Applied Mathematics, 2015, 6, 274-294
Published Online February 2015 in SciRes. http://www.scirp.org/journal/am
http://dx.doi.org/10.4236/am.2015.62026
Complete Semigroups of Binary Relations
Defined by Semilattices of the Class
Σ1 ( X ,10 )
Shota Makharadze1, Neşet Aydın2, Ali Erdoğan3
1
Shota Rustavelli University, Batum, Georgia
Çanakkale Onsekiz Mart University, Çanakkale, Turkey
3
Hacettepe University, Ankara, Turkey
Email: [email protected], [email protected], [email protected]
2
Received 10 January 2015; accepted 28 January 2015; published 4 February 2015
Copyright © 2015 by authors and Scientific Research Publishing Inc.
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 formulas by means of which we can calculate the numbers of idempotent elements of the respective semigroup.
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 arbitrary mapping of the set X in the set D. To each such a mapping f we put into correspondence a binary relation
α f on the set X that satisfies the condition
=
αf
 ({ x} × f ( x ) )
x∈ X
The set of all such α f ( f : X → D ) is denoted by BX ( D ) . It is easy to prove that BX ( 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 an X-semilattice of unions D.
How to cite this paper: Makharadze, S., Aydın, N. and Erdoğan, A. (2015) Complete Semigroups of Binary Relations Defined
by Semilattices of the Class Σ1 ( X ,10 ) . Applied Mathematics, 6, 274-294. http://dx.doi.org/10.4236/am.2015.62026
S. Makharadze et al.
Recall that we denote by ∅ an empty binary relation or empty subset of the set X. The condition
will be written
in the form xαy. Further let x, y ∈ X , Y ⊆ X , α ∈ BX ( D ) , T ∈ D , ∅ ≠ D ′ ⊆ D ,

D = ∪ D and t ∈ D . Then by symbols we denoted the following sets:
(x, y ) ∈ α
2 X \ {∅} ,
yα =
{ x ∈ X yα x} , Yα =
{Y Y ⊆ X } , X ∗ =
 y∈Y yα , 2 X =
V ( D, α ) =
{Yα Y ∈ D} , DT′ ={T ′ ∈ D′ T ⊆ T ′} , DT′ ={T ′ ∈ D′ T ′ ⊆ T } ,
Dt′ =∈
∪ ( D′ \ DT′ ) .
{Z ′ D′ t ∈ Z ′} , l ( D′, T ) =
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, Dt ) ∈ D for any t ∈ D ;
b)=
Z  t∈Z Λ ( D, Dt ) 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 ) = ∅ .
T } . A representation of a binary relation
Definition 3. Let α ∈ BX ( D ) , T ∈ V X ∗ , α , YTα =∈
{ y X yα =
α
α of=
the form α T ∈V X ∗ ,α YT × T is called quasinormal.
)(
(
Note
that, if α T ∈V X ∗ ,α
=
(
lowing conditions are true:
1) X = T ∈V X ∗ ,α YTα ;
(
)
) (Y × T )
α
T
)
(
(
)
is a quasinormal representation of the binary relation α , then the fol-
)
2) YT ∩ YT ′ =
∅ 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; δ and q be respectively the number of basic sources and the number of all
s , then
automorphisms of the semilattice D. If X = n ≥ δ and Σ n ( X , m ) =
α
α
s=
where C kj =
j!
( k !) ⋅ ( j − k )!
p + i +1
⋅ Cmp−−δδ ⋅ C δp ⋅ (δ !) ⋅ ( ( p − δ ) !) ⋅ i n
1 m  p +1  ( −1)
⋅ ∑ ∑
 i 1
q=p δ=
( i − 1)! ⋅ ( p − i + 1)!
 




(see Theorem 11.5.1 [1]).
Theorem 5. Let D be a complete X-semilattice of unions. The semigroup BX ( 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
 . A binary relation α having a quasinormal
=
Q V ( D, α ) \ {∅} which are nonlimiting elements of the set Q
T
representation
=
α T ∈V ( D ,α ) (YTα × T ) is an idempotent element of this semigroup iff
a) V ( D, α ) is complete XI-semilattice of unions;
b) T ′∈D (α ) YTα′ ⊇ T for any T ∈ D (α ) ;
T
 (α ) (see Theorem 6.3.9 [1]).
c) YTα ∩ T ≠ ∅ for any nonlimiting element of the set D
T
r
Theorem 7. Let D, Σ ( D ) , E X( ) ( 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 BX ( D′ ) and the set of
r
all idempotents of the semigroup BX ( D ) . Then for the sets E X( ) ( D′ ) and I the following statements are true:
1) if ∅ ∈ D and Σ∅ ( D=
) D′ ∈ Σ ( D ) ∅ ∈ D′ then
r
r
∅ for any elements D′ and D′′ of the set Σ∅ ( D ) that satisfy the condition
a) E X( ) ( D ′ ) ∩ E X( ) ( D ′′ ) =
D ′ ≠ D ′′ ;
b) I =  D′∈Σ ( D )E X( r ) ( D ′ )
{
}
∅
c) the equality I = ∑ D′∈Σ
∅ ( D)
r
E X( ) ( D ′ ) is fulfilled for the finite set X.
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2) if ∅ ∉ D , then
r
r
∅ for any elements D′ and D ′′ of the set Σ ( D ) that satisfy the condition
a) E X( ) ( D ′ ) ∩ E X( ) ( D ′′ ) =
D ′ ≠ D ′′ ;
b) I =  D′∈Σ( D ) E X( r ) ( D ′ )
c) the equality I = ∑ D′∈Σ( D ) E X( r ) ( D ′ ) is fulfilled for the finite set X (see Theorem 6.2.3 [1]).
{
}
Corollary 1. Let Y = { y1 , y2 , , yk } and D j = T1 , T2 , , T j 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 j that T j ∈ D ′j
k
can be calculated by the formula s ( k , j ) = j k − ( j − 1) (see Corollary 1.18.1 [1]).
2. Idempotent Elements of the Semigroups B X ( D ) Defined by Semilattices of the
Class Σ1 ( 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 9 , Z8 , Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D that satisfies the conditions:


Z 9 ⊂ Z 4 ⊂ Z1 ⊂ D, Z 9 ⊂ Z 5 ⊂ Z1 ⊂ D,


Z 9 ⊂ Z 6 ⊂ Z1 ⊂ D, Z 9 ⊂ Z 6 ⊂ Z 2 ⊂ D,


Z 9 ⊂ Z 6 ⊂ Z 3 ⊂ D, Z 9 ⊂ Z 7 ⊂ Z 3 ⊂ D,

Z 9 ⊂ Z8 ⊂ Z 3 ⊂ D, Z1 \ Z 2 ≠ ∅, Z 2 \ Z1 ≠ ∅,
{
Z1 \ Z 3 ≠ ∅, Z 3 \ Z1 ≠ ∅, Z 2 \ Z 3 ≠ ∅,
Z 3 \ Z 2 ≠ ∅, Z 4 \ Z 5 ≠ ∅, Z 5 \ Z 4 ≠ ∅,
Z 4 \ Z 6 ≠ ∅, Z 6 \ Z 4 ≠ ∅, Z 4 \ Z 7 ≠ ∅,
Z 7 \ Z 4 ≠ ∅ , Z 4 \ Z 8 ≠ ∅ , Z 8 \ Z 4 ≠ ∅,
Z 5 \ Z 6 ≠ ∅ , Z 6 \ Z 5 ≠ ∅ , Z 5 \ Z 7 ≠ ∅,
Z 7 \ Z 5 ≠ ∅, Z 5 \ Z 8 ≠ ∅, Z 8 \ Z 5 ≠ ∅,
Z 6 \ Z 7 ≠ ∅, Z 7 \ Z 6 ≠ ∅, Z 6 \ Z 8 ≠ ∅,
Z 8 \ Z 6 ≠ ∅, Z 7 \ Z 8 ≠ ∅, Z 8 \ Z 7 ≠ ∅,
Z1 ∪ Z 2 = Z1 ∪ Z 3 = Z 2 ∪ Z 3 = Z 4 ∪ Z 2
= Z 4 ∪ Z 3 = Z 4 ∪ Z 7 = Z 4 ∪ Z8 = Z 5 ∪ Z 2
= Z 5 ∪ Z 3 = Z 5 ∪ Z 7 = Z 5 ∪ Z8 = Z 7 ∪ Z1

= Z 7 ∪ Z 2 = Z8 ∪ Z1 = Z8 ∪ Z 2 = D,
Z 4 ∪ Z 5 = Z 4 ∪ Z 6 = Z 5 ∪ Z 6 = Z1 ,
Z 6 ∪ Z 7 = Z 6 ∪ Z8 = Z 7 ∪ Z8 = Z 3 .
}
(1)
An X-semilattice that satisfies conditions (1) is shown in Figure 1.
Let C ( D ) = { P0 , P1 , P2 , P3 , P4 , P5 , P6 , P7 , P8 , P9 } be a family of sets, where P0, P1, P2, P3, P4, P5, P6, P7, P8, P9
Figure 1. Diagram of D.
276
S. Makharadze et al.

D
are pairwise disjoint subsets of the set X and ϕ = 
 P0
Z1 Z 2
P1 P2
Z3
P3
Z4
P4
Z5
P5
Z6
P6
Z7
P7
Z9 
 be a mapP9 
Z8
P8
ping of the semilattice D onto the family sets C ( D ) . Then for the formal equalities of the semilattice D we
have a form:

D = P0 ∪ P1 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ,
Z1 = P0 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ,
Z 2 = P0 ∪ P1 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ,
Z 3 = P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ,
Z 4 = P0 ∪ P2 ∪ P3 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ,
(2)
Z 5 = P0 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ,
Z 6 = P0 ∪ P4 ∪ P5 ∪ P7 ∪ P8 ∪ P9 ,
Z 7 = P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P8 ∪ P9 ,
Z8 = P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P9 ,
Z 9 = P0 .
Here the elements P1, P2, P3, P4, P5, P6, P7, P8 are basis sources, the elements P0, P6, P9 are sources of completeness of the semilattice D. Therefore X ≥ 7 and δ = 7 (see [2]).
s and X ≥ δ ≥ 7 . If X is a finite set, then
Lemma 1. Let D ∈ Σ1 ( X ,10 ) , Σ1 ( X ,10 ) =
s=
(
)
1
( −1) × 4n + 7 × 5n − 21× 6n + 35 × 7n − 35 × 8n + 21× 9n + 11n .
8
Proof. In this case we have: m = 10, δ = 7. Notice that an X-semilattice given in Figure 1 has eight automorphims. By Theorem 1.1 it follows that
p + i +1
⋅ C3p − 7 ⋅ C p7 ⋅ ( 7!) ⋅ ( ( p − 7 )!) ⋅ i n
1 10  p +1  ( −1)
⋅ ∑ ∑
i 1
8=p 7=
( i − 1)! ⋅ ( p − i + 1)!
 
s=
where C kj =

 ,


j!
and that
k ! ⋅ ( j − k )!
s=
)
(
1
( −1) × 4n + 7 × 5n − 21× 6n + 35 × 7n − 35 × 8n + 21× 9n + 11n .
8
Example 8. Let n = 7, 8, 9, 10 Then:
BX ( D ) = 107 , 108 , 109 , 1010 .
Lemma 2. Let D ∈ Σ1 ( X ,10 ) . Then
 the following sets are all proper subsemilattices of the semilattice
D = Z 9 , Z8 , Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D :

1) {Z 9 } , {Z8 } , {Z 7 } , {Z 6 } , {Z 5 } , {Z 4 } , {Z 3 } , {Z 2 } , {Z1} , D
{
}
{ }
(see diagram 1 of the Figure 2);
2)

{Z9 , Z8 } , {Z9 , Z7 } , {Z9 , Z6 } , {Z9 , Z5 } , {Z9 , Z 4 } , {Z9 , Z3} , {Z9 , Z 2 } , {Z9 , Z1} , {Z9 , D} , {Z8 , Z3} ,




{Z8 , D} , {Z7 , Z3} , {Z7 , D} , {Z6 , Z3} , {Z6 , Z 2 } , {Z6 , Z1} , {Z6 , D} , {Z5 , Z1} , {Z5 , D} , {Z 4 , Z1} ,




{Z , D} , {Z , D} , {Z , D} , {Z , D}
4
3
2
1
(see diagram 2 of the Figure 2);


3) {Z 9 , Z8 , Z 3 } , Z 9 , Z8 , D , {Z 9 , Z 7 , Z 3 } , Z 9 , Z 7 , D , {Z 9 , Z 6 , Z 3 } , {Z 9 , Z 6 , Z 2 } , {Z 9 , Z 6 , Z1} ,





Z 9 , Z 6 , D , {Z 9 , Z 5 , Z1} , Z 9 , Z 5 , D , {Z 9 , Z 4 , Z1} , Z 9 , Z 4 , D , Z 9 , Z 3 , D , Z 9 , Z 2 , D ,
{
}
{
}
{
}
{
}
277
{
} {
} {
}
S. Makharadze et al.








{Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D}
9
1
8
3
7
3
6
3
6
2
6
1
5
1
4
1
(see diagram 3 of the Figure 2);





4) Z 9 , Z 4 , Z1 , D , Z 9 , Z 5 , Z1 , D , Z 9 , Z 6 , Z1 , D , Z 9 , Z 6 , Z 2 , D , Z 9 , Z 6 , Z 3 , D ,


Z 9 , Z 7 , Z 3 , D , Z 9 , Z8 , Z 3 , D
{
{
} {
} {
} {
}
} {
} {
}
(see diagram 4 of the Figure 2);
5)
{Z9 , Z5 , Z 4 , Z1} , {Z9 , Z 6 , Z 4 , Z1} , {Z9 , Z 6 , Z5 , Z1} , {Z9 , Z 7 , Z 6 , Z3} , {Z9 , Z8 , Z 6 , Z3} , {Z9 , Z8 , Z 7 , Z3} ,






{Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} ,






{Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} ,






{Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} ;
9
8
4
9
8
5
9
7
2
9
7
4
9
7
5
9
8
1
9
8
2
9
4
2
9
4
3
9
5
2
9
5
3
9
7
1
9
2
1
9
3
1
9
6
2
1
3
2
6
3
1
6
3
2
(see diagram 5 of the Figure 2);




6) Z 9 , Z 5 , Z 4 , Z1 , D , Z 9 , Z 6 , Z 4 , Z1 , D , Z 9 , Z 6 , Z 5 , Z1 , D , Z 9 , Z 7 , Z 6 , Z 3 , D ,


Z 9 , Z8 , Z 6 , Z 3 , D , Z 9 , Z8 , Z 7 , Z 3 , D
{
{
} {
} {
} {
}
} {
}
(see diagram 6 of the Figure 2);



7) Z 9 , Z 6 , Z 2 , Z1 , D , Z 9 , Z 6 , Z 3 , Z1 , D , Z 9 , Z 6 , Z 3 , Z 2 , D
{
} {
} {
}
(see diagram 7 of the Figure 2);




8) Z 9 , Z8 , Z 6 , Z 3 , Z 2 , D , Z 9 , Z8 , Z 6 , Z 3 , Z1 , D , Z 9 , Z 7 , Z 6 , Z 3 , Z 2 , D , Z 9 , Z 7 , Z 6 , Z 3 , Z1 , D ,




Z9 , Z 6 , Z5 , Z3 , Z1 , D , Z9 , Z 6 , Z5 , Z 2 , Z1 , D , Z9 , Z 6 , Z 4 , Z3 , Z1 , D , Z9 , Z 6 , Z 4 , Z 2 , Z1 , D
{
{
} {
} {
} {
} {
(see diagram 8 of the Figure 2);
9)
} {
} {


}
}



{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 , D} , {Z , Z , D}
{Z 8 , Z 7 , Z 3 } , {Z 8 , Z 6 , Z 3 } ,
8
6
8
5
8
4
7
5
7
4
7
2
7
1
6
5
5
2
4
3
4
2
3
2
3
1
8
1
2
6
4
2
1
8
1
5
4
1
7
6
5
3
1
(see diagram 9 of the Figure 2);






10) Z8 , Z 6 , Z 3 , D , Z8 , Z 7 , Z 3 , D , Z 7 , Z 6 , Z 3 , D , Z 5 , Z 4 , Z1 , D , Z 6 , Z 4 , Z1 , D , Z 6 , Z 5 , Z1 , D
{
} {
} {
} {
} {
} {
}
(see diagram 10 of the Figure 3);




11) Z 6 , Z 5 , Z 3 , Z1 , D , Z 6 , Z 5 , Z 2 , Z1 , D , Z 6 , Z 4 , Z 3 , Z1 , D , Z 6 , Z 4 , Z 2 , Z1 , D ,




Z 7 , Z 6 , Z 3 , Z 2 , D , Z 7 , Z 6 , Z 3 , Z1 , D , Z8 , Z 6 , Z 3 , Z 2 , D , Z8 , Z 6 , Z 3 , Z1 , D
{
{
} {
} {
} {
} {
} {
} {
(see diagram 11 of the Figure 2);
12)

}
}



{Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} ,





{Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D}
{Z 6 , Z5 , Z 4 , Z1} , {Z8 , Z 7 , Z6 , Z3} ,
7
2
1
7
4
2
8
7
2
5
1
3
2
2
8
4
1
4
2
3
8
5
2
5
3
2
2
(see diagram 12 of the Figure 2);






13) Z 7 , Z 5 , Z 3 , D , Z 4 , Z 2 , Z1 , D , Z8 , Z 3 , Z1 , D , Z8 , Z 3 , Z 2 , D , Z8 , Z 4 , Z1 , D , Z 4 , Z 3 , Z1 , D ,






Z 5 , Z 2 , Z1 , D , Z 5 , Z 3 , Z1 , D , Z 7 , Z 3 , Z1 , D , Z 7 , Z 3 , Z 2 , D , Z 7 , Z 4 , Z1 , D , Z 7 , Z 4 , Z 3 , D ,




Z 7 , Z 5 , Z1 , D , Z8 , Z 4 , Z 3 , D , Z8 , Z 5 , Z1 , D , Z8 , Z 5 , Z 3 , D
{
{
{
} {
} {
} {
} {
} {
} {
} {
} {
}
} {
} {
} {
(see diagram 13 of the Figure 2);
14)
{Z9 , Z6 , Z5 , Z 4 , Z1} , {Z9 , Z8 , Z 7 , Z6 , Z3} ,
} {
} {

}
}


{Z , Z , Z , Z , D} , {Z , Z , Z , Z , D} , {Z , Z , Z , Z , D} ,
6
3
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1
9
3
2
1
9
4
3
2
3
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

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

{Z , Z , Z , Z , D} , {Z , Z , Z , Z , D} , {Z , Z , Z , Z , D} , {Z , Z , Z , Z , D} , {Z , Z , Z , Z , D} ,



{Z , Z , Z , Z , D} , {Z , Z , Z , Z , D} , {Z , Z , Z , Z , D}
9
5
3
2
9
7
2
1
9
7
4
2
9
8
2
1
9
8
4
2
9
8
5
2
9
7
4
3
9
7
5
2
(see diagram 14 of the Figure 2);





15) Z 9 , Z 4 , Z 2 , Z1 , D , Z 9 , Z 4 , Z 3 , Z1 , D , Z 9 , Z 5 , Z 2 , Z1 , D , Z 9 , Z 5 , Z 3 , Z1 , D , Z 9 , Z 7 , Z 3 , Z1 , D ,





Z 9 , Z 7 , Z 3 , Z 2 , D , Z 9 , Z 7 , Z 4 , Z1 , D , Z 9 , Z 7 , Z5 , Z1 , D , Z9 , Z 7 , Z5 , Z3 , D , Z9 , Z8 , Z3 , Z1 , D ,





Z 9 , Z8 , Z 3 , Z 2 , D , Z 9 , Z8 , Z 4 , Z1 , D , Z 9 , Z8 , Z 4 , Z 3 , D , Z 9 , Z8 , Z 5 , Z1 , D , Z 9 , Z8 , Z 5 , Z 3 , D
{
{
{
} {
} {
} {
} {
} {
} {
} {
} {
} {
} {
} {
} {
}
}
}
(see diagram 15 of the Figure 2);





16) Z 5 , Z 4 , Z 3 , Z1 , D , Z 5 , Z 4 , Z 2 , Z1 , D , Z 7 , Z 5 , Z 4 , Z1 , D , Z8 , Z 5 , Z 4 , Z1 , D , Z8 , Z 7 , Z3 , Z 2 , D ,



Z8 , Z 7 , Z 3 , Z1 , D , Z8 , Z 7 , Z 5 , Z 3 , D , Z8 , Z 7 , Z 4 , Z 3 , D
{
{
} {
} {
} {
} {
} {
}
} {
}
(see diagram 16 of the Figure 2);





17) Z 5 , Z 3 , Z 2 , Z1 , D , Z 4 , Z 3 , Z 2 , Z1 , D , Z 7 , Z 4 , Z 2 , Z1 , D , Z 7 , Z 3 , Z 2 , Z1 , D , Z 7 , Z 5 , Z 3 , Z 2 , D ,





Z 7 , Z 5 , Z 2 , Z1 , D , Z 7 , Z 4 , Z3 , Z 2 , D , Z8 , Z 3 , Z 2 , Z1 , D , Z8 , Z5 , Z3 , Z 2 , D , Z8 , Z 5 , Z 2 , Z1 , D ,


Z8 , Z 4 , Z 3 , Z 2 , D , Z8 , Z 4 , Z 2 , Z1 , D
{
{
{
} {
} {
} {
} {
} {
}
} {
} {
} {
} {
}
}
(see diagram 17 of the Figure 2);




18) Z 7 , Z 4 , Z 3 , Z1 , D , Z 7 , Z 5 , Z 3 , Z1 , D , Z8 , Z 5 , Z 3 , Z1 , D , Z8 , Z 4 , Z 3 , Z1 , D
{
} {
} {
} {
}
(see diagram 18 of the Figure 2);


19) Z 6 , Z 5 , Z 4 , Z1 , D , Z8 , Z 7 , Z 6 , Z 3 , D .
{
} {
}
(see diagram 19 of the Figure 2);




20) Z8 , Z 7 , Z 5 , Z 3 , Z 2 , D , Z8 , Z 7 , Z 4 , Z 3 , Z 2 , D , Z8 , Z 5 , Z 4 , Z 2 , Z1 , D , Z 7 , Z 5 , Z 4 , Z 2 , Z1 , D ,


Z8 , Z 7 , Z 3 , Z 2 , Z1 , D , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D
{
{
} {
} {
} {
}
} {
}
(see diagram 20 of the Figure 2);




21) Z 6 , Z 5 , Z 4 , Z 2 , Z1 , D , Z8 , Z 7 , Z 6 , Z 3 , Z 2 , D , Z8 , Z 7 , Z 6 , Z 3 , Z1 , D , Z 6 , Z 5 , Z 4 , Z 3 , Z1 , D
{
} {
} {
} {
}
(see diagram 21 of the Figure 2);




22) Z 9 , Z 5 , Z 4 , Z 2 , Z1 , D , Z 9 , Z8 , Z 7 , Z 4 , Z 3 , D , Z 9 , Z8 , Z 7 , Z 3 , Z1 , D , Z 9 , Z8 , Z 5 , Z 4 , Z1 , D ,




Z 9 , Z 7 , Z 5 , Z 4 , Z1 , D , Z9 , Z5 , Z 4 , Z3 , Z1 , D , Z9 , Z8 , Z 7 , Z3 , Z 2 , D , Z9 , Z8 , Z 7 , Z5 , Z3 , D
{
{
} {
} {
} {
} {
} {
} {
}
}
(see diagram 22 of the Figure 2);


23) Z 9 , Z8 , Z 7 , Z 6 , Z 3 , D , Z 9 , Z 6 , Z 5 , Z 4 , Z1 , D
{
} {
}
(see diagram 23 of the Figure 2);




24) Z 9 , Z8 , Z 5 , Z 3 , Z 2 , D , Z 9 , Z8 , Z 5 , Z 2 , Z1 , D , Z 9 , Z8 , Z 4 , Z 3 , Z 2 , D , Z 9 , Z8 , Z 4 , Z 2 , Z1 , D ,




Z 9 , Z8 , Z 3 , Z 2 , Z1 , D , Z 9 , Z 7 , Z 5 , Z 3 , Z 2 , D , Z 9 , Z 7 , Z 5 , Z 2 , Z1 , D , Z 9 , Z 7 , Z 4 , Z 3 , Z 2 , D ,




Z 9 , Z 7 , Z 4 , Z 2 , Z1 , D , Z9 , Z 7 , Z3 , Z 2 , Z1 , D , Z9 , Z 5 , Z 3 , Z 2 , Z1 , D , Z 9 , Z 4 , Z 3 , Z 2 , Z1 , D
{
{
{
} {
} {
} {
} {
} {
} {
} {
} {
} {
}
}
}
(see diagram 24 of the Figure 2);




25) Z 9 , Z8 , Z 5 , Z 3 , Z1 , D , Z 9 , Z8 , Z 4 , Z 3 , Z1 , D , Z 9 , Z 7 , Z 5 , Z 3 , Z1 , D , Z 9 , Z 7 , Z 4 , Z 3 , Z1 , D
{
} {
} {
(see diagram 25 of the Figure 2);

26) Z 9 , Z 6 , Z 3 , Z 2 , Z1 , D
{
}
(see diagram 26 of the Figure 2);
279
} {
}
S. Makharadze et al.
27)




{Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D}
8
7
5
3
1
8
7
4
3
1
8
5
4
3
1
7
5
4
3
1
(see diagram 27 of the Figure 2);




28) Z8 , Z 6 , Z 5 , Z 3 , Z1 , D , Z8 , Z 6 , Z 4 , Z 3 , Z1 , D , Z8 , Z 5 , Z 3 , Z 2 , Z1 , D , Z 7 , Z 6 , Z 5 , Z 3 , Z1 , D ,

Z 7 , Z 6 , Z 4 , Z 3 , Z1 , D
{
{
} {
}
} {
} {
}
(see diagram 28 of the Figure 2);




29) Z8 , Z 6 , Z 3 , Z 2 , Z1 , D , Z 7 , Z 6 , Z 3 , Z 2 , Z1 , D , Z 6 , Z 5 , Z 3 , Z 2 , Z1 , D , Z 6 , Z 4 , Z 3 , Z 2 , Z1 , D
{
} {
} {
} {
}
(see diagram 29 of the Figure 2);



30) Z8 , Z 4 , Z 3 , Z 2 , Z1 , D , Z 7 , Z 5 , Z 3 , Z 2 , Z1 , D , Z 7 , Z 4 , Z 3 , Z 2 , Z1 , D
{
} {
} {
}
(see diagram 30 of the Figure 2);

31) Z8 , Z 7 , Z 5 , Z 4 , Z 3 , Z1 , D
{
}
(see diagram 31 of the Figure 2);


32) Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D , Z8 , Z 7 , Z 6 , Z 3 , Z 2 , Z1 , D
{
} {
}
(see diagram 32 of the Figure 2);



33) Z 7 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D , Z8 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D , Z8 , Z 7 , Z 4 , Z 3 , Z 2 , Z1 , D ,

Z8 , Z 7 , Z 5 , Z 3 , Z 2 , Z1 , D
{
{
} {
}
} {
}
(see diagram 33 of the Figure 2);



34) Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z1 , D , Z8 , Z 6 , Z 5 , Z 4 , Z 3 , Z1 , D , Z8 , Z 7 , Z 6 , Z 4 , Z 3 , Z1 , D ,

Z8 , Z 7 , Z 6 , Z 5 , Z 3 , Z1 , D
{
{
} {
}
} {
}
(see diagram 34 of the Figure 2);



35) Z 9 , Z 6 , Z 4 , Z 3 , Z 2 , Z1 , D , Z 9 , Z 6 , Z 5 , Z 3 , Z 2 , Z1 , D , Z 9 , Z 7 , Z 6 , Z 3 , Z 2 , Z1 , D ,


Z9 , Z8 , Z 6 , Z 3 , Z 2 , Z1 , D , Z 9 , Z8 , Z 6 , Z 3 , Z 2 , Z1 , D
{
{
} {
} {
} {
}
}
(see diagram 35 of the Figure 2);



36) Z 9 , Z 6 , Z 5 , Z 4 , Z 3 , Z1 , D , Z 9 , Z8 , Z 7 , Z 6 , Z 3 , Z1 , D , Z 9 , Z8 , Z 7 , Z 6 , Z 3 , Z1 , D
{
} {
} {
}
(see diagram 36 of the Figure 2);



37) Z 9 , Z 7 , Z 4 , Z 3 , Z 2 , Z1 , D , Z 9 , Z 7 , Z 5 , Z 3 , Z 2 , Z1 , D , Z 9 , Z8 , Z 4 , Z 3 , Z 2 , Z1 , D ,

Z 9 , Z8 , Z 5 , Z 3 , Z 2 , Z1 , D
{
{
} {
}
} {
}
(see diagram 37 of the Figure 2);



38) Z 9 , Z 7 , Z 5 , Z 4 , Z 3 , Z1 , D , Z 9 , Z8 , Z 5 , Z 4 , Z 3 , Z1 , D , Z 9 , Z8 , Z 7 , Z 4 , Z 3 , Z1 , D
{
} {
} {
}
(see diagram 38 of the Figure 2);

39) Z 9 , Z 7 , Z 6 , Z 5 , Z 3 , Z1 , D
{
}
(see diagram 39 of the Figure 2);



40) Z 9 , Z 7 , Z 5 , Z 4 , Z 2 , Z1 , D , Z 9 , Z8 , Z 5 , Z 4 , Z 2 , Z1 , D , Z 9 , Z8 , Z 7 , Z 4 , Z3 , Z 2 , D ,


Z 9 , Z8 , Z 7 , Z 5 , Z 3 , Z 2 , D , Z 9 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D
{
{
} {
} {
} {
}
(see diagram 40 of the Figure 2);


41) Z 9 , Z 6 , Z 5 , Z 4 , Z 2 , Z1 , D , Z 9 , Z8 , Z 7 , Z 6 , Z 3 , Z 2 , D
{
} {
}
(see diagram 41 of the Figure 2);
280
}
42)
S. Makharadze et al.

{Z , Z , Z , Z , Z , Z , Z , D}
7
6
5
4
3
2
1
(see diagram 42 of the Figure 2);



43) Z8 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D , Z8 , Z 7 , Z 6 , Z 4 , Z 3 , Z 2 , Z1 , D , Z8 , Z 7 , Z 6 , Z 5 , Z 3 , Z 2 , Z1 , D
{
} {
} {
}
(see diagram 43 of the Figure 2);

44) Z8 , Z 7 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D
{
}
(see diagram 44 of the Figure 2);

45) Z 9 , Z8 , Z 7 , Z 5 , Z 4 , Z 3 , Z1 , D
{
}
(see diagram 45 of the Figure 2);


46) Z 9 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D , Z 9 , Z8 , Z 7 , Z 6 , Z 3 , Z 2 , Z1 , D
{
} {
}
(see diagram 46 of the Figure 2);



47) Z 9 , Z 7 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D , Z 9 , Z8 , Z 7 , Z 5 , Z 3 , Z 2 , Z1 , D , Z 9 , Z 7 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D ,

Z 9 , Z8 , Z 7 , Z 4 , Z 3 , Z 2 , Z1 , D
{
{
} {
}
} {
}
(see diagram 47 of the Figure 2);



48) Z 9 , Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z1 , D , Z 9 , Z8 , Z 6 , Z 5 , Z 4 , Z 3 , Z1 , D , Z 9 , Z8 , Z 7 , Z 6 , Z 4 , Z 3 , Z1 , D ,

Z 9 , Z8 , Z 7 , Z 6 , Z 5 , Z 3 , Z1 , D
{
{
} {
}
} {
(see diagram 48 of the Figure 2);
Figure 2. Diagram of all subsemilattices of D.
281
}
S. Makharadze et al.

{Z , Z , Z , Z , Z , Z , Z , Z , D}
49)
8
7
6
5
4
3
2
1
(see diagram 49 of the Figure 2);



50) Z 9 , Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D , Z 9 , Z8 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D , Z 9 , Z8 , Z 7 , Z 6 , Z 4 , Z 3 , Z 2 , Z1 , D ,

Z 9 , Z8 , Z 7 , Z 6 , Z 5 , Z 3 , Z 2 , Z1 , D
{
{
} {
}
} {
}
(see diagram 50 of the Figure 2);

51) Z 9 , Z8 , Z 7 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D
{
}
(see diagram 51 of the Figure 2);

52) Z 9 , Z8 , Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z1 , D
{
}
(see diagram 52 of the Figure 2);
Diagrams of subsemilattices of the semilattice D.
Lemma 3. Let D ∈ Σ1 ( X ,10 ) . Then the following sets are all XI-subsemi-lattices of the given semilattice D:

1) {Z 9 } , {Z8 } , {Z 7 } , {Z 6 } , {Z 5 } , {Z 4 } , {Z 3 } , {Z 2 } , {Z1} , D
{ }
(see diagram 1 of the Figure 2);

2) Z 9 , D , {Z 9 , Z8 } , {Z 9 , Z 7 } , {Z 9 , Z 6 } , {Z 9 , Z 5 } , {Z 9 , Z 4 } , {Z 9 , Z 3 } , {Z 9 , Z 2 } , {Z 9 , Z1} , {Z8 , Z 3 } ,




Z8 , D , {Z 7 , Z 3 } , Z 7 , D , {Z 6 , Z 3 } , {Z 6 , Z 2 } , {Z 6 , Z1} , Z 6 , D , {Z 5 , Z1} , Z 5 , D , {Z 4 , Z1} ,




Z 4 , D , Z 3 , D , Z 2 , D , Z1 , D
{
{
{
}
}
} {
{
} {
}
} {
{
}
}
{
}
(see diagram 2 of the Figure 2);








3) Z 9 , Z8 , D , Z 9 , Z 7 , D , Z 9 , Z 6 , D , Z 9 , Z 5 , D , Z 9 , Z 4 , D , Z 9 , Z 3 , D , Z 9 , Z 2 , D , Z 9 , Z1 , D ,
{
} {
} {
} {
} {
} {
} {
} {
{Z9 , Z8 , Z3} , {Z9 , Z 7 , Z3} , {Z9 , Z 6 , Z3} , {Z9 , Z 6 , Z 2 } , {Z9 , Z 6 , Z1} , {Z9 , Z5 , Z1} , {Z9 , Z 4 , Z1} ,







{Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D} , {Z , Z , D}
8
3
7
3
6
3
6
2
6
1
5
1
4
(see diagram 3 of the Figure 2);





4) Z 9 , Z 4 , Z1 , D , Z 9 , Z 5 , Z1 , D , Z 9 , Z 6 , Z1 , D , Z 9 , Z 6 , Z 2 , D , Z 9 , Z 6 , Z 3 , D ,


Z 9 , Z 7 , Z 3 , D , Z 9 , Z8 , Z 3 , D
{
{
} {
} {
} {
}
} {
} {
}
(see diagram 4 of the Figure 2);
5)
{Z9 , Z5 , Z 4 , Z1} , {Z9 , Z 6 , Z 4 , Z1} , {Z9 , Z 6 , Z5 , Z1} , {Z9 , Z 7 , Z 6 , Z3} , {Z9 , Z8 , Z 6 , Z3} ,




{Z9 , Z8 , Z7 , Z3} , {Z9 , Z8 , Z 4 , D} , {Z9 , Z8 , Z5 , D} , {Z9 , Z7 , Z 2 , D} , {Z9 , Z7 , Z 4 , D} ,





{Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} ,





{Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} ,




{Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D} , {Z , Z , Z , D}
9
7
5
9
8
1
9
8
2
9
4
2
9
4
9
5
2
9
5
3
9
7
1
9
2
1
9
3
9
3
2
6
2
1
6
3
1
6
3
2
3
1
(see diagram 5 of the Figure 2);




6) Z 9 , Z 5 , Z 4 , Z1 , D , Z 9 , Z 6 , Z 4 , Z1 , D , Z 9 , Z 6 , Z 5 , Z1 , D , Z 9 , Z 7 , Z 6 , Z 3 , D ,


Z 9 , Z8 , Z 6 , Z 3 , D , Z 9 , Z8 , Z 7 , Z 3 , D
{
{
} {
} {
} {
}
} {
(see diagram 6 of the Figure 2);



7) Z 9 , Z 6 , Z 2 , Z1 , D , Z 9 , Z 6 , Z 3 , Z1 , D , Z 9 , Z 6 , Z 3 , Z 2 , D
{
} {
} {
}
(see diagram 7 of the Figure 2);
282
}
1
}
8)


S. Makharadze et al.


{Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D} ,




{Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , D}
9
8
6
3
2
9
8
6
3
1
9
7
6
9
6
5
3
1
9
6
5
2
1
9
6
4
3
3
2
9
1
9
7
6
6
3
4
2
1
1
(see diagram 8 of the Figure 2);
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



{Z 8 , Z 7 , Z 3 } , {Z 8 , Z 6 , Z 3 } , Z 8 , Z 6 , D , Z 8 , Z 5 , D , Z 8 , Z 4 , D ,




Z8 , Z 2 , D , Z8 , Z1 , D , {Z 7 , Z 6 , Z 3 } , Z 7 , Z 5 , D , Z 7 , Z 4 , D ,


Z 7 , Z 2 , D , Z 7 , Z1 , D , {Z 6 , Z 5 , Z1} , {Z 6 , Z 4 , Z1} , {Z 5 , Z 4 , Z1} ,





Z5 , Z3 , D , Z5 , Z 2 , D , Z 4 , Z3 , D , Z 4 , Z 2 , D , Z3 , Z 2 , D ,


Z 3 , Z1 , D , Z 2 , Z1 , D .
{
{
{
{
}
}
}
}
{
{
{
{
}
}
} {
}
{
} {
{
} {
} {
}
}
} {
} {
}
(see diagram 9 of the Figure 2);






Z8 , Z 6 , Z 3 , D , Z8 , Z 7 , Z 3 , D , Z 7 , Z 6 , Z 3 , D , Z 5 , Z 4 , Z1 , D , Z 6 , Z 4 , Z1 , D , Z 6 , Z 5 , Z1 , D
{
} {
} {
} {
} {
} {
}
(see diagram 10 of the Figure 2);
are XI-semilattices iff the intersection of minimal elements of the given semilattices is empty set. From the formal equalities (1) of the given semilattice D we have
Z8 ∩ Z 7 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P9 ) ∪ ( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P8 ∪ P9 ) ≠ ∅
Z8 ∩ Z 6 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P9 ) ∪ ( P0 ∪ P4 ∪ P5 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z8 ∩ Z 5 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z8 ∩ Z 4 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z8 ∩ Z 2 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P9 ) ∪ ( P0 ∪ P1 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z8 ∩ Z1 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z7 ∩ Z6 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P4 ∪ P5 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z7 ∩ Z5 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z7 ∩ Z4 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z7 ∩ Z2 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P1 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z 7 ∩ Z1 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z6 ∩ Z5 =
( P0 ∪ P4 ∪ P5 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z6 ∩ Z4 =
( P0 ∪ P4 ∪ P5 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z5 ∩ Z 4 =
( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z5 ∩ Z3 =
( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z5 ∩ Z 2 =
( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P1 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z 4 ∩ Z3 =
( P0 ∪ P2 ∪ P3 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
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Z4 ∩ Z2 =
( P0 ∪ P2 ∪ P3 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P1 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z3 ∩ Z 2 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P1 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z 3 ∩ Z1 =
( P0 ∪ P1 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
Z 2 ∩ Z1 =
( P0 ∪ P1 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ∪ ( P0 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7 ∪ P8 ∪ P9 ) ≠ ∅
From the equalities given above it follows that the semilattices 9 and 10 are not XI-semilattices. 
The semilattices 11




{Z6 , Z5 , Z3 , Z1 , D} , {Z6 , Z5 , Z 2 , Z1 , D} , {Z6 , Z 4 , Z3 , Z1 , D} , {Z6 , Z 4 , Z 2 , Z1 , D} ,




{Z7 , Z6 , Z3 , Z 2 , D} , {Z7 , Z6 , Z3 , Z1 , D} , {Z8 , Z6 , Z3 , Z 2 , D} , {Z8 , Z6 , Z3 , Z1 , D}.
(see diagram 1-8 of the Figure 3);
are not XI-semilattice since we have the following inequalities
Z 5 ∩ Z 3 ≠ ∅, Z 5 ∩ Z 2 ≠ ∅, Z 4 ∩ Z 3 ≠ ∅, Z 4 ∩ Z 2 ≠ ∅,
Z 7 ∩ Z 2 ≠ ∅, Z 7 ∩ Z1 ≠ ∅, Z8 ∩ Z 2 ≠ ∅, Z8 ∩ Z1 ≠ ∅.
The semilattices 12 to 52 are never XI-semilattices. We prove that the semilattice, diagram 52 of the Figure 2,
is not an XI-semilattice (see Figure 4). Indeed, let Q = {T0 , T1 , T2 , T3 , T4 , T5 , T6 , T7 , T8 } and
C ( Q ) = { P0′, P1′, P2′, P3′, P4′, P5′, P6′, P7′, P8′}
be a family of sets, where P0′, P1′, P2′, P3′, P4′, P5′, P6′, P7′, P8′ are pairwise disjoint subsets of the set X. Let
 T0 T1 T2 T3 T4 T5 T6 T7 T8 

 P0′ P1′ P2′ P3′ P4′ P5′ P6′ P7′ P8′ 
ϕ =
be a mapping of the semilattice Q onto the family of sets C ( Q ) . Then for the formal equalities of the semilattice Q we have a form:
Figure 3. Diagram of all subsemilattices which are
isomorphic to 11 in Figure 2.
Figure 4. Diagram of subsemilattice 52 in Figure 2.
284
T0 = P0′ ∪ P1′∪ P2′ ∪ P3′ ∪ P4′ ∪ P5′ ∪ P6′ ∪ P7′ ∪ P8′,
T1 = P0′ ∪ P2′ ∪ P3′ ∪ P4′ ∪ P5′ ∪ P6′ ∪ P7′ ∪ P8′,
T2 = P0′ ∪ P1′∪ P3′ ∪ P4′ ∪ P5′ ∪ P6′ ∪ P7′ ∪ P8′,
T3 = P0′ ∪ P2′ ∪ P4′ ∪ P5′ ∪ P6′ ∪ P7′ ∪ P8′,
S. Makharadze et al.
T4 = P0′ ∪ P2′ ∪ P3′ ∪ P5′ ∪ P6′ ∪ P7′ ∪ P8′,
T5 = P0′ ∪ P3′ ∪ P4′ ∪ P6′ ∪ P7′ ∪ P8′,
T6 = P0′ ∪ P1′∪ P3′ ∪ P4′ ∪ P5′ ∪ P7′ ∪ P8′,
T7 = P0′ ∪ P1′∪ P3′ ∪ P4′ ∪ P5′ ∪ P6′ ∪ P8' ,
(3)
T8 = P0′.
Here the elements P1′ , P2′ , P3′ , P4′ , P6′ , P7′ are basis sources, the elements P0′ , P5′ , P8′ are sources of completeness of the semilattice D. Therefore X ≥ 6 and δ = 7 (see [2]). Then of the formal equalities we have:
Q,
 T ,T ,T ,T ,
{ 7 6 2 0 }
{T4 , T3 , T1 , T0 } ,

{T7 , T6 , T5 , T4 , T2 , T1 , T0 } ,

Qt = {T7 , T6 , T5 , T3 , T2 , T1 , T0 } ,
{T , T , T , T , T , T , T } ,
 7 6 4 3 2 1 0
{T7 , T5 , T4 , T3 , T2 , T1 , T0 } ,
 T ,T ,T ,T ,T ,T ,T ,
{ 6 5 4 3 2 1 0 }
{T8 , T7 , T6 , T5 , T4 , T3 , T2 , T1 , T0 } ,
T8 ,

T8 ,
T8 ,

T8 ,
Λ ( Q, Qt =
) T8 ,
T ,
 8
T8 ,

T8 ,
T8 ,
t ∈ P0′,
t ∈ P1′,
t ∈ P2′,
t ∈ P3′,
t ∈ P4′,
t ∈ P5′,
t ∈ P6′,
if t ∈ P7′,
if t ∈ P8′.
if
if
if
if
if
if
if
t ∈ P0′,
t ∈ P1′,
t ∈ P2′,
t ∈ P3′,
t ∈ P4′,
if t ∈ P5′,
if t ∈ P6′,
if t ∈ P7′,
if
if
if
if
if
if t ∈ P8′.
We have, that Q ∧ = {T8 } and Λ ( Q, Qt ) ∈ 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 the Figure 5);
b) Q2 = {T , T ′} , where T , T ′ ∈ D and T ⊂ T ′ (see diagram 2 of the Figure 5);
c) Q3 = {T , T ′, T ′′} , where T , T ′, T ′′ ∈ D and T ⊂ T ′ ⊂ T ′′ (see diagram 3 of the Figure 5);


d) Q4 = Z 9 , T , T ′, D , where T , T ′ ∈ D and Z 9 ⊂ T ⊂ T ′ ⊂ D (see diagram 4 of the Figure 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 the Figure 5);

Z 9 , T , T ′, T ∪ T ′, D , where T , T ′ ∈ D , Z 9 ⊂ T , Z 9 ⊂ T ′ , T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ (see diagram
=
f) Q6
6 of the Figure 5);


g) Q7 = Z 9 , Z 6 , T , T ′, D , where T , T ′ ∈ D , Z 6 ⊂ T , Z 6 ⊂ T ′ , T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , T ∪ T ′ =
D
(see diagram 7 of the Figure 5);
{
{
}
}
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S. Makharadze et al.
Figure 5. Diagram of all XI-subsemilattices of D.

=
Z 9 , T , T ′, T ∪ T ′, Z , D , where Z 9 ⊂ T ′ ⊂ Z , T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , (T ∪ T ′ ) \ Z ≠ ∅ ,
h) Q8
Z \ (T ∪ T ′ ) ≠ ∅ (see diagram 8 of the Figure 5);
Note that the semilattices in Figure 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-semilatices of unions D that every element of this set contains an empty set if ∅ ∈ D or denotes a set of all XI-subsemilatices 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 ′ϑXI D ′′ iff there
exists some complete isomorphism ϕ between the semilatices D ′ and D ′′ . One can easily verify that the
binary relation ϑXI is an equivalence relation on the set Σ′XI ( X , D ) .
By the symbol QiϑXI denote the ϑXI -equivalence class of the set Σ′XI ( X , D ) , where every element is isomorphic to the X-semilattice Qi ( 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 semigroup BX ( D′ ) , and
}
{
I ∗ ( Qi ) =
∑
D ′∈Qiϑ XI
I ( D′ )
where i = 1, 2, ,8 .
Lemma 4. If X is a finite set, then the following equalities hold
a) I ( Q1 ) = 1
b) I ( Q=
2)
c) I ( Q=
3)
d) I ( Q=
4)
e) I ( Q=
5)
f) I ( Q=
6)
g) I ( Q7 ) =
h) I ( Q8=
)
(2
(2
(2
(2
(2
T ′ \T
T ′ \T
T \ Z9
T ′ \ T ′′
T ′ \ T ′′
(
(2
2
)
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 2
− 1) ⋅ ( 2
T ′′ \ T ′
−2
T ′ \T
−2
T ′′ \ T ′
T ′′ \ T ′
)
− 1) ⋅ ( 2
Z 6 \ Z9
T \Z
X \T ′
−1 ⋅ 2
−1 ⋅ 2
T ′′ \ T ′
X \ T ′′
)
− 1) ⋅ ( 5
−1 ⋅ 4
(T ∩T ′ ) \ Z6
T ′ \T
T ′ \T
)⋅3
) ⋅(4
(
)(
−3

D \T ′
)
⋅4

X \D
X \ (T ′ ∪T ′′ )
⋅ 3
−1 ⋅ 3

D \T ′

D \ (T ∪T ′′ )
T \T ′
−2
Z \ (T ∪T ′ )
−4
T \T ′
−2

D \ (T ∪T ′′ )
)(
⋅ 3
T ′ \T
Z \ (T ∪T ′ )
)⋅5
−2
)⋅6

X \D
T ′ \T
)
⋅5

X \D

X \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 ∈ BX ( D ) . Binary relation α is an idempotent relation of the
semmigroup BX ( D ) iff binary relation α satisfies only one conditions of the following conditions:
a) α= X × T , where T ∈ D ;
b) α = YTα × T ∪ YTα′ × T ′ , where T , T ′ ∈ D , T ⊂ T ′ , YTα , YTα′ ∉ {∅} , and satisfies the conditions: YTα ⊇
T, YTα′ ∩ T ′ ≠ ∅ ;
c) α = YTα × T ∪ YTα′ × T ′ ∪ YTα′′ × T ′′ , where T , T ′, T ′′ ∈ D , T ⊂ T ′ ⊂ T ′′ , YTα , YTα′ , YTα′′ ∉ {∅} , and sa-
(
) (
(
) (
)
) (
)
tisfies the conditions: YT ⊇ T , YT ∪ YT ′ ⊇ T ′ , YTα′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ ;
α
α
α
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

d) α = Y9α × Z 9 ∪ YTα × T ∪ YTα′ × T ′ ∪ Y0α × D , where T , T ′ ∈ D , Z 9 ⊂ T ⊂ T ′ ⊂ D , Y9α , YTα , YTα′ ,
(
) (
) (
) (
)
α
α
α
α
α
α
Y0 ∉ {∅} , and satisfies
 the conditions: Y9 ⊇ Z 9 , Y9 ∪ YT ⊇ T , Y9 ∪ YT ∪ YT ′ ⊇ T ′ , YT ∩ T ≠ ∅ ,
α
α
YT ′ ∩ T ′ ≠ ∅ , Y0 ∩ D ≠ ∅ ;
e) α = YTα × T ∪ YTα′ × T ′ ∪ YTα′′ × T ′′ ∪ YTα′∪T ′′ × (T ′ ∪ T ′′ ) , where T , T ′ , T ′′ ∈ D , T ⊂ T ′ , T ⊂ T ′′ ,
α
α
(
) (
) (
) (
)
) (
) (
T ′ \ T ′′ ≠ ∅ , T ′′ \ T ′ ≠ ∅ , YT , YT ′ , YT ′′ ∉ {∅} and satisfies the conditions: YTα ∪ YTα′ ⊇ T ′ , YTα ∪ YTα′′ ⊇ T ′′ ,
YTα′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ ;

f) α = Y9α × Z 9 ∪ YTα × T ∪ YTα′ × T ′ ∪ YTα∪T ′ × (T ∪ T ′ ) ∪ Y0α × D , where Z 9 ⊂ T , Z 9 ⊂ T ′ ,
α
(
) (
α
α
) (
)
T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , YT , YT ′ , Y0 ∉ {∅} and satisfies the conditions: Y9α ∪ YTα ⊇ T , Y9α ∪ YTα′ ⊇ T ′ ,
YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ , Y0α ∩ D ≠ ∅ ;

g) α = Y9α × Z 9 ∪ Y6α × Z 6 ∪ YTα × T ∪ YTα′ × T ′ ∪ Y0α × D , where T , T ′ ∈ D , Z 6 ⊂ T , Z 6 ⊂ T ′ ,

T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , T ∪T ′ =
D , Y6α , YTα , YTα′ ∉ {∅} and satisfies the conditions: Y9α ⊇ Z 9 ,
Y9α ∪ Y6α ⊇ Z 6 , Y9α ∪ Y6α ∪ YTα ⊇ T , Y9α ∪ Y6α ∪ YTα′ ⊇ T ′ , Y6α ∩ Z 6 ≠ ∅ , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ;

h) α = Y9α × Z 9 ∪ YTα × T ∪ YTα′ × T ′ ∪ YTα∪T ′ × (T ∪ T ′ ) ∪ YZα × Z ∪ Y0α × D , where Z 9 ⊂ T ′ ⊂ Z ,
α
α
α
(
) (
) (
) (
(
) (
) (
) (
) (
)
) (
) (
)
T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , (T ∪ T ′ ) \ Z ≠ ∅ , Z \ (T ∪ T ′ ) ≠ ∅ , YT , YT ′ , YZ ∉ {∅} and satisfies the conditions: Y9α ∪ YTα ⊇ T , Y9α ∪ YTα′ ⊇ T ′ Y9α ∪ YTα′ ∪ YZα ⊇ Z , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ , YZα ∩ Z ≠ ∅ .
Proof. By Lemma 3 we know that 1 to 8 are an XI-semilattices. We prove only statement g. Indeed, if

α = Y9α × Z 9 ∪ Y6α × Z 6 ∪ YTα × T ∪ YTα′ × T ′ ∪ Y0α × D ,
α
) (
(
) (
α
α
) (
) (
)
where Y6 , YT , YT ′ ∉ {∅} , then it is easy to see, that the set D (α ) = {Z 9 , Z 6 , T , T1′} is a generating set of the
semilattice Z 9 , Z 6 , T , T ′, D . Then the following equalities hold
 (α )
 (α )
=
D
Z }, D
{=
{Z , Z } ,
α
{
α
α
}
9
Z9
 (α )
=
D
T
Z6
9
 (α )
Z9 , Z6 , T } , D
{=
T′
6
{Z9 , Z 6 , T ′} .
By statement a of the Theorem 6.2.1 (see [1]) we have:
Y9α ⊇ Z 9 , Y9α ∪ Y6α ⊇ Z 6 , Y9α ∪ Y6α ∪ YTα ⊇ T , Y9α ∪ Y6α ∪ YTα′ ⊇ T ′ .
Further, one can see, that the equalities are true:
(
) (
)
(
)
 (α ) , Z = ∪ D
 (α ) \ {Z } = Z , Z \ l D
 (α ) , Z = Z \ Z ≠ ∅,
l D
6
6
9
6
6
6
9
Z
Z
Z
6
6
6
 (α ) , T ) = ∪ ( D
 (α ) \ {T } ) = Z , T \ l ( D
 (α ) , T ) = T \ Z ≠ ∅,
l (D
6
6
T
T
T
 (α ) , T ′ ) = ∪ ( D
 (α ) \ {T ′} ) = Z , T ′ \ l ( D
 (α ) , T ′ ) = T ′ \ Z ≠ ∅,
l (D
6
6
T′
T′
T′
 (α ) , D
 (α ) respectively.
 (α ) , D
We have the elements Z6, T, T' are nonlimiting elements of the sets D
Z
T
T′
6
By statement b of the Theorem 6.2.1 [1] it follows, that the conditions Y6α ∩ Z 6 ≠ ∅ , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠
∅ hold. Therefore, the statement g is proved. Rest of statements can be proved analogously.
Lemma 5. Let D ∈ Σ1 ( X ,10 ) and Z 9 ≠ ∅ . If X is a finite set, then the number I ∗ ( Q1 ) may be calculated by the formula I ∗ ( Q1 ) = 10 .
Lemma 6. Let D ∈ Σ1 ( X ,10 ) and Z 9 ≠ ∅ . If X is a finite set, then the number I ∗ ( Q2 ) may be calculated by formula
 2 Z8 \ Z 9 − 1  ⋅ 2 X \ Z8 +  2 Z 7 \ Z 9 − 1  ⋅ 2 X \ Z 7 +  2 Z 6 \ Z 9 − 1  ⋅ 2 X \ Z 6 +  2 Z 5 \ Z 9 − 1  ⋅ 2 X \ Z 5
I ∗ (Q
=








2)








+  2

Z 4 \ Z9
Z3 \ Z9
Z 3 \ Z8
Z3 \ Z 7
Z3 \ Z 6
X \Z
X \Z
− 1 ⋅ 2 4 +  2
+2
+2
+2
− 4  ⋅ 2 3



+  2

Z 2 \ Z9
+2

Z 2 \ Z6

Z1 \ Z9
Z1 \ Z 6
Z1 \ Z5
Z1 \ Z 4
X \Z
X \Z
− 2  ⋅ 2 2 +  2
+2
+2
+2
− 4  ⋅ 2 1










D \ Z8
D \ Z7
D \ Z6
D \ Z5
D \ Z4
D \ Z3
D \ Z2
D \ Z1
 D \ Z9
 X \ D
+2
+2
+2
+2
+2
+2
+2
+2
+2
− 9⋅ 2 .


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Lemma 7. Let D ∈ Σ1 ( X ,10 ) and Z 9 ≠ ∅ . If X is a finite set, then the number I ∗ ( Q3 ) may be calculated by formula






 2 Z8 \ Z 9 − 1  ⋅  3 D \ Z8 − 2 D \ Z8  ⋅ 3 X \ D +  2 Z 7 \ Z 9 − 1  ⋅  3 D \ Z 7 − 2 D \ Z 7  ⋅ 3 X \ D
I (Q
=

 

 
3)



 

 


∗


D \ Z5
 X \ D  Z5 \ Z9
 D \ Z5
+2
− 1 ⋅  3
−2
⋅3

 



D \ Z3
 X \ D  Z3 \ Z9
 D \ Z3
+2
− 1 ⋅  3
−2
⋅3

 



D \ Z1
 X \ D  Z1 \ Z9
 D \ Z1
+2
− 1 ⋅  3
−2
⋅3

 

D \ Z6
 D \ Z6
− 1 ⋅  3
−2
 


 X \ D
⋅3



 X \ D
⋅3



+  2

Z 6 \ Z9
+  2

Z 4 \ Z9
+  2

Z 2 \ Z9
+  2

Z8 \ Z 9
Z 3 \ Z8
Z 3 \ Z8
− 1 ⋅  3
−2
 
+  2

Z 6 \ Z9
Z3 \ Z 6
Z3 \ Z 6
 X \ Z3 +  2 Z 6 \ Z9 − 1  ⋅  3 Z 2 \ Z 6 − 2 Z 2 \ Z 6  ⋅ 3 X \ Z 2
− 1 ⋅  3
−2
⋅3

 

 


 

+  2

Z 6 \ Z9
Z1 \ Z 6
Z1 \ Z 6
− 1 ⋅  3
−2
 
+  2

Z 4 \ Z9
+  2

Z3 \ Z 7
+  2

Z 2 \ Z6
D \ Z4
 D \ Z4
− 1 ⋅  3
−2
 
D \ Z2
 D \ Z2
− 1 ⋅  3
−2
 
 X \ D
⋅3

 X \ Z3 +  2 Z 7 \ Z9 − 1  ⋅  3 Z3 \ Z 7 − 2 Z3 \ Z 7
⋅3

 


 
 X \ Z1 +  2 Z5 \ Z9 − 1 ⋅  3 Z1 \ Z5 − 2 Z1 \ Z5
⋅3

 


 


Z1 \ Z 4
Z1 \ Z 4
 ⋅ 3 X \ Z1 +  2 Z3 \ Z8 − 1 ⋅  3 D \ Z3 − 2 D \ Z3
− 1 ⋅  3
−2


 

 

 


D \ Z3
 X \ D  Z3 \ Z6
 D \ Z3
+2
− 1 ⋅  3
−2
⋅3

 



D \ Z1
 X \ D  Z1 \ Z6
 D \ Z1
+2
− 1 ⋅  3
−2
⋅3

 

D \ Z3
 D \ Z3
− 1 ⋅  3
−2
 
D \ Z2
 D \ Z2
− 1 ⋅  3
−2
 


 X \ Z3
⋅3

 X \ Z1
⋅3

 X \ D
⋅3



 X \ D
⋅3



 X \ D
⋅3





D \ Z1 
D \ Z1 
Z1 \ Z5
Z1 \ Z 4
 D \ Z1
 D \ Z1
X \D
X \D
+  2
− 1 ⋅  3
−2
+  2
− 1 ⋅  3
−2
⋅3
⋅3 .

 

 


Lemma 8. Let D ∈ Σ1 ( X ,10 ) and Z 9 ≠ ∅ . If X is a finite set, then the number I ∗ ( Q4 ) may be
calculated by formula
I ∗ (Q
=
4)
(2
Z8 \ Z 9
(
+ (2
+ (2
+ (2
+ (2
+ (2
+ 2
)(
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
−1 ⋅ 3
Z 7 \ Z9
Z 6 \ Z9
Z 6 \ Z9
Z 6 \ Z9
Z5 \ Z9
Z 4 \ Z9
Z 3 \ Z8
−2
Z 3 \ Z8
)⋅(4
) ⋅(4
) ⋅(4
)⋅(4
)⋅(4
) ⋅(4
) ⋅(4

D \ Z3
)⋅4
−3
)⋅4
−3
)⋅4
−3
)⋅4
−3
)⋅4
−3
)⋅4
−3
)⋅4
−3
Z3 \ Z 7
−2
Z3 \ Z 7

D \ Z3
Z3 \ Z 6
−2
Z3 \ Z 6

D \ Z3
Z 2 \ Z6
−2
Z 2 \ Z6

D \ Z2
Z1 \ Z 6
−2
Z1 \ Z 6

D \ Z1
Z1 \ Z5
−2
Z1 \ Z5

D \ Z1
Z1 \ Z 4
−2
Z1 \ Z 4

D \ Z1

D \ Z3

X \D

D \ Z3

X \D

D \ Z3

X \D

D \ Z2

X \D

D \ Z1

X \D

D \ Z1

X \D

D \ Z1

X \D
.
Lemma 9. Let D ∈ Σ1 ( X ,10 ) and Z 9 ≠ ∅ . If X is a finite set, then the number I ∗ ( Q5 ) may be
calculated by formula
288
I ∗ (Q
=
5)
(2
) ( − 1) ⋅ 4 + ( 2 − 1) ⋅ ( 2 − 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ (2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ 2 ⋅(2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ 2 ⋅(2
− 1) ⋅ ( 2
− 1) ⋅ 4
+ 2 ⋅(2
− 1) ⋅ ( 2
− 1) ⋅ 4 .
−1 ⋅ 2
Z5 \ Z 4
Z 4 \ Z5
Z6 \ Z 4
X \ Z1
Z 4 \ Z6
X \ Z1
Z 6 \ Z5
Z5 \ Z 6
X \ Z1
Z7 \ Z6
Z6 \ Z7
X \ Z3
Z8 \ Z 6
Z 6 \ Z8
X \ Z3
Z8 \ Z 7
Z 7 \ Z8
X \ Z3
Z8 \ Z 4
Z 4 \ Z8

X \D
Z8 \ Z 5
Z 5 \ Z8

X \D
Z7 \ Z 2
Z 2 \ Z7

X \D
Z7 \ Z 4
Z 4 \ Z7

X \D
Z 7 \ Z5
Z5 \ Z 7

X \D
Z8 \ Z1
Z1 \ Z8
Z8 \ Z 2
Z 2 \ Z8

X \D
Z4 \ Z2
Z2 \ Z4

X \D
Z 4 \ Z3
Z3 \ Z 4

X \D
Z5 \ Z 2
Z 2 \ Z5

X \D
Z5 \ Z3
Z3 \ Z5

X \D
Z 7 \ Z1
Z1 \ Z 7
Z 2 \ Z1
Z1 \ Z 2

X \D
Z3 \ Z 2
Z 2 \ Z3

X \D
S. Makharadze et al.
Z3 \ Z1

X \D

X \D
Z1 \ Z3

X \D
Lemma 10. Let D ∈ Σ1 ( X ,10 ) and Z 9 ≠ ∅ . If X is a finite set, then the number I ∗ ( Q6 ) may be calculated by formula
I ∗ ( Q6 ) = 1 + 3 + 3 + 3 + 3 + 1 = 14
Lemma 11. Let D ∈ Σ1 ( X ,10 ) and Z 9 ≠ ∅ . If X is a finite set, then the number I ∗ ( Q7 ) may be calculated by formula
(
I ∗ ( Q7 ) = 2
Z 6 \ Z9
(
+ (2
+ 2
)
−1 ⋅ 2
Z 6 \ Z9
Z 6 \ Z9
(
( Z 2 ∩ Z1 ) \ Z6
) (
(
− 1) ⋅ 2
−1 ⋅ 2
⋅ 3
Z3 ∩ Z1 ) \ Z 6
Z3 ∩ Z 2 ) \ Z 6
−2
Z 2 \ Z1
(
⋅ (3
⋅ 3
Z3 \ Z1
Z3 \ Z 2
Z 2 \ Z1
−2
) ⋅ (3
) ⋅ (3
) ⋅ (3
Z1 \ Z 2
Z3 \ Z1
−2
−2
Z1 \ Z3
Z3 \ Z 2
Z1 \ Z 2
−2
Z 2 \ Z3
)⋅5
)⋅5
)⋅5

X \D

X \D
Z1 \ Z3
−2
Z 2 \ Z3

X \D
.
Lemma 12. Let D ∈ Σ1 ( X ,10 ) and Z 9 ≠ ∅ . If X is a finite set, then the number I ∗ ( Q8 ) may be calculated by formula
I ∗ (Q
=
8)
(2
Z8 \ Z 2
(
+ (2
+ (2
+ (2
+ 2
)(
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
−1 ⋅ 2
Z8 \ Z1
Z7 \ Z 2
Z 7 \ Z1
Z5 \ Z3
Z 6 \ Z8
)(
Z 6 \ Z8
Z6 \ Z7
Z6 \ Z7
(
Z5 \ Z 2
−1 ⋅ 2
+ 2
Z 4 \ Z3
−1 ⋅ 2
+ 2
(
+ (2
Z4 \ Z2
)(
− 1) ⋅ ( 2
)(
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
−1 ⋅ 3
Z 6 \ Z5
Z 6 \ Z5
Z6 \ Z 4
Z6 \ Z 4
Figure 6 shows all XI-subsemilattices with six elements.
289
Z 2 \ Z3
−2
Z1 \ Z3
Z 2 \ Z3
Z 2 \ Z3
−2
)⋅6
)⋅6
)⋅6
)⋅6
)⋅6
)⋅6
)⋅6
)⋅6
Z1 \ Z3
−2
Z 2 \ Z3

X \D

X \D

X \D
Z1 \ Z3
−2
Z1 \ Z3

X \D
Z3 \ Z1
−2
Z3 \ Z1

X \D
Z 2 \ Z1
−2
Z 2 \ Z1

X \D
Z3 \ Z1
−2
Z3 \ Z1

X \D
Z 2 \ Z1
−2
Z 2 \ Z1

X \D
.
S. Makharadze et al.
Figure 6. Diagram of all subsemilattices which are isomorphic.
Theorem 11. Let D ∈ Σ1 ( X ,10 ) , Z 9 ≠ ∅ . If X is a finite set and I D is a set of all idempotent elements of
the semigroup BX ( D ) . Then I D = ∑ i =1 I ∗ ( Qi ) .
8
Example 12. Let X = {1, 2,3, 4,5, 6, 7,8} ,
=
P0
P5 =
6} , P1 {1=
{=
} , P2 {2=
} , P3 {3=
} , P4 {4} ,
{5} , P7 = {7} , P8 = {8} , P9 = P6 = ∅.

Then D = {1, 2,3, 4,5, 6, 7,8} , Z1 = {2,3, 4,5, 6, 7,8} , Z 2 = {1,3, 4,5, 6, 7,8} , Z 3 = {1, 2, 4,5, 6, 7,8} ,
Z 4 = {2,3,5, 6, 7,8} , Z 5 = {2,3, 4, 6, 7,8} , Z 6 = {4,5, 6, 7,8} , Z 7 = {1, 2, 4,5, 6,8} , Z8 = {1, 2, 4,5, 6, 7} and
Z 9 = {6} .
D = {{1, 2,3, 4,5, 6, 7,8} , {2,3, 4,5, 6, 7,8} , {1,3, 4,5, 6, 7,8} , {1, 2, 4,5, 6, 7,8} , {2,3,5, 6, 7,8} ,
{2,3, 4, 6, 7,8} , {4,5, 6, 7,8} , {1, 2, 4,5, 6,8} , {1, 2, 4,5, 6, 7} , {6}}
We have Z 9 ≠ ∅ . Where I ∗ ( Q1 ) = 10 , I ∗ ( Q2 ) = 1169 , I ∗ ( Q3 ) = 2154 , I ∗ ( Q4 ) = 349 ,
I ∗ ( Q5 ) = 122 , I ∗ ( Q6 ) = 14 , I ∗ ( Q7 ) = 90 , I ( Q8 ) = 8 , I D = 3916 .
3. Results
Lemma 13. Let D ∈ Σ1 ( X ,10 ) and Z 9 = ∅
 . Then the following sets exhaust all subsemilattices of the semilattice D = Z 9 , Z8 , Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D which contains the empty set:
{
1)
}
{∅}
(see diagram 1 of the Figure 2);

2) ∅, D , {∅, Z8 } , {∅, Z 7 } , {∅, Z 6 } , {∅, Z 5 } , {∅, Z 4 } , {∅, Z 3 } , {∅, Z 2 } , {∅, Z1}
{
}
(see diagram 2 of the Figure 2);








3) ∅, Z8 , D , ∅, Z 7 , D , ∅, Z 6 , D , ∅, Z 5 , D , ∅, Z 4 , D , ∅, Z 3 , D , ∅, Z 2 , D , ∅, Z1 , D ,
{
} {
} {
} {
} {
} {
} {
} {
{∅, Z8 , Z3} , {∅, Z 7 , Z3} , {∅, Z 6 , Z3} , {∅, Z 6 , Z 2 } , {∅, Z 6 , Z1} , {∅, Z5 , Z1} , {∅, Z 4 , Z1}
(see diagram 3 of the Figure 2);
290
}
4)




S. Makharadze et al.

{∅, Z , Z , D} , {∅, Z , Z , D} , {∅, Z , Z , D} , {∅, Z , Z , D} , {∅, Z , Z , D} ,


{∅, Z , Z , D} , {∅, Z , Z , D}
4
1
5
1
7
3
8
3
6
1
6
2
6
3
(see diagram 4 of the Figure 2);
5)
{∅, Z5 , Z 4 , Z1} , {∅, Z 6 , Z 4 , Z1} , {∅, Z 6 , Z5 , Z1} , {∅, Z 7 , Z 6 , Z3} , {∅, Z8 , Z 6 , Z3} , {∅, Z8 , Z 7 , Z3} ,






{∅, 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 , Z , D} , {∅, Z , Z , D} , {∅, Z , Z , D} , {∅, Z , Z , D} ,



{∅, Z , Z , D} , {∅, Z , Z , D} , {∅, Z , Z , D}
8
4
8
5
7
2
7
4
7
5
8
1
8
2
4
2
4
3
5
2
5
3
7
1
2
1
3
1
3
2
(see diagram 5 of the Figure 2);




6) ∅, Z 5 , Z 4 , Z1 , D , ∅, Z 6 , Z 4 , Z1 , D , ∅, Z 6 , Z 5 , Z1 , D , ∅, Z 7 , Z 6 , Z 3 , D ,


∅, Z 8 , Z 6 , Z 3 , D , ∅, Z 8 , Z 7 , Z 3 , D
{
{
} {
} {
} {
}
} {
}
(see diagram 6 of the Figure 2);



7) ∅, Z 6 , Z 2 , Z1 , D , ∅, Z 6 , Z 3 , Z1 , D , ∅, Z 6 , Z 3 , Z 2 , D
} {
{
} {
}
(see diagram 7 of the Figure 2);




8) ∅, Z8 , Z 6 , Z 3 , Z 2 , D , ∅, Z8 , Z 6 , Z 3 , Z1 , D , ∅, Z 7 , Z 6 , Z 3 , Z 2 , D , ∅, Z 7 , Z 6 , Z 3 , Z1 , D ,




∅, Z 6 , Z 5 , Z 3 , Z1 , D , ∅, Z 6 , Z 5 , Z 2 , Z1 , D , ∅, Z 6 , Z 4 , Z 3 , Z1 , D , ∅, Z 6 , Z 4 , Z 2 , Z1 , D
{
{
} {
} {
} {
} {
} {
} {
}
}
(see diagram 8 of the Figure 2);
Theorem 13. Let D ∈ Σ1 ( X ,10 ) , Z 9 = ∅ and α ∈ BX ( D ) . Binary relation α is an idempotent relation
of the semmigroup BX ( D ) iff binary relation α satisfies only one conditions of the following conditions:
a) α = ∅ ;
α Y9α × ∅ ∪ YTα × T , where T ∈ D , ∅ ≠ T , YTα ≠ ∅ , and satisfies the conditions: YTα ∩ T ≠ ∅ ;
b) =
c)
(
α = (Y
α
9
) (
× ∅ ) ∪ (Y
α
T
)
× T ) ∪ (Y
α
T′
)
× T ′ , where T , T ′ ∈ D , ∅ ≠ T ⊂ T ′ , YTα , YTα′ ∉ {∅} , and satisfies the
conditions: Y9α ∪ YTα ⊇ T , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ;


d) α = Y9α × ∅ ∪ YTα × T ∪ YTα′ × T ′ ∪ Y0α × D , where T , T ′ ∈ D , ∅ ≠ T ⊂ T ′ ⊂ D , YTα , YTα′ , Y0α ∉
(
) (
) (
{∅} , and
 satisfies the conditions:
α
) (
)
Y9 ∪ YT ⊇ T , Y9α ∪ YTα ∪ YTα′ ⊇ T ′ , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ,
α
α
Y0 ∩ D ≠ ∅ ;
e) α = Y9α × ∅ ∪ YTα × T ∪ YTα′ × T ′ ∪ YTα∪T ′ × (T ∪ T ′ ) , where T , T ′ ∈ D , T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , YTα ,
(
) (
) (
) (
)
YT ′ ∉ {∅} and satisfies the conditions: Y9 ∪ YT ⊇ T , Y9 ∪ YTα′ ⊇ T ′ , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ;

f) α = Y9α × ∅ ∪ YTα × T ∪ YTα′ × T ′ ∪ YTα∪T ′ × (T ∪ T ′ ) ∪ Y0α × D , where T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , YTα ,
α
α
(
) (
) (
α
α
) (
) (
)
YT ′ , Y0 ∉ {∅} and satisfies the conditions: Y9 ∪ YT ⊇ T , Y9 ∪ YTα′ ⊇ T ′ , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ,
Y0α ∩ D ≠ ∅ .

g) α = Y9α × ∅ ∪ Y6α × Z 6 ∪ YTα × T ∪ YTα′ × T ′ ∪ Y0α × D , where T , T ′ ∈ D , Z 6 ⊂ T , Z 6 ⊂ T ′ ,

T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , T ∪T ′ =
D , Y6α , YTα , YTα′ ∉ {∅} and satisfies the conditions: Y9α ∪ Y6α ⊇ Z 6 ,
Y9α ∪ Y6α ∪ YTα ⊇ T , Y9α ∪ Y6α ∪ YTα′ ⊇ T ′ , Y6α ∩ Z 6 ≠ ∅ , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ;

h) α = Y9α × ∅ ∪ YTα × T ∪ YTα′ × T ′ ∪ YTα∪T ′ × (T ∪ T ′ ) ∪ YZα × Z ∪ Y0α × D , where T ′ ⊂ Z ,
α
α
α
(
) (
) (
) (
(
) (
) (
) (
α
) (
α
)
) (
) (
)
T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , (T ∪ T ′ ) \ Z ≠ ∅ , Z \ (T ∪ T ′ ) ≠ ∅ , YT , YT ′ , YZ ∉ {∅} and satisfies the conditions:
Y9α ∪ YTα ⊇ T , Y9α ∪ YTα′ ⊇ T ′ , Y9α ∪ YTα′ ∪ YZα ⊇ Z , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ , YZα ∩ Z ≠ ∅ ;
α
α
α
Lemma 14. Let D ∈ Σ1 ( X ,10 ) and Z 9 = ∅ . If X is a finite set, then I ∗ ( Q1 ) = 1 .
Lemma 15. Let D ∈ Σ1 ( X ,10 ) and Z 9 = ∅ . If X is a finite set, then the number I ∗ ( Q2 ) may be calcu-
291
S. Makharadze et al.
lated by formula

Z7
Z6
 D
 X \ D  Z8
X \Z
X \Z
X \Z
+  2 − 1 ⋅ 2 8 +  2 − 1 ⋅ 2 7 +  2 − 1 ⋅ 2 6
I ( Q2 )=  2 − 1 ⋅ 2








Z5
Z
Z
3
4
X \Z
X \Z
X \Z
+  2 − 1 ⋅ 2 5 +  2 − 1 ⋅ 2 4 +  2 − 1 ⋅ 2 3






Z2
Z1
X \Z
X \Z
+  2 − 1 ⋅ 2 2 +  2 − 1 ⋅ 2 1 .




∗
Lemma 16. Let D ∈ Σ1 ( X ,10 ) and Z 9 = ∅ . If X is a finite set, then the number I ∗ ( Q3 ) may be calculated by formula



+  2

Z6

D \ Z7
 X \ D  Z7
 D \ Z7
+  2 − 1 ⋅  3
−2
⋅3

 

D \ Z8
Z8
 D \ Z8
I ∗ ( Q3 =
)  2 − 1 ⋅  3 − 2

 












 X \ D
⋅3

D \ Z5
 X \ D  Z5
 D \ Z5
+  2 − 1 ⋅  3
−2
⋅3

 

D \ Z6
 D \ Z6
− 1 ⋅  3
−2
 
 X \ D
⋅3



D \ Z4 
D \ Z3 
Z3
Z4
 D \ Z4
 D \ Z3
X \D
X \D
+  2 − 1 ⋅  3
−2
+  2 − 1 ⋅  3
−2
⋅3
⋅3

 

 




D \ Z2 
D \ Z1 
Z2
Z1
 D \ Z2
 D \ Z1
X \D
X \D
+  2 − 1 ⋅  3
−2
+  2 − 1 ⋅  3
−2
⋅3
⋅3

 

 


+  2

Z8
Z 3 \ Z8
Z 3 \ Z8
 ⋅ 3 X \ Z3 +  2 Z 7 − 1  ⋅  3 Z3 \ Z 7 − 2 Z3 \ Z 7  ⋅ 3 X \ Z3
− 1 ⋅  3
−2


 

 


 

+  2

Z6
Z3 \ Z 6
Z3 \ Z 6
− 1 ⋅  3
−2
 
+  2

Z6
Z1 \ Z 6
Z1 \ Z 6
 ⋅ 3 X \ Z1 +  2 Z5 − 1 ⋅  3 Z1 \ Z5 − 2 Z1 \ Z5  ⋅ 3 X \ Z1
− 1 ⋅  3
−2


 

 


 

+  2

Z4
Z1 \ Z 4
Z1 \ Z 4
 ⋅ 3 X \ Z1 .
− 1 ⋅  3
−2

 

 ⋅ 3 X \ Z3 +  2 Z 6 − 1  ⋅  3 Z 2 \ Z 6 − 2 Z 2 \ Z 6


 


 
 ⋅ 3 X \ Z2


Lemma 17. Let D ∈ Σ1 ( X ,10 ) and Z 9 = ∅ . If X is a finite set, then the number I ∗ ( Q4 ) may be calculated by formula
I ∗ ( Q4 =
)
(2
Z4
(
+ (2
+ (2
+ (2
+ (2
+ (2
+ 2
)(
−1 ⋅ 3
Z5
Z6
Z6
Z6
Z7
Z8
Z1 \ Z 4
)(
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
−1 ⋅ 3
−2
Z1 \ Z 4
)⋅4
) ⋅(4 − 3 ) ⋅ 4
) ⋅(4 − 3 )⋅ 4
)⋅(4 − 3 )⋅ 4
)⋅(4 − 3 )⋅ 4
) ⋅(4 − 3 ) ⋅ 4
)⋅(4 − 3 )⋅ 4
) ⋅(4
Z1 \ Z5
−2
Z1 \ Z5
Z1 \ Z 6
−2
Z1 \ Z 6
Z 2 \ Z6
−2
Z 2 \ Z6
Z3 \ Z 6
−2
Z3 \ Z 6
Z3 \ Z 7
−2
Z3 \ Z 7
Z 3 \ Z8
−2
Z 3 \ Z8

D \ Z1
−3

D \ Z1

X \D

D \ Z1

D \ Z1

X \D

D \ Z1

D \ Z1

X \D

D \ Z2

D \ Z2

X \D

D \ Z3

D \ Z3

X \D

D \ Z3

D \ Z3

X \D

D \ Z3

D \ Z3

X \D
.
Lemma 18. Let D ∈ Σ1 ( X ,10 ) and Z 9 = ∅ . If X is a finite set, then the number I ∗ ( Q5 ) may be calcu-
292
S. Makharadze et al.
lated by formula
I ∗ (Q
=
5)
(2
Z5 \ Z 4
(
+ (2
+ (2
+ (2
+ (2
+ (2
+ (2
+ (2
+ (2
+ (2
+ 2
)(
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
−1 ⋅ 2
Z 6 \ Z5
Z8 \ Z 6
Z8 \ Z 4
Z7 \ Z 2
Z 7 \ Z5
Z8 \ Z 2
Z 4 \ Z3
Z5 \ Z3
Z 2 \ Z1
Z3 \ Z 2
Z 4 \ Z5
)
−1 ⋅ 4
Z5 \ Z 6
Z 6 \ Z8
Z 4 \ Z8
Z 2 \ Z7
Z5 \ Z 7
Z 2 \ Z8
Z3 \ Z 4
Z3 \ Z5
X \ Z1
)
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
−1 ⋅ 4
(
+ 2
X \ Z1
X \ Z3

X \D
Z 2 \ Z3
(
+ (2
+ (2
+ (2
+ (2
+ (2
+ (2
+ (2
+ (2
+ 2

X \D

X \D
Z7 \ Z6
Z8 \ Z 7
Z7 \ Z 4
Z8 \ Z1

X \D
Z4 \ Z2

X \D
Z5 \ Z 2

X \D
Z 7 \ Z1
Z3 \ Z1

X \D
)(
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
−1 ⋅ 2
Z8 \ Z 5

X \D
Z1 \ Z 2
Z6 \ Z 4
Z 4 \ Z6
)
−1 ⋅ 4
X \ Z1
)
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
− 1) ⋅ 4
Z6 \ Z7
−1 ⋅ 4
Z 7 \ Z8
X \ Z3
X \ Z3

X \D
Z 5 \ Z8
Z 4 \ Z7

X \D

X \D
Z1 \ Z8
Z2 \ Z4
Z 2 \ Z5

X \D

X \D

X \D
Z1 \ Z 7

X \D
Z1 \ Z3
.
Lemma 19. Let D ∈ Σ1 ( X ,10 ) and Z 9 = ∅ . If X is a finite set, then the number I ∗ ( Q6 ) may be calculated by formula
I ∗ (Q
=
6)
(2
Z5 \ Z 4
)(
−1 ⋅ 2
Z 4 \ Z5
+ 2
(
Z6 \ Z 4
−1 ⋅ 2
+ 2
(
Z 6 \ Z5
−1 ⋅ 2
+ 2
(
Z7 \ Z6
−1 ⋅ 2
+ 2
(
Z8 \ Z 6
−1 ⋅ 2
(
Z8 \ Z 7
−1 ⋅ 2
+ 2
)(

D \ Z1
−1 ⋅ 5
)(

D \ Z1
)(

D \ Z1
)(

D \ Z3
)(

D \ Z3
)(

D \ Z3
)(
Z 4 \ Z6
−1 ⋅ 5
)(
Z5 \ Z 6
−1 ⋅ 5
)(
Z6 \ Z7
−1 ⋅ 5
)(
Z 6 \ Z8
−1 ⋅ 5
)(
Z 7 \ Z8
−1 ⋅ 5
)⋅5
−4
)⋅5
−4
)⋅5
−4
)⋅5
−4
)⋅5
−4
)⋅5
−4

D \ Z1

X \D

D \ Z1

X \D

D \ Z1

X \D

D \ Z3

X \D

D \ Z3

X \D

D \ Z3

X \D
.
Lemma 20. Let D ∈ Σ1 ( X ,10 ) and Z 9 = ∅ . If X is a finite set, then the number I ∗ ( Q7 ) may be calculated by formula
(
I ∗ ( Q7 ) =2
Z6
(
+ (2
+ 2
)
−1 ⋅ 2
Z6
Z6
( Z 2 ∩ Z1 ) \ Z6
) (
(
− 1) ⋅ 2
−1 ⋅ 2
(
⋅ 3
Z3 ∩ Z1 ) \ Z 6
Z3 ∩ Z 2 ) \ Z 6
Z 2 \ Z1
(
⋅ (3
⋅ 3
−2
Z3 \ Z1
Z3 \ Z 2
Z 2 \ Z1
−2
)(
) ⋅ (3
) ⋅ (3
⋅ 3
Z3 \ Z1
−2
Z3 \ Z 2
Z1 \ Z 2
−2
Z1 \ Z3
Z 2 \ Z3
Z1 \ Z 2
−2
)
⋅5
Z1 \ Z3
−2

X \D
)⋅5
)⋅5
Z 2 \ Z3

X \D

X \D
.
Lemma 21. Let D ∈ Σ1 ( X , 7 ) and Z 9 = ∅ . If X is a finite set, then the number I ∗ ( Q8 ) may be calcu-
293
S. Makharadze et al.
lated by formula
I ∗ (Q
=
8)
(2
Z8 \ Z 2
(
+ (2
+ (2
+ (2
+ (2
+ (2
+ (2
+ 2
)(
−1 ⋅ 2
Z8 \ Z1
Z7 \ Z 2
Z 7 \ Z1
Z5 \ Z3
Z5 \ Z 2
Z 4 \ Z3
Z4 \ Z2
Z 6 \ Z8
)(
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 2
−1 ⋅ 2
)(
−1 ⋅ 3
Z 6 \ Z8
Z6 \ Z7
Z6 \ Z7
Z 6 \ Z5
Z 6 \ Z5
Z6 \ Z 4
Z6 \ Z 4
Z 2 \ Z3
)(
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
− 1) ⋅ ( 3
−1 ⋅ 3
−2
Z1 \ Z3
Z 2 \ Z3
Z 2 \ Z3
−2
)⋅6
)⋅6
)⋅6
)⋅6
)⋅6
)⋅6
)⋅6
)⋅6
Z1 \ Z3
−2
Z 2 \ Z3

X \D

X \D

X \D
Z1 \ Z3
−2
Z1 \ Z3

X \D
Z3 \ Z1
−2
Z3 \ Z1

X \D
Z 2 \ Z1
−2
Z 2 \ Z1

X \D
Z3 \ Z1
−2
Z3 \ Z1

X \D
Z 2 \ Z1
−2
Z 2 \ Z1

X \D
.
Theorem 14. Let D ∈ Σ1 ( X ,10 ) , Z 9 = ∅ . If X is a finite set and I D is a set of all idempotent elements of
the semigroup BX ( D ) , then I D = ∑ i =1 I ∗ ( Qi ) .
8
Example 15. Let X = {1, 2,3, 4,5, 6, 7} ,
P1 = {1} , P2 = {2} , P3 = {3} , P4 = {4} , P5 = {5} , P7 = {6} , P8 = {7} , P0 = P6 = P9 = ∅ .

Then D = {1, 2,3, 4,5, 6, 7} , Z1 = {2,3, 4,5, 6, 7} , Z 2 = {1,3, 4,5, 6, 7} , Z 3 = {1, 2, 4,5, 6, 7} ,
Z 4 = {2,3,5, 6, 7} , Z 5 = {2,3, 4, 6, 7} , Z 6 = {4,5, 6, 7} , Z 7 = {1, 2, 4,5, 7} , Z8 = {1, 2, 4,5, 6} and Z 9 = ∅ .
D = {{1, 2,3, 4,5, 6, 7} , {2,3, 4,5, 6, 7} , {1,3, 4,5, 6, 7} , {1, 2, 4,5, 6, 7} , {2,3,5, 6, 7} ,
{2,3, 4, 6, 7} , {4,5, 6, 7} , {1, 2, 4,5, 7} , {1, 2, 4,5, 6} , ∅}
We have Z 9 = ∅ . Where I ∗ ( Q1 ) = 1 , I ∗ ( Q2 ) = 1121 , I ∗ ( Q3 ) = 2141 , I ∗ ( Q4 ) = 349 , I ∗ ( Q5 ) = 119 ,
I ∗ ( Q6 ) = 14 , I ∗ ( Q7 ) = 90 , I ( Q8 ) = 8 , I D = 3843 .
It was seen in ([4], Theorem 2) that if α and β are regular elements of BX ( D ) then V ( D, α  β ) is an
XI-subsemilattice of D. Therefore α  β is regular elements of BX ( D ) . That is the set of all regular elements
of BX ( D ) is a subsemigroup of BX ( D ) .
References
[1]
Diasamidze, Ya. and Makharadze, Sh. (2013) Complete Semigroups of Binary Relations. Monograph. Kriter, Turkey,
620 p.
[2]
Diasamidze, Ya. and Makharadze, Sh. (2010) Complete Semigroups of Binary Relations. Monograph. M., Sputnik+,
657 p. (In Russian)
[3]
Diasamidze, Ya., Makharadze, Sh. and Diasamidze, Il. (2008) Idempotents and Regular Elements of Complete Semigroups of Binary Relations. Journal of Mathematical Sciences, Plenum Publ. Cor., New York, 153, 481-499.
[4]
Diasamidze, Ya. and Bakuridze, Al. (to appear) On Some Properties of Regular Elements of Complete Semigroups
Defined by Semilattices of the Class Σ 4 ( X ,8 ) .
294