z Z D8, 7, 5, 3,

} {

, Z Z Z Z D8, 7, 4, 3,

}

  

17)

{

Z Z Z Z D5, 3, 2, 1,

} {

, Z Z Z Z D4, 3, 2, 1,

} {

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

} {

, Z Z Z Z D7, 3, 2, 1,

} {

, Z Z Z Z D7, 5, 3, 2,

}

,

    

{

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

} {

, Z Z Z Z D7, 4, 3, 2,

} {

, Z Z Z Z D8, 3, 2, 1,

} {

, Z Z Z Z D8, 5, 3, 2,

} {

, Z Z Z Z D8, 5, 2, 1,

}

,

    

{

Z Z Z Z D8, 4, 3, 2,

} {

, Z Z Z Z D8, 4, 2, 1,

}

 

18)

{

Z Z Z Z D₇, ₄, ₃, ₁, 

} {

, Z Z Z Z D₇, ₅, ₃, ₁,

} {

, Z Z Z Z D₈, ₅, ₃, ₁, 

} {

, Z Z Z Z D₈, ₄, ₃, ₁,

}

(see diagram 18 of the Figure 2); 19)

{

Z Z Z Z D₆, ₅, ₄, ₁, 

} {

, Z Z Z Z D₈, ₇, ₆, ₃,

}

.

20)

{

Z Z Z Z Z D₈, ₇, ₅, ₃, ₂,

} {

, Z Z Z Z Z D₈, ₇, ₄, ₃, ₂, 

} {

, Z Z Z Z Z D₈, ₅, ₄, ₂, ₁, 

} {

, Z Z Z Z Z D₇, ₅, ₄, ₂, ₁,

}

,

{

Z Z Z Z Z D8, 7, 3, 2, 1,

} {

, Z Z Z Z Z D5, 4, 3, 2, 1,

}

 

21)

{

Z Z Z Z Z D₆, ₅, ₄, ₂, ₁,

} {

, Z Z Z Z Z D₈, ₇, ₆, ₃, ₂,

} {

, Z Z Z Z Z D₈, ₇, ₆, ₃, ₁,

} {

, Z Z Z Z Z D₆, ₅, ₄, ₃, ₁, 

}

22)

{

Z Z Z Z Z D9, 5, 4, 2, 1,

} {

, Z Z Z Z Z D9, 8, 7, 4, 3,

} {

, Z Z Z Z Z D9, 8, 7, 3, 1,

} {

, Z Z Z Z Z D9, 8, 5, 4, 1,

}

,

   

{

Z Z Z Z Z D9, 7, 5, 4, 1,

} {

, ,Z Z Z Z Z D9 5, 4, 3, 1,

} {

, ,Z Z Z Z Z D9 8, 7, 3, 2,

} {

, ,Z Z Z Z Z D9 8, 7, 5, 3,

}

   

23)

{

Z Z Z Z Z D₉, ₈, ₇, ₆, ₃,

} {

, Z Z Z Z Z D₉, ₆, ₅, ₄, ₁,

}

(see diagram 23 of the Figure 2);

24)

{

Z Z Z Z Z D9, 8, 5, 3, 2,

} {

, Z Z Z Z Z D9, 8, 5, 2, 1,

} {

, Z Z Z Z Z D9, 8, 4, 3, 2,

} {

, Z Z Z Z Z D9, 8, 4, 2, 1,

}

,

   

{

Z Z Z Z Z D9, 8, 3, 2, 1,

} {

, Z Z Z Z Z D9, 7, 5, 3, 2,

} {

, Z Z Z Z Z D9, 7, 5, 2, 1,

} {

, Z Z Z Z Z D9, 7, 4, 3, 2,

}

,

   

{

Z Z Z Z Z D9, 7, 4, 2, 1,

} {

, Z Z Z Z Z D9, 7, 3, 2, 1,

} {

, Z Z Z Z Z D9, 5, 3, 2, 1,

} {

, Z Z Z Z Z D9, 4, 3, 2, 1,

}

   

25)

{

Z Z Z Z Z D₉, ₈, ₅, ₃, ₁, 

} {

, Z Z Z Z Z D₉, ₈, ₄, ₃, ₁,

} {

, Z Z Z Z Z D₉, ₇, ₅, ₃, ₁, 

} {

, Z Z Z Z Z D₉, ₇, ₄, ₃, ₁, 

}

{

Z Z Z Z Z D9, 6, 3, 2, 1,

}



(7)

27)

{

Z Z Z Z Z D₈, ₇, ₅, ₃, ₁,

} {

, Z Z Z Z Z D₈, ₇, ₄, ₃, ₁, 

} {

, Z Z Z Z Z D₈, ₅, ₄, ₃, ₁,

} {

, Z Z Z Z Z D₇, ₅, ₄, ₃, ₁,

}

28)

{

Z Z Z Z Z D8, 6, 5, 3, 1,

} {

, Z Z Z Z Z D8, 6, 4, 3, 1,

} {

, Z Z Z Z Z D8, 5, 3, 2, 1,

} {

, Z Z Z Z Z D7, 6, 5, 3, 1,

}

,

   

{

Z Z Z Z Z D7, 6, 4, 3, 1,

}



29)

{

Z Z Z Z Z D₈, ₆, ₃, ₂, ₁, 

} {

, Z Z Z Z Z D₇, ₆, ₃, ₂, ₁, 

} {

, Z Z Z Z Z D₆, ₅, ₃, ₂, ₁, 

} {

, Z Z Z Z Z D₆, ₄, ₃, ₂, ₁,

}

30)

{

Z Z Z Z Z D8, 4, 3, 2, 1,

} {

, Z Z Z Z Z D7, 5, 3, 2, 1,

} {

, Z Z Z Z Z D7, 4, 3, 2, 1,

}

  

{

Z Z Z Z Z Z D8, 7, 5, 4, 3, 1,

}



32)

{

Z Z Z Z Z Z D₆, ₅, ₄, ₃, ₂, ₁,

} {

, Z Z Z Z Z Z D₈, ₇, ₆, ₃, ₂, ₁,

}

33)

{

Z Z Z Z Z Z D₇, ₅, ₄, ₃, ₂, ₁, 

} {

, Z Z Z Z Z Z D₈, ₅, ₄, ₃, ₂, ₁,

} {

, Z Z Z Z Z Z D₈, ₇, ₄, ₃, ₂, ₁,

}

,

{

Z Z Z Z Z Z D8, 7, 5, 3, 2, 1,

}



34)

{

Z Z Z Z Z Z D7, 6, 5, 4, 3, 1,

} {

, Z Z Z Z Z Z D8, 6, 5, 4, 3, 1,

} {

, Z Z Z Z Z Z D8, 7, 6, 4, 3, 1,

}

,

  

{

Z Z Z Z Z Z D8, 7, 6, 5, 3, 1,

}



35)

{

Z Z Z Z Z Z D9, 6, 4, 3, 2, 1,

} {

, Z Z Z Z Z Z D9, 6, 5, 3, 2, 1,

} {

, Z Z Z Z Z Z D9, 7, 6, 3, 2, 1,

}

,

  

{

Z Z Z Z Z Z D9, 8, 6, 3, 2, 1,

} {

, Z Z Z Z Z Z D9, 8, 6, 3, 2, 1,

}

 

36)

{

Z Z Z Z Z Z D9, 6, 5, 4, 3, 1,

} {

, Z Z Z Z Z Z D9, 8, 7, 6, 3, 1,

} {

, Z Z Z Z Z Z D9, 8, 7, 6, 3, 1,

}

  

37)

{

Z Z Z Z Z Z D9, 7, 4, 3, 2, 1,

} {

, Z Z Z Z Z Z D9, 7, 5, 3, 2, 1,

} {

, Z Z Z Z Z Z D9, 8, 4, 3, 2, 1,

}

,

  

{

Z Z Z Z Z Z D9, 8, 5, 3, 2, 1,

}



38)

{

Z Z Z Z Z Z D9, 7, 5, 4, 3, 1,

} {

, Z Z Z Z Z Z D9, 8, 5, 4, 3, 1,

} {

, Z Z Z Z Z Z D9, 8, 7, 4, 3, 1,

}

  

(see diagram 38 of the Figure 2); 39)

{

Z Z Z Z Z Z D9, 7, 6, 5, 3, 1,

}



40)

{

Z Z Z Z Z Z D9, 7, 5, 4, 2, 1,

} {

, Z Z Z Z Z Z D9, 8, 5, 4, 2, 1,

} {

, Z Z Z Z Z Z D9, 8, 7, 4, 3, 2,

}

,

  

{

Z Z Z Z Z Z D9, 8, 7, 5, 3, 2,

} {

, Z Z Z Z Z Z D9, 5, 4, 3, 2, 1,

}

 

41)

{

Z Z Z Z Z Z D9, 6, 5, 4, 2, 1,

} {

, Z Z Z Z Z Z D9, 8, 7, 6, 3, 2,

}

 

(8)

42)

{

Z Z Z Z Z Z Z D₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

43)

{

Z Z Z Z Z Z Z D8, 6, 5, 4, 3, 2, 1,

} {

, Z Z Z Z Z Z Z D8, 7, 6, 4, 3, 2, 1,

} {

, Z Z Z Z Z Z Z D8, 7, 6, 5, 3, 2, 1,

}

  

{

Z Z Z Z Z Z Z D8, 7, 5, 4, 3, 2, 1,

}



{

Z Z Z Z Z Z Z D₉, ₈, ₇, ₅, ₄, ₃, ₁,

}

46)

{

Z Z Z Z Z Z Z D₉, ₆, ₅, ₄, ₃, ₂, ₁, 

} {

, Z Z Z Z Z Z Z D₉, ₈, ₇, ₆, ₃, ₂, ₁, 

}

47)

{

Z Z Z Z Z Z Z D₉, ₇, ₅, ₄, ₃, ₂, ₁, 

} {

, Z Z Z Z Z Z Z D₉, ₈, ₇, ₅, ₃, ₂, ₁, 

} {

, Z Z Z Z Z Z Z D₉, ₇, ₅, ₄, ₃, ₂, ₁,

}

,

{

Z Z Z Z Z Z Z D9, 8, 7, 4, 3, 2, 1,

}



48)

{

Z Z Z Z Z Z Z D₉, ₇, ₆, ₅, ₄, ₃, ₁,

} {

, Z Z Z Z Z Z Z D₉, ₈, ₆, ₅, ₄, ₃, ₁, 

} {

, Z Z Z Z Z Z Z D₉, ₈, ₇, ₆, ₄, ₃, ₁,

}

,

{

Z Z Z Z Z Z Z D9, 8, 7, 6, 5, 3, 1,

}



[image:8.595.161.470.369.713.2]

(9)

49)

{

Z Z Z Z Z Z Z Z D₈, ₇, ₆, ₅, ₄, ₃, ₂, ₁, 

}

50)

{

Z Z Z Z Z Z Z Z D₉, ₇, ₆, ₅, ₄, ₃, ₂, ₁,

} {

, Z Z Z Z Z Z Z Z D₉, ₈, ₆, ₅, ₄, ₃, ₂, ₁, 

} {

, Z Z Z Z Z Z Z Z D₉, ₈, ₇, ₆, ₄, ₃, ₂, ₁, 

}

,

{

Z Z Z Z Z Z Z Z D9, 8, 7, 6, 5, 3, 2, 1,

}



{

Z Z Z Z Z Z Z Z D₉, ₈, ₇, ₅, ₄, ₃, ₂, ₁, 

}

(see diagram 51 of the Figure 2); 52)

{

Z Z Z Z Z Z Z Z D₉, ₈, ₇, ₆, ₅, ₄, ₃, ₁, 

}

Diagrams of subsemilattices of the semilattice D.

(

X,10

)

. Then the following sets are all XI-subsemi-lattices of the given semilattice D: 1)

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

Z₉ , , , , , , , , , Z₈ Z₇ Z₆ Z₅ Z₄ Z₃ Z₂ Z₁

{ }

D

2)

{

Z D₉,

}

, ,

{

Z Z₉ ₈

} {

, ,Z Z₉ ₇

} {

, ,Z Z₉ ₆

} {

, ,Z Z₉ ₅

} {

, ,Z Z₉ ₄

} {

, ,Z Z₉ ₃

} {

, ,Z Z₉ ₂

} {

, ,Z Z₉ ₁

} {

, ,Z Z₈ ₃

}

,

{

Z D8,

}

, ,

{

Z Z7 3

}

, ,

{

Z D7

}

, ,

{

Z Z6 3

} {

, ,Z Z6 2

} {

, ,Z Z6 1

}

, ,

{

Z D6

}

, ,

{

Z Z5 1

}

, ,

{

Z D5

}

, ,

{

Z Z4 1

}

,

   

{

Z D4,

} {

, ,Z D3

} {

, ,Z D2

} { }

, ,Z D1

   

3)

{

Z Z D₉, ₈,

} {

, Z Z D₉, ₇, 

} {

, Z Z D₉, ₆,

} {

, Z Z D₉, ₅, 

} {

, Z Z D₉, ₄,

} {

, Z Z D₉, ₃,

} {

, Z Z D₉, ₂,

} {

, Z Z D₉, ₁, 

}

,

{

Z Z Z9, 8, 3

} {

, ,Z Z Z9 7, 3

} {

, ,Z Z Z9 6, 3

} {

, ,Z Z Z9 6, 2

} {

, ,Z Z Z9 6, 1

} {

, ,Z Z Z9 5, 1

} {

, ,Z Z Z9 4, 1

}

,

{

Z Z D8, 3,

} {

, ,Z Z D7 3,

} {

, ,Z Z D6 3,

} {

, ,Z Z D6 2,

} {

, ,Z Z D6 1,

} {

, ,Z Z D5 1,

} {

, ,Z Z D4 1,

}

      

4)

{

Z Z Z D₉, ₄, ₁,

} {

, Z Z Z D₉, ₅, ₁, 

} {

, Z Z Z D₉, ₆, ₁,

} {

, Z Z Z D₉, ₆, ₂,

} {

, Z Z Z D₉, ₆, ₃, 

}

,

{

Z Z Z D9, 7, 3,

} {

, Z Z Z D9, 8, 3,

}

 

5)

{

Z Z Z Z₉, ₅, ₄, ₁

} {

, ,Z Z Z Z₉ ₆, ₄, ₁

} {

, ,Z Z Z Z₉ ₆, ₅, ₁

} {

, ,Z Z Z Z₉ ₇, ₆, ₃

} {

, ,Z Z Z Z₉ ₈, ₆, ₃

}

,

{

Z Z Z Z9, 8, 7, 3

}

, ,

{

Z Z Z D9 8, 4,

} {

, ,Z Z Z D9 8, 5,

} {

, ,Z Z Z D9 7, 2,

} {

, ,Z Z Z D9 7, 4,

}

,

   

{

Z Z Z D9, 7, 5,

} {

, Z Z Z D9, 8, 1,

} {

, Z Z Z D9, 8, 2,

} {

, Z Z Z D9, 4, 2,

} {

, Z Z Z D9, 4, 3,

}

,

    

{

Z Z Z D9, 5, 2,

} {

, Z Z Z D9, 5, 3,

} {

, Z Z Z D9, 7, 1,

} {

, Z Z Z D9, 2, 1,

} {

, Z Z Z D9, 3, 1,

}

,

    

{

Z Z Z D9, 3, 2,

} {

, ,Z Z Z D6 2, 1,

} {

, ,Z Z Z D6 3, 1,

} {

, ,Z Z Z D6 3, 2,

}

   

6)

{

Z Z Z Z D₉, ₅, ₄, ₁, 

} {

, Z Z Z Z D₉, ₆, ₄, ₁,

} {

, Z Z Z Z D₉, ₆, ₅, ₁, 

} {

, Z Z Z Z D₉, ₇, ₆, ₃,

}

,

{

Z Z Z Z D9, 8, 6, 3,

} {

, Z Z Z Z D9, 8, 7, 3,

}

 

7)

{

Z Z Z Z D9, 6, 2, 1,

} {

, Z Z Z Z D9, 6, 3, 1,

} {

, Z Z Z Z D9, 6, 3, 2,

}

  

(10)

8)

{

Z Z Z Z Z D₉, ₈, ₆, ₃, ₂, 

} {

, Z Z Z Z Z D₉, ₈, ₆, ₃, ₁,

} {

, Z Z Z Z Z D₉, ₇, ₆, ₃, ₂,

} {

, Z Z Z Z Z D₉, ₇, ₆, ₃, ₁, 

}

,

{

Z Z Z Z Z D9, 6, 5, 3, 1,

} {

, Z Z Z Z Z D9, 6, 5, 2, 1,

} {

, Z Z Z Z Z D9, 6, 4, 3, 1,

} {

, Z Z Z Z Z D9, 6, 4, 2, 1,

}

   

Proof.It is well know (see [1]), that the semilattices 1 to 8, which are given by lemma 2 are always XI-semilattices. The semilattices 9 and 10 which are given by Lemma 2

{

} {

}

{

} {

}

{

} {

}

{

}

{

} {

}

{

} {

}

{

} {

}

{

} {

}

8 7 3 8 6 3 8 6 8 5 8 4

8 2 8 1 7 6 3 7 5 7 4

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

5 3 5 2 4 3 4 2 3 2

3

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

,

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

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

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

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

Z Z

  

   

 

    

{

1,D

} {

, ,Z Z D2 1,

}

.

 

(see diagram 9 of theFigure 2);

{

Z Z Z D8, 6, 3,

} {

, ,Z Z Z D8 7, 3,

} {

, ,Z Z Z D7 6, 3,

} {

, ,Z Z Z D5 4, 1,

} {

, ,Z Z Z D6 4, 1,

} {

, ,Z Z Z D6 5, 1,

}

     

(see diagram 10 of theFigure 2);

are XI-semilattices iff the intersection of minimal elements of the given semilattices is empty set. From the

for-mal equalities (1) of the given semilattice D we have

(

) (

)

8 7 0 1 2 4 5 6 7 9 0 1 2 4 5 6 8 9

Z ∩Z = P ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P ∪ P ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P ≠ ∅

(

) (

)

8 6 0 1 2 4 5 6 7 9 0 4 5 7 8 9

Z ∩Z = P ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P ∪ P ∪ ∪ ∪ ∪ ∪P P P P P ≠ ∅

(

) (

)

8 5 0 1 2 4 5 6 7 9 0 2 3 4 6 7 8 9

(

) (

)

8 4 0 1 2 4 5 6 7 9 0 2 3 5 6 7 8 9

(

) (

)

8 2 0 1 2 4 5 6 7 9 0 1 3 4 5 6 7 8 9

Z ∩Z = P ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P ∪ P ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P P ≠ ∅

(

) (

)

8 1 0 1 2 4 5 6 7 9 0 2 3 4 5 6 7 8 9

(

) (

)

7 6 0 1 2 4 5 6 8 9 0 4 5 7 8 9

Z ∩Z = P ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P ∪ P ∪ ∪ ∪ ∪ ∪P P P P P ≠ ∅

(

) (

)

7 5 0 1 2 4 5 6 8 9 0 2 3 4 6 7 8 9

(

) (

)

7 4 0 1 2 4 5 6 8 9 0 2 3 5 6 7 8 9

(

) (

)

7 2 0 1 2 4 5 6 8 9 0 1 3 4 5 6 7 8 9

(

) (

)

7 1 0 1 2 4 5 6 8 9 0 2 3 4 5 6 7 8 9

(

) (

)

6 5 0 4 5 7 8 9 0 2 3 4 6 7 8 9

Z ∩Z = P ∪ ∪ ∪ ∪ ∪P P P P P ∪ P ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P ≠ ∅

(

) (

)

6 4 0 4 5 7 8 9 0 2 3 5 6 7 8 9

Z ∩Z = P ∪ ∪ ∪ ∪ ∪P P P P P ∪ P ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P ≠ ∅

(

) (

)

5 4 0 2 3 4 6 7 8 9 0 2 3 5 6 7 8 9

(

) (

)

5 3 0 2 3 4 6 7 8 9 0 1 2 4 5 6 7 8 9

(

) (

)

5 2 0 2 3 4 6 7 8 9 0 1 3 4 5 6 7 8 9

(

) (

)

4 3 0 2 3 5 6 7 8 9 0 1 2 4 5 6 7 8 9

(11)

(

) (

)

4 2 0 2 3 5 6 7 8 9 0 1 3 4 5 6 7 8 9

(

) (

)

3 2 0 1 2 4 5 6 7 8 9 0 1 3 4 5 6 7 8 9

Z ∩Z = P ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P P ∪ P ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪P P P P P P P P ≠ ∅

(

) (

)

3 1 0 1 2 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9

(

) (

)

2 1 0 1 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9

From the equalities given above it follows that the semilattices 9 and 10 are not XI-semilattices. _

The semilattices 11

{

} {

}

{

} {

}

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

7 6 3 2 7 6 3 1 8 6 3 2 8 6 3 1

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

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

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

   

(see diagram 1-8 of theFigure 3);

are not XI-semilattice since we have the following inequalities

5 3 5 2 4 3 4 2

7 2 7 1 8 2 8 1

, , , , , , , .

Z Z Z Z Z Z Z Z

∩ ≠ ∅ ∩ ≠ ∅ ∩ ≠ ∅ ∩ ≠ ∅

The semilattices 12 to 52 are never XI-semilattices. We prove that the semilattice, diagram 52 of theFigure 2,

is not an XI-semilattice (seeFigure 4). Indeed, let Q=

{

T T T T T T T T T₀, ,₁ ₂, ₃, ₄, ₅, ₆, ₇, ₈

}

and

( ) {

0, 1, 2, 3, 4, 5, 6, 7, 8

}

C Q = P P P P P P P P P′ ′ ′ ′ ′ ′ ′ ′ ′

be a family of sets, where P P P P P P P P P₀′, , , , , , , , ₁′ ′₂ ₃′ ′₄ ₅′ ₆′ ₇′ ₈′ are pairwise disjoint subsets of the set X. Let

0 1 2 3 4 5 6 7 8

T T T T T T T T T

P P P P P P P P P

ϕ=  _′ _′ _′ _′ _′ _′ _′ _′ _′

 

be a mapping of the semilattice Q onto the family of sets C Q

( )

. Then for the formal equalities of the

[image:11.595.214.412.455.571.2]

semilat-tice Q we have a form:

Figure 3. Diagram of all subsemilattices which are isomorphic to 11 in Figure 2.

[image:11.595.251.377.610.703.2]

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0 0 1 2 3 4 5 6 7 8

1 0 2 3 4 5 6 7 8

2 0 1 3 4 5 6 7 8

3 0 2 4 5 6 7 8

4 0 2 3 5 6 7 8

5 0 3 4 6 7 8

6 , , , , , ,

T P P P P P P P P P T P P P P P P P P T P P P P P P P P T P P P P P P P T P P P P P P P T P P P P P P T ′ ′ ′ ′ ′ ′ ′ ′ ′ = ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ′ ′ ′ ′ ′ ′ ′ ′ = ∪ ∪ ∪ ∪ ∪ ∪ ∪ ′ ′ ′ ′ ′ ′ ′ ′ = ∪ ∪ ∪ ∪ ∪ ∪ ∪ ′ ′ ′ ′ ′ ′ ′ = ∪ ∪ ∪ ∪ ∪ ∪ ′ ′ ′ ′ ′ ′ ′ = ∪ ∪ ∪ ∪ ∪ ∪ ′ ′ ′ ′ ′ ′ = ∪ ∪ ∪ ∪ ∪

0 1 3 4 5 7 8

'

7 0 1 3 4 5 6 8

8 0

, , .

P P P P P P P T P P P P P P P T P ′ ′ ′ ′ ′ ′ ′ = ∪ ∪ ∪ ∪ ∪ ∪ ′ ′ ′ ′ ′ ′ = ∪ ∪ ∪ ∪ ∪ ∪ ′ = (3)

Here the elements P P P P P P₁′, ₂′ ′, ₃, ₄′, ₆′, ₇′ are basis sources, the elements P₀′, P₅′, P₈′ are sources of

com-pleteness of the semilattice D. Therefore X ≥6 and

δ

=7 (see [2]). Then of the formal equalities we have:

{

}

{

}

{

}

{

}

{

}

{

}

0

7 6 2 0 1

4 3 1 0 2

7 6 5 4 2 1 0 3

7 6 5 3 2 1 0 4

7 6 4 3 2 1 0 5

7 5 4 3 2 1 0 6

6 5 4 3 2 1

, if ,

, , , , if ,

, , , , , , , if ,

, , , , ,

t

Q t P

T T T T t P

T T T T T T T t P

Q

T T T T T T T t P

T T T T T T

′ ∈ ′ ∈ ′ ∈ ′ ∈ ′ ∈ = ′ ∈ ′ ∈

{

}

{

8 7 6 5 4 3 02 1 0

}

78

, , if ,

, , , , , , , , , if .

T t P

T T T T T T T T T t P

            _∈ _′   ∈ ′ 

(

)

8 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8

, if ,

, , if ,

, if ,

, if .

t

T t P

Q Q T t P

T t P

′ ∈   _∈ _′   ∈ ′  _∈ _′  _′

Λ =_ ∈

 _∈ _′  ′ ∈   _∈ _′   ∈ ′ 

We have, that Q∧ =

{ }

T₈ and Λ

(

Q Q, t

)

∈Q for any t∈Q. But elements T7, T6, T5, T4, T3, T2, T1, T0 are not

union of some elements of the set Q∧. Therefore from the Definition 1 it follows that Q is not an XI-semilattice

of unions. Statements 12 to 51 can be proved analogously. We denoted the following semitattices by symbols:

a) Q₁=

{ }

T , where T∈D (see diagram 1 of theFigure 5);

b) Q₂=

{

T T, ′

}

, where T T, ′∈D and T⊂T′ (see diagram 2 of theFigure 5);

c) Q₃=

{

T T T, ′ ′′,

}

, where T T, , ′ T′′∈D and T⊂T′⊂T′′ (see diagram 3 of theFigure 5); d) Q₄=

{

Z T T D₉, , ′, 

}

, where T T, ′∈D and Z₉⊂ ⊂T T′⊂D (see diagram 4 of theFigure 5);

e) Q₅=

{

T T T T, ′ ′′ ′, , ∪T′′

}

where T T, , ′ T′′∈D, T ⊂T′, T⊂T′′, T′\T′′ ≠ ∅, T′′\T′ ≠ ∅, (see dia-

gram 5 of theFigure 5);

f) Q₆ =

{

Z T T T₉, , ′, ∪T D′, 

}

, where T T, ′∈D, Z₉ ⊂T, Z₉⊂T′, T T\ ′ ≠ ∅, T′\T ≠ ∅ (see diagram

6 of theFigure 5);

g) Q₇=

{

Z Z T T D₉, ₆, , ′, 

}

, where , T T′∈D, Z₆ ⊂T , Z₆⊂T′, T T\ ′ ≠ ∅, T′\T ≠ ∅, T∪T′=D

[image:12.595.83.543.89.524.2] [image:12.595.220.423.255.529.2]

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[image:13.595.169.483.82.188.2]

Figure 5. Diagram of all XI-subsemilattices of D.

h) Q8=

{

Z T T T9, , ′, ∪T Z D′, ,

}



, where Z₉⊂T′⊂Z, T T\ ′ ≠ ∅, T′\T ≠ ∅,

(

T∪T′

)

\Z≠ ∅,

(

)

\

Z T∪T′ ≠ ∅ (see diagram 8 of theFigure 5);

Note that the semilattices inFigure 5 are all XI-semilattices (see [1] and Lemma 1.2.3).

Definition 9. Let us assume that by the symbol Σ′XI

(

X D,

)

denote a set of all XI-subsemilatices of X-semila-

tices of unions D that every element of this set contains an empty set if ∅ ∈D or denotes a set of all XI-sub- semilatices of D.

Further, let D D′, ,′′∈ Σ′XI

(

X D

)

and ϑXI ⊆ Σ′XI

(

X D,

)

× Σ′XI

(

X D,

)

. It is assumed that D′ϑXID′′ iff there

exists some complete isomorphism

ϕ

between the semilatices D′ and D′′. One can easily verify that the

binary relation ϑ_XI is an equivalence relation on the set Σ′XI

(

X D,

)

.

By the symbol Q_iϑ_XI denote the ϑ_XI-equivalence class of the set Σ′XI

(

X D,

)

, where every element is iso-

morphic to the X-semilattice Q_i

(

i=1, 2,,8

)

.

Let D' be an XI-subsemilattice of the semilattice D. By I D

( )

′ we denoted the set of all right units of the

se-migroup B_X

( )

D′ , and

( )

i XI

i D Q

I Q I D

ϑ ∗

′∈

′

=

∑

where i=1, 2,,8.

Lemma 4. If X is a finite set, then the following equalities hold a) I Q

( )

₁ =1

b) I Q

( )

₂ =

(

2T T′\ − ⋅1 2

)

X T\ ′

c) I Q

( )

₃ =

(

2T T′\ − ⋅1

) (

3T′′ ′\T −2T′′ ′\T

)

⋅3X T\ ′′

d)

( )

(

\9

) (

\ \

)

(

\ \

)

\

4 2 1 3 2 4 3 4

D T D T X D

T Z T T T T

I Q = − ⋅ ′ − ′ ⋅ ′ − ′ ⋅

  

e) I Q

( )

₅ =

(

2T T′ ′′\ − ⋅1

) (

2T′′ ′\T − ⋅1 4

)

X\(T′∪T′′)

f) I Q

( )

₆ =

(

2T T′ ′′\ − ⋅1

) (

2T T′′ ′\ − ⋅1

)

(

5D T\( ∪T′′)−4D T\( ∪T′′)

)

⋅5X D\

  

g)

( )

(

6\ 9

)

( )\ 6

(

\ \

) (

\ \

)

\

7 2 1 2 3 2 3 2 5

T T Z X D

Z Z T T T T T T T T

I Q = − ⋅ ∩ ′ ⋅ ′ − ′ ⋅ ′ − ′ ⋅ 

h) I Q

( )

₈ =

(

2T Z\ − ⋅1

) (

2T T′\ − ⋅1

)

(

3Z T\( ∪T′) −2Z T\( ∪T′)

)

⋅6X D\



Proof. This lemma immediately follows from Theorem 13.1.2, 13.3.2, and 13.7.2 of the [1]. 

Theorem 10. Let D∈Σ₁

(

X,10

)

and α ∈B_X

( )

D . Binary relation

α

is an idempotent relation of the semmigroup B_X

( )

D iff binary relation

α

satisfies only one conditions of the following conditions:

a)

α

= ×X T, where T∈D;

b)

α

=

(

Y_Tα×T

) (

∪ Y_Tα′×T′

)

, where T T, ′∈D, T ⊂T′, YT , YT

{ }

α α

′ ∉ ∅ , and satisfies the conditions: YT

α _⊇

T, Y_Tα′∩T′≠ ∅;

c)

(

YT T

) (

YT T

) (

YT T

)

α α α

α

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

{ }

α α α

′ ′′∉ ∅ , and sa-

tisfies the conditions: Y_Tα ⊇T , Y_Tα∪Y_Tα′ ⊇T′, YT T

α

′ ∩ ′≠ ∅, YT T

α