Gen. Math. Notes, Vol. 27, No. 2, April 2015, pp. 17-36
ISSN 2219-7184; Copyright © ICSRS Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Idempotent Elements of the Semigroups BX(D)
Defined by Semilattices of the Class Ʃ2(X, 8),
When Z7ᴖZ6 = Ø
Nino Tsinaridze
Faculty of Mathematics, Physics and Computer Sciences
Department of Mathematics, Shota Rustaveli Batumi State University
35, Ninoshvili St., Batumi 6010, Georgia
E-mail: ninocinaridze@mail.ru
(Received: 13-2-15 / Accepted: 8-4-15)
Abstract
By the symbol Ʃ2(X, 8) we denote the class of all X- semilattices of unions
whose every element is isomorphic to an X- semilattice of form D = {Z7, Z6, Z5,
Z4, Z3, Z2, Z1, D}, where
⌣
⌣
⌣
Z 6 ⊂ Z 3 ⊂ Z1 ⊂ D⌣, Z6 ⊂ Z 4 ⊂ Z1 ⊂ D⌣, Z6 ⊂ Z 4 ⊂ Z 2 ⊂ D, Z7 ⊂ Z 4 ⊂ Z1 ⊂ D,
Z 7 ⊂ Z 4 ⊂ Z 2 ⊂ D , Z7 ⊂ Z 5 ⊂ Z 2 ⊂ D;
Z i \ Z j ≠ ∅, ( i, j ) ∈ {( 7,6 ) , ( 6,7 ) , ( 5,4 ) , ( 4, 5 ) , ( 5, 3 ) , ( 3, 5 ) , ( 4, 3 ) , ( 3, 4 ) , ( 2,1) , (1, 2 )}.
The paper gives description of idempotent elements of the semigroup BX(D) which
are defined by semilattices of the class Ʃ2(X, 8), for which intersection the
minimal elements is empty. When X is a finite set, the formulas are derived, by
means of which the number of idempotent elements of the semigroup is
calculated.
Keywords: Semilattice, Semigroup, Binary Relation, Idempotent Element.
18
1
Nino Tsinaridze
Introduction
Let X be an arbitrary nonempty set, D be a X − semilattice of unions, i.e. a
nonempty set of subsets of the set X that is closed with respect to the settheoretic operations of unification of elements from D , f be an arbitrary mapping from X into D . To each such a mapping f there corresponds a binary
relation α f on the set X that satisfies the condition α f = ∪ ({ x} × f ( x ) ) . The set
x∈ X
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 a X −
semilattice of unions D (see ([1], Item 2.1).
By ∅ we denote an empty binary relation or empty subset of the set X . The
condition ( x, y ) ∈ α will be written in the form xα y . Let x, y∈X , Y ⊆ X , α ∈ BX ( D ) ,
⌣
T ∈ D , ∅ ≠ D′ ⊆ D and t ∈ D = ∪ Y . Then by symbols we denote the following
Y ∈D
sets:
yα = { x ∈ X | yα x} , Yα = ∪ yα , V ( D,α ) = {Yα | Y ∈ D} ,
y∈Y
X ∗ = {T | ∅ ≠ T ⊆ X } , Dt′ = {Z ′ ∈ D′ | t ∈ Z ′} , YTα = {x ∈ X | xα = T },
ɺɺ′ = {Z ′ ∈ D′ | Z ′ ⊆ T } .
DT′ = {Z ′ ∈ D′ | T ⊆ Z ′} , D
T
By symbol ∧ ( D, Dt ) we mean an exact lower bound of the set D ′ in the
semilattice D .
Definition 1.1: Let ε ∈ BX ( D ) . If ε ε = ε , then ε is called an idempotent element of
the semigroup BX ( D ) and ε is called right unit if α ε = α for any α ∈ BX ( D )
(see [1], [2], [3]).
Definition 1.2: We say that a complete X- semilattice of unions D is an XI −
semilattice of unions if it satisfies the following two conditions:
a)
⌣
∧ ( D, Dt ) ∈ D for any t ∈ D ;
b)
Z = ∪ ∧ ( D, Dt ) for any nonempty element Z of D . (see ([1], definition
t∈Z
1.14.2), ([2] definition 1.14.2), [3], or [4]).
Definition
1.3:
Let
α ∈BX ( D) ,
(
T ∈V X ∗ ,α
)
and
YTα = { y ∈ X | yα = T } .
A
representation of a binary relation α of the form α = ∪ T ∈V ( X ∗ ,α ) (YTα × T ) is colled
quasinormal.
Idempotent Elements of the Semigroups BX(D)...
19
α
( X ,α ) (YT × T ) is a quasinormal representation of a binary
relation α , then the following conditions are true:
Note that, if α = ∪ T ∈V
∗
α
( X ,α ) YT ;
1)
X = ∪T ∈V
2)
YTα ∩ YTα′ = ∅ , for T , T ′ ∈V ( X ∗ , α ) and T ≠ T ′ ;
∗
Let ∑n ( X , m ) denote the class of all complete X − semilattice of unions where
every element is isomorphic to a fixed semilattice D (see [1]).
Definition 1.4: 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 ) = ∅ (see ([1], Definition 1.13.1 and 1.13.2), ([2], Definition 1.13.1
and 1.13.2]).
Theorem 1.1: 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 ([1],
Theorem 6.1.3), ([2] Theorem6.1.3), or [5]).
Theorem 1.2: Let X be a finite set and D (α ) be the set of all those elements T of
the semilattice Q = V ( D,α ) \ {∅} which are nonlimiting elements of the set QɺɺT . A
binary relation α having a quasinormal representation α =
∪ α (Y α × T )
T ∈V ( D ,
)
T
is an
idempotent element of this semigroup iff
a)
b)
V ( D, α ) is complete XI − semilattise of unions;
c)
for any nonlimiting element of the set Dɺɺ (α )T (see ([1], Theorem
6.3.9), ([2], Theorem 6.3.9) or [5]).
∪
ɺɺ (α )
T ′∈D
T
YTα′ ⊇ T for any T ∈ D (α ) ;
YTα ∩ T ≠ ∅
Theorem1.3. Let D , Σ ( D ) , E X( r ) ( 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 all idempotents of the
semigroup BX ( D ) . Then for the sets E X( r ) ( D′ ) and I the following statements are
true:
b)
if ∅ ∉ D , then
1)
for any elements D ′ and D′′ of the set Σ ( D ) that satisfy
the condition D′ ≠ D′′ ;
I = ∪ E X( r ) ( D′ ) ;
2)
EX( ) ( D′) ∩EX( ) ( D′ ) =∅
r
r
D ′∈Σ ( D )
20
Nino Tsinaridze
The equality I =
3)
∑
D′∈Σ( D)
EX( r) ( D′)
is fulfilled for the finite set X (see ([1],
statement b) Theorem (6.2.3) , ([2] statement b) Theorem 6.2.3), or [5]).
2
Results
By the symbol Σ2 ( X ,8) we denote the class of all X − semilattices of unions whose
every element
is
isomorphic to
an
X − semilattice
of form
⌣
D = {Z7 , Z6 , Z5 , Z4 , Z3 , Z2 , Z1, D} , where
⌣
⌣
Z 6 ⊂ Z3 ⊂ Z1 ⊂ D⌣ , Z 6 ⊂ Z 4 ⊂ Z1 ⊂ D⌣, Z 6 ⊂ Z 4 ⊂ Z 2 ⊂ D
⌣,
Z 7 ⊂ Z 4 ⊂ Z1 ⊂ D, Z7 ⊂ Z 4 ⊂ Z 2 ⊂ D, Z 7 ⊂ Z5 ⊂ Z 2 ⊂ D,
Z1 \ Z 2 ≠ ∅, Z 2 \ Z1 ≠ ∅, Z3 \ Z 4 ≠ ∅, Z 4 \ Z3 ≠ ∅, Z3 \ Z5 ≠ ∅,
Z5 \ Z3 ≠ ∅, Z 4 \ Z5 ≠ ∅, Z5 \ Z 4 ≠ ∅, Z 6 \ Z 7 ≠ ∅, Z 7 \ Z 6 ≠ ∅.
(1)
The semilattice satisfying the conditions (1) is shown in Figure 1. Let
are
C(D) = {P0 , P1 , P2 , P3 , P4 , P5 , P6 , P7 } is a family sets, where P0 , P1 , P2 , P3 , P4 , P5 , P6 , P7
pairwise disjoint subsets of the set X and
⌣
 D Z1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 
ϕ =

 P0 P1 P2 P3 P4 P5 P6 P7 
is a mapping of the semilattice D onto the family sets C ( D ) . Then for the formal
equalities of the semilattice D we have a form:
⌣
D
Z1
Z2
Z3
Z 4 Z5
Z6
Z7
⌣
D = P0 ∪ P1 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7
Z1 = P0 ∪ P2 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7
Z 2 = P0 ∪ P1 ∪ P3 ∪ P4 ∪ P5 ∪ P6 ∪ P7
Z 3 = P0 ∪ P2 ∪ P4 ∪ P5 ∪ P6 ∪ P7
Z 4 = P0 ∪ P3 ∪ P5 ∪ P6 ∪ P7
Z 5 = P0 ∪ P1 ∪ P3 ∪ P4 ∪ P6 ∪ P7
Z 6 = P0 ∪ P5 ∪ P7
Z 7 = P0 ∪ P3 ∪ P6
(2)
Fig. 1
Here the elements P1 , P2 , P3 , P5 are basis sources, the element P0 , P4 , P6 , P7 is sources
of completenes of the semilattice D . Therefore X ≥ 4 and δ = 4 (see ([1], Item
11.4), ([2], Item 11.4) or [3]).
Now assume that D ∈ Σ2 ( X ,8) . We introduce the following notation:
1)
2)
3)
Q1 = {T } , where T ∈ D (see diagram 1 in figure 2);
Q2 = {T , T ′} , where T , T ′ ∈ D and T ⊂ T ′ (see diagram 2 in figure 2);
Q3 = {T , T ′, T ′′} , where T , T ′, T ′′ ∈ D and T ⊂ T ′ ⊂ T ′′ (see diagram 3 in figure
2);
Idempotent Elements of the Semigroups BX(D)...
4)
21
⌣
⌣
Q4 = T , T ′, T ′′, D , where T , T ′, T ′′ ∈ D and T ⊂ T ′ ⊂ T ′′ ⊂ D (see diagram 4 in
{
}
figure 2);
5)
Q5 = {T , T ′, T ′′, T ′ ∪ T ′′} , where T , T ′, T ′′ ∈ D ,
T ′′ \ T ′ ≠ ∅ (see
T ⊂ T′ ,
T ⊂ T ′′
diagram 5 in figure 2);
T ∈ {Z 7 , Z 6 } ,
where
}
and T′ \ T′′ ≠∅,
6)
⌣
Q6 = T , Z 4 , Z , Z ′, D ,
7)
Z \ Z′ ≠ ∅, Z′\Z ≠ ∅ (see diagram 6 in figure 2);
⌣
Q7 = T , T ′, T ′′, T ′ ∪ T ′′, D , where T , T ′, T ′′ ∈ D , T ⊂ T ′ , T ⊂ T ′′ and T ′ \ T ′′ ≠ ∅ ,
8)
T ′′ \ T ′ ≠ ∅ (see diagram 7 in figure
⌣
Q8 = T , T ′, Z4 , Z4 ∪T ′, Z, D , where
{
{
Z , Z ′ ∈ {Z 2 , Z1} , Z ≠ Z ′ ,
}
{
}
2);
T ∈ {Z 7 , Z 6 } , T ′ ∈ {Z 5 , Z3 } , Z 4 ∪ T ′, Z ∈ {Z 2 , Z1} , Z 4 ∪ T ′ ≠ Z , T ⊂ T′
9)
10)
11)
12)
13)
14)
and
T ′ \ Z 4 ≠ ∅ , Z 4 \ T ′ ≠ ∅ , ( Z 4 ∪ T ′ ) \ Z ≠ ∅, Z \ ( Z 4 ∪ T ′ ) ≠ ∅ (see diagram 8 in
figure 2);
Q9 = {T , T ′, T ∪ T ′} , where T , T ′ ∈ D , T \ T′ ≠∅, T ′ \ T ≠ ∅ and T ∩ T ′ = ∅ (see
diagram 9 in figure 2);
Q10 = {T , T ′, T ∪ T ′, T ′′} , where T , T ′, T ′′ ∈ D , T ∪ T ′ ⊂ T ′′ , T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ and
T ∩ T ′ = ∅ (see diagram 10 in figure2);
⌣
Q11 = {Z 7 , Z 6 , Z 4 , Z , D} , where Z ∈ {Z 2 , Z1} and Z7 ∩Z6 =∅ (see diagram 11 in
figure 2);
⌣
Q12 = {Z 7 , Z 6 , Z 4 , Z 2 , Z1 , D} , where Z7 ∩Z6 =∅ (see diagram 12 in figure 2);
Q13 = {T , T ′, T ∪ T ′, T ′′, Z } ,
where T,T′,T′′, Z ∈D ,
(T ∪T ′) ⊂ Z , T ′ ⊂ T ′′ ⊂ Z ,
(T ∪T′) \ T ′′ ≠ ∅ , T ′′ \ (T ∪T′) ≠ ∅ and T ∩ T ′′ = ∅ (see diagram 13 in figure 2);
{
}
Z4 \ Z ≠∅ , Z \ Z4 ≠∅
15)
⌣
⌣
Q14 = T , T ′, Z 4 , Z , Z ′, D , where T , T ′, Z , Z ′ ∈ D ,
and
T ∩Z = ∅
⌣
Q15 = T ′, T , Z 4 , T ′′, Z , T ′′ ∪ Z 4 , D ,
{
T ′′ ∈ {Z 5 , Z 3 } ,
Z4 ⊂ Z ,
}
(T ∪ T ′) ⊂ Z ′ , T ′ ⊂ Z ⊂ Z ′ ⊂ D ,
(see diagram 14 in figure 2);
where T , T ′ ∈ {Z 7 , Z 6 } , T ≠ T ′ ,
⌣
Z ∪ T ′′ ∪ Z 4 = D ,
(T ′′ ∪ Z 4 ) \ Z ≠ ∅ ,
T ⊂ T ′′ ,
Z \ (T ′′ ∪ Z 4 ) ≠ ∅ ,
T′ \ Z4 ≠∅ , Z4 \ T′ ≠∅
16)
and T′ ∩T′ =∅ (see diagram 15 in figure 2);
⌣
Q16 = {Z 7 , Z 6 , Z5 , Z 4 , Z 3 , Z 2 , Z1 , D} , where Z5 ∩Z3 =∅ (see diagram 16 in figure
⌣
⌣
2).
⌣
T′
T
1
T ∪T′
T ′′ T ′
T′
T′
T
2
T
3
T
4
⌣
D
T ∪T′
Z Z
1
T T′ T T′
10
Z6 Z7
11
D
D
Z′
T ′′ Z
Z4
T
5
T ′′
Z4
9
T ′ ∪ T ′′
⌣
D
T ′′
Z6 Z7
12
T′
Z
T ′′
T T′
13
Fig.2
1.1.
Z4 ∪ T ′
Z4
T
8
⌣
D
Z ′ T ′′ ∪ Z 4
T ′′
Z
T T′
14
Z
Z4
T′
T
7
T
6
⌣
D
Z2 T ∪ T ′
Z4
T′ ∪T′
T ′′
D
⌣
D
Z
Z1
Z4 Z
3
T T′
15
⌣
D
Z2
Z 4 Z5
Z6 Z7
16
22
Nino Tsinaridze
Denote by the symbol ∑ ( Qi ) ( i = 1, 2,...,16 ) the set of all XI -subsemilattices of the
semilattice D isomorphic to Qi . Assume that D′ ∈ ∑ ( Qi ) and denote by the symbol
I ( D′) the set of all idempotent elements α of the semigroup BX ( D ′ ) , for which the
semilattices V ( D, α ) and Qi are mutually α isomorphic and V ( D, α ) = Qi .
Definition 2.1: Let the symbol ∑′XI ( X , D ) denote the set of all XI -subsemilattices
of the semilattice D .
Let, further, D, D′ ∈ ∑′ ( X , D ) and ϑXI ⊆ ∑′XI ( X , D ) × ∑′XI ( X , D ) . It is assumed that
DϑXI D′ if and only if there exists some complete isomorphism ϕ between the
semilattices D and D ′ . One can easily verify that the binary relation ϑXI is an
equivalence relation on the set ∑′XI ( X , D ) .
Let D ′ be an XI -subsemilattices of the semilattice . By I ( D′) we denoted the set
of all idempotent elements of the semigroup BX ( D′ ) and I ∗ ( Qi ) =
where i = 1, 2,...,16 .
Lemma 2.1: If D ∈ Σ 2 ( X ,8) , then the following equalities are true:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
I ( Q1 ) = 1 ;
( − 1) ⋅ 2 ;
I (Q ) = ( 2
− 1) ⋅ ( 3
−2
)⋅3 ;
I (Q ) = ( 2
− 1) ⋅ ( 3
−2
)⋅(4 − 3 )⋅ 4 ;
(
)
I (Q ) = ( 2
− 1) ⋅ ( 2
− 1) ⋅ 4
;
)
I (Q ) = ( 2
− 1) ⋅ 2 (
⋅ (3
−2
) ⋅ (3 − 2 ) ⋅ 5
(
)
(
)
I (Q ) = ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 5
−4
)⋅5 ;
(
)
(
)
I (Q ) = ( 2
− 1) ⋅ ( 2
− 1) ⋅ ( 3
−2
)⋅6 ;
I (Q 2 ) = 2
T ′ \T
X \T ′
T ′\T
T ′′ \ T ′
T ′′ \ T ′
X \ T ′′
3
T ′ \T
T ′′ \ T ′
⌣
D \ T ′′
T ′′ \ T ′
⌣
D \ T ′′
⌣
X \D
4
T ′ \ T ′′
X \ T ′ ∪ T ′′
T ′′ \ T ′
5
Z2 ∩ Z1 \ Z 4
Z4 \ T
Z \Z′
Z \Z′
Z ′\ Z
⌣
X \D
Z ′\ Z
6
T ′ \ T ′′
⌣
D \ T ′∪T ′′
T ′′ \ T ′
⌣
D \ T ′ ∪T ′′
⌣
X \D
7
T ′\ Z
Z \ Z 4 ∪T ′
Z 4 \T ′
Z \ Z 4 ∪T ′
8
I ( Q9 ) = 3
X \ (T ∪T ′ )
(
I ( Q10 ) = 4
(
) = (4
11)
I ( Q11 ) = 4
Z \ Z4
12)
I ( Q12
Z1 \ Z 2
13)
I ( Q13 ) = 2
14)
I ( Q14 ) = 2
(
(
;
T ′′ \ (T ∪ T ′ )
−3
−3
Z \ Z4
T ′′ \ (T ∪ T ′ )
) ⋅ (5
)⋅(4
⌣
D\Z
− 3 Z1 \ Z 2
T ′′ \ (T ′∪ T )
Z \ Z4
)⋅ 4
−4
Z 2 \ Z1
⌣
D\Z
X \ T ′′
)
⋅5
;
⌣
X \D
)
− 3 Z 2 \ Z1 ⋅ 6
)
;
⌣
X \D
;
− 1 ⋅ 5 X \Z ;
)(
−1 ⋅ 6
⌣
D \Z ′
−5
⌣
D \Z ′
)⋅6
⌣
X \D
;
⌣
X \D
;
∑
D ′∈QiϑXI
I ( D′ ) ,
Idempotent Elements of the Semigroups BX(D)...
15)
16)
(
I ( Q15 ) = 2 T
(
I ( Q16 ) = 2
)(
′′ \ Z
−1 ⋅ 4
Z 5 \ Z1
Z \ (T ′′ ∪ Z 4 )
)(
−1 ⋅ 2
−3
Z \ (T ′′ ∪ Z 4 )
)
−1 ⋅8
Z3 \ Z 2
⌣
X \D
23
)⋅ 7
⌣
X \D
;
(see [6] Lemma 3.3).
Theorem 2.1: Let D ∈ Σ2 ( X ,8) , Z7 ∩ Z6 = ∅, Z7 ∩ Z3 ≠ ∅, Z6 ∩ Z5 ≠ ∅ and α ∈ BX ( D ) .
The binary relation α is an idempotent relation of the semmigroup BX ( D ) iff
binary relation α satisfies only one condition of the following conditions:
1) α = X × T , where T ∈ D ;
2) α = (YTα ×T ) ∪ (YTα′ ×T ′) ,
YTα , YTα′ ∉ {∅}
T,T′∈D, T ⊂ T ′ ,
where
and
satisfies
the
conditions: YT ⊇ T , YT ′ ∩ T ′ ≠ ∅ ;
α
α
3) α = (YTα × T ) ∪ (YTα′ × T ′ ) ∪ (YTα′′ × T ′′ ) ,
where T,T′,T′′ ∈ D , T ⊂ T ′ ⊂ T ′′ , YTα , YTα′ , YTα′′ ∉ {∅} and
satisfies the conditions: YTα ⊇ T , YTα ∪ YTα′ ⊇ T ′ , YTα′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ ;
⌣
4) α = (YTα × T ) ∪ ( YTα′ × T ′ ) ∪ (YTα′′ × T ′′ ) ∪ Y0α × D
(
YTα , YTα′ , YTα′′ , Y0α ∉ {∅ }
YTα ∪ YTα′ ∪ YTα′′
) , where T , T ′, T ′′ ∈ D
YTα ⊇ T
and
satisfies the conditions:
⌣
α
⊇ T ′′ , YT ′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ , Y0α ∩ D ≠ ∅ ;
5) α = (YTα × T ) ∪ (YTα′ × T ′ ) ∪ (YTα′′ × T ′′ ) ∪ ( YTα′∪T ′′ × T ′ ∪ T ′′ ) ,
⌣
and T ⊂ T ′ ⊂ T ′′ ⊂ D ,
,
YTα ∪ YTα′ ⊇ T ′ ,
T , T ′, T ′′ ∈ D, T ⊂ T ′ ,
where
T ′′ \ T ′ ≠ ∅, Y , Y , Y ∉ {∅} and satisfies the conditions:
T ⊂ T ′′ , T′ \ T′′ ≠∅,
α
T
α
T′
α
T ′′
YTα ∪ YTα′ ⊇ T ′ , YTα ∪ YTα′′ ⊇ T ′′ , YTα′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ ;
⌣
6) α = (YTα × T ) ∪ (Y4α × Z 4 ) ∪ (YZα × Z ) ∪ (YZα′ × Z ′ ) ∪ Y0α × D
Z , Z ′ ∈ {Z 2 , Z1} , Z ≠ Z ′ ,
(
Z \ Z′ ≠ ∅, Z′\Z ≠ ∅,
),
Y ,Y4 ,Y ,Y ∉{∅}
and
α
α
T ∈ {Z 7 , Z 6 } ,
where
α
α
T
α
Z
α
Z′
satisfies the
conditions: YT ⊇ T , YT ∪ Y4 ⊇ Z 4 , YT ∪ Y4 ∪ YZ ⊇ Z , YT ∪ Y4 ∪ YZ ′ ⊇ Z ′ , Y4α ∩ Z 4 ≠ ∅ ,
YZα ∩ Z Z ≠ ∅ , YZα′ ∩ Z ′ ≠ ∅
α
α
α
α
α
α
α
⌣
7) α = (YTα × T ) ∪ (YTα′ × T ′ ) ∪ (YTα′′ × T ′′ ) ∪ (YTα′∪T ′′ × (T ′ ∪ T ′′ ) ) ∪ Y0α × D
(
),
where T , T ′, T ′′ ∈ D
and T ⊂ T ′ , T ⊂ T ′′ , T ′ \ T ′′ ≠ ∅ , T ′′ \ T ′ ≠ ∅ , YTα , YTα′ , YTα′′ , Y0α ∉ {∅} and satisfies the
⌣
conditions: YTα ∪ YTα′ ⊇ T ′ , YTα ∪ YTα′′ ⊇ T ′′ , YTα′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ , Y0α ∩ D ≠ ∅ ;
⌣
8) α = (YTα ×T ) ∪(YTα′ ×T ′) ∪(Y4α × Z4 ) ∪ YTα′∪Z4 × (T ′ ∪ Z4 ) ∪(YZα × Z ) ∪ Y0α × D , where T ∈{Z7 , Z6 } ,
T ′ ∈{Z5 , Z3 } ,
Z4 ∪T ′, Z ∈{Z2 , Z1} ,
( Z4 ∪ T ′) \ Z ≠ ∅ ,
(
)
Z4 ∪ T ′ ≠ Z ,
(
T ⊂T′ ,
)
T ′ \ Z4 ≠ ∅ ,
Z 4 \ T ′ ≠ ∅,
Z \ ( Z 4 ∪ T ′) ≠ ∅, YTα , YTα′ , Y4α , YZα ∉ {∅} and satisfies the conditions:
YTα ∪ YTα′ ⊇ T ′ , YTα ∪ Y4α ⊇ Z 4 , YTα ∪ Y4α ∪ YZα ⊇ Z , YTα′ ∩ T ′ ≠ ∅ , Y4α ∩ Z 4 ≠ ∅ , YZα ∩ Z ≠ ∅
;
24
Nino Tsinaridze
9 ) α = (YTα × T ) ∪ (YTα′ × T ′ ) ∪ (YTα∪T ′ × (T ∪ T ′ ) ) , where T,T′ ∈ D ,
T \T′ ≠ ∅
, T′\T ≠ ∅ ,
YTα , YTα′ ∉ {∅} and satisfies the conditions: YTα ⊇ T , YTα′ ⊇ T ′ ;
10 ) α = (YTα ×T ) ∪(YTα′ ×T ′) ∪(YTα∪T′ × (T ∪T ′) ) ∪(YTα′′ ×T ′′) ,
T ′ \ T ≠ ∅ , YT ,YT ′ ,YT ′′ ∉{∅}
α
α
α
T,T′,T′′ ∈D ,
where
T \T′ ≠ ∅
,
and satisfies the conditions: YT ⊇ T , YT ′ ⊇ T ′ , YT ′′ ∩ T ′′ ≠ ∅ ;
α
α
α
⌣
11) α = (Y7α × Z7 ) ∪(Y6α × Z6 ) ∪(Y4α × Z4 ) ∪(YZα × Z ) ∪ Y0α × D , where Z ∈{Z2 , Z1} , Y7α ,Y6α , YZα ,Y0α ∉{∅}
(
)
and satisfies the conditions:
⌣
Y7α ⊇ Z7 , Y6α ⊇ Z6 , Y7α ∪ Y6α ∪ Y4α ∪ YZα ⊇ Z , YZα ∩ Z ≠ ∅, Y0α ∩ D ≠ ∅;
⌣
12 ) α = (Y7α × Z7 ) ∪(Y6α × Z6 ) ∪(Y4α × Z4 ) ∪(Y2α × Z2 ) ∪(Y1α × Z1 ) ∪ Y0α × D ,
(
and satisfies the conditions:
Y2α ∩ Z 2 ≠ ∅ ,
)
where Y7α ,Y6α ,Y2α ,Y1α ∉{∅}
Y7α ⊇ Z 7 , Y6α ⊇ Z 6 , Y7α ∪ Y6α ∪ Y2α ⊇ Z 2 , Y7α ∪ Y6α ∪ Y1α ⊇ Z 1 ,
Y1α ∩ Z1 ≠ ∅.
Proof: In this case, when Z 7 ∩ Z 6 = ∅ , from the Lemma 2.3 in [6] it follows that
diagrams 1-12 given in fig.1 exhibit all diagrams of XI − subsemilattices of the
semilattices D , a quasinormal representation of idempotent elements of the
semigroup BX ( D ) , which are defined by these XI − semilattices, may have one of
the forms listed above. The statements 1)-4) immediately follows from the
Corollary 13.1.1 in [1], Corollary 13.1.1 in [2], the statements 5)-7) immediately
follows from the Corollary 13.3.1 in [1], Corollary 13.3.1 in [2] and the statement
8) immediately follows from the Theorems 13.7.2 in [1], Theorems 13.7.2 in [2],
The statements 9)-11) immediately follows from the Corollary 13.2.1 in [1],
13.2.1 in [2], the statement 12) immediately follows from the Corollary 13.5.1 in
[1], 13.5 in [2].
The Theorem is proved.
Lemma 2.2: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 ≠ ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q9 ) can be calculated by the formula
I ∗ ( Q9 ) = 3
X \ Z4
.
Proof: By definition of the given semilattice D we have Q9θ XI = {{Z 7 , Z 6 , Z 4 }} . If the
following equality is hold D1′ = {Z7 , Z6 , Z4 } , then I ∗ ( Q9 ) = I ( D1′ ) . From this equality
and statement (9) of Lemma 2.1 we obtain validity of Lemma 2.2.
The Lemma is proved.
Lemma 2.3: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 ≠ ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q10 ) can be calculated by the formula
Idempotent Elements of the Semigroups BX(D)...
(
I ∗ ( Q10 ) = 4
Z2 \ Z4
−3
Z2 \ Z 4
)⋅4
X \ Z2
(
+ 4
−3
Z1 \ Z 4
Z1 \ Z 4
)⋅4
25
X \ Z1
(
+ 4
⌣
D \ Z4
−3
⌣
D \ Z4
)
⋅4
⌣
X \D
.
Proof: By definition of the given semilattice D we have
{
}} .
⌣
Q10θXI = {Z7 , Z6 , Z4 , Z2} ,{Z7 , Z6 , Z4 , Z1} , Z7 , Z6 , Z4 , D
{
If the following equalities are hold
⌣
D1′ = {Z7 , Z6 , Z4 , Z2 } , D2′ = {Z7 , Z6 , Z4 , Z1} , D3′ = Z7 , Z6 , Z4 , D ,
{
}
then
I ∗ ( Q10 ) = I ( D1′ ) + I ( D2′ ) + I ( D3′ )
From this equality and statement (10) of Lemma 2.1 we obtain validity of Lemma
2.3.
The Lemma is proved.
Lemma 2.4: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 ≠ ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q11 ) can be calculated by the formula
(
I ∗ ( Q11 ) = 4
Z2 \ Z4
−3
Z2 \ Z4
) ⋅ (5
⌣
D \ Z2
−4
⌣
D \ Z2
)⋅5
⌣
X \D
(
+ 4
Z1 \ Z 4
−3
Z1 \ Z 4
) ⋅ (5
⌣
D \ Z1
−4
⌣
D \ Z1
)⋅5
⌣
X \D
.
Proof: By definition of the given semilattice D we have
Q11θ XI =
{{Z , Z , Z , Z , D} ,{Z , Z , Z , Z , D}} .
⌣
7
6
4
⌣
2
7
6
4
1
⌣
⌣
If the following equalities are hold D1′ = {Z7 , Z6 , Z4 , Z2 , D} , D2′ = {Z7 , Z6 , Z4 , Z1 , D} , then
I ∗ ( Q11 ) = I ( D1′ ) + I ( D2′ )
From this equality and statement (11) of Lemma 2.1 we obtain validity of Lemma
2.4.
The Lemma is proved.
Lemma 2.5: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 ≠ ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q12 ) can be calculated by the formula
I ∗ ( Q12 ) = 3 (
Z 2 ∩ Z1 ) \ Z 4
(
)(
)
⋅ 4 Z1 \ Z 2 − 3 Z1 \ Z 2 ⋅ 4 Z 2 \ Z1 − 3 Z 2 \ Z1 ⋅ 6
⌣
X \D
.
Proof: By definition of the given semilattice D we have
26
Nino Tsinaridze
Q12θ XI =
{{Z , Z , Z , Z , Z , D}} .
⌣
7
6
4
2
1
⌣
If the following equality is hold D1′ = {Z7 , Z6 , Z4 , Z2 , Z1 , D} , then I ∗ ( Q12 ) = I ( D1′ )
From this equality and statement (12) of Lemma 2.1 we obtain validity of Lemma
2.5.
The Lemma is proved.
8
It was seen in [6] that k1 = ∑ I ∗ ( Qi ) . Now, let us assume that
i =1
k2 = I ∗ ( Q9 ) + I ∗ ( Q10 ) + I ∗ ( Q11 ) + I ∗ ( Q12 ) =
=3
(
X \ Z4
+ 4
+3
(
+ 4
Z2 \ Z4
Z2 \ Z4
−3
( Z 2 ∩ Z1 ) \ Z 4
Z2 \ Z4
(
⋅ 4
−3
Z2 \ Z4
) ⋅ (5
Z1 \ Z 2
)⋅4
⌣
D \ Z2
−3
−4
Z1 \ Z 2
⌣
D \ Z2
)⋅(4
(
+ 4
X \ Z2
)⋅5
Z 2 \ Z1
Z1 \ Z 4
−3
⌣
X \D
(
−3
+ 4
Z 2 \ Z1
Z1 \ Z 4
Z1 \ Z 4
)⋅6
)⋅4
X \ Z1
−3
Z1 \ Z 4
⌣
X \D
(
) ⋅ (5
+ 4
⌣
D \ Z4
⌣
D \ Z1
−3
−4
⌣
D \ Z4
⌣
D \ Z1
)⋅4
)⋅5
⌣
X \D
⌣
X \D
+
+
Theorem 2.2: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 ≠ ∅, Z6 ∩ Z5 ≠ ∅ . If X is a
finite set and I D is the set of all idempotent elements of the semigroup BX ( D ) , then
ID = k1 + k2 .
Proof: This Theorem immediately follows from the Theorem 2.1.
The Theorem is proved.
Theorem 3.1: Let D ∈ Σ2 ( X ,8) , Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 ≠ ∅ and α ∈ BX ( D ) .
The binary relation α is an idempotent relation of the semmigroup BX ( D ) iff
binary relation α satisfies only one condition of the Theorem 2.1 and only one
following conditions:
13) α = (Y7α × Z 7 ) ∪ (Y6α × Z 6 ) ∪ (Y4α × Z 4 ) ∪ (Y3 × Z 3 ) ∪ (Y1α × Z1 ) , where
Y7α , Y6α , Y3α ∉ {∅}
and satisfies the conditions: Y7α ⊇ Z 7 , Y6α ⊇ Z 6 , Y6α ∪ Y3α ⊇ Z 3 , Y3α ∩ Z 3 ≠ ∅;
⌣
14) α = (Y7α × Z7 ) ∪ (Y6α × Z6 ) ∪ (Y4α × Z4 ) ∪ (Y3α × Z3 ) ∪ (Y1α × Z1 ) ∪ (Y0α × D) , where
Y7α , Y6α , Y3α , Y0α ∉ {∅} and satisfies the conditions:
⌣
Y7α ⊇ Z 7 , Y6α ⊇ Z 6 , Y6α ∪ Y3α ⊇ Z3 , Y3α ∩ Z 3 ≠ ∅, Y0α ∩ D ≠ ∅;
⌣
15) α = (Y7α × Z7 ) ∪(Y6α × Z6 ) ∪(Y4α × Z4 ) ∪(Y3α × Z3 ) ∪(Y2α × Z2 ) ∪(Y1α × Z1 ) ∪(Y0α × D) , where
Y7α ,Y6α ,Y3α ,Y2α ∉{∅} and satisfies the conditions:
Idempotent Elements of the Semigroups BX(D)...
27
Y7α ⊇ Z 7 , Y6α ⊇ Z 6 , Y6α ∪ Y3α ⊇ Z 3 , Y6α ∪ Y3α ∪ Y2α ⊇ Z 2 , Y3α ∩ Z 3 ≠ ∅, Y2α ∩ Z 2 ≠ ∅;
Proof: In this case, when Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 ≠ ∅ , from the Lemma 2.4 in
[6] it follows that diagrams 1-15 given in fig.1 exhibit all diagrams of XI −
subsemilattices of the semilattices D , a quasinormal representation of idempotent
elements of the semigroup BX ( D ) , which are defined by these XI − semilattices,
may have one of the forms listed above. The statements 13) , 14 ) immediately
follows from the Corollary 13.4.1 in [1], 13.4.1 in [2], the statement 15)
immediately follows from the Theorem 13.10.2 in [1].
The Theorem is proved.
Lemma 3.1: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q9 ) can be calculated by the formula
I ∗ ( Q9 ) = 3
X \ Z1
+3
X \ Z4
.
Proof: By definition of the given semilattice D we have
Q9θ XI = {{Z 7 , Z 6 , Z 4 } , {Z 7 , Z 3 , Z1 }} .
equalities are hold D1′ = {Z7 , Z6 , Z4 } , D2′ = {Z7 , Z3 , Z1} , then
I ∗ ( Q9 ) = I ( D1′ ) + I ( D2′ ) . From this equality and statement (9) of Lemma 2.1 we
obtain validity of Lemma 3.1.
If
the
following
The Lemma is proved.
Lemma 3.2: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q10 ) can be calculated by the formula
(
I ∗ ( Q10 ) = 4
(
+ 4
⌣
D \ Z1
Z1 \ Z 4
−3
−3
⌣
D \ Z1
Z1 \ Z 4
)⋅4
)⋅4
⌣
X \D
(
+ 4
Z2 \ Z4
(
+ 4
X \ Z1
⌣
D \ Z4
−3
Z2 \ Z4
−3
⌣
D \ Z4
)⋅4
)⋅4
X \ Z2
+
⌣
X \D
Proof: By definition of the given semilattice D we have
Q10θ XI =
{{Z , Z , Z , Z } ,{Z , Z , Z , Z } ,{Z , Z , Z , D} ,{Z , Z , Z , D}} .
⌣
7
6
4
2
7
6
4
1
7
6
4
⌣
7
3
1
If the following equalities are hold
⌣
⌣
D1′ = {Z7 , Z6 , Z4 , Z2 } , D2′ = {Z7 , Z6 , Z4 , Z1} , D3′ = Z7 , Z6 , Z4 , D , D4′ = Z7 , Z3 , Z1, D ,
{
}
{
}
28
Nino Tsinaridze
Then I ∗ ( Q10 ) = I ( D1′ ) + I ( D2′ ) + I ( D3′ ) + I ( D4′ ) . From this equality and statement
(10) of Lemma 2.1 we obtain validity of Lemma 3.2.
The Lemma is proved.
Lemma 3.3: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q13 ) can be calculated by the formula
(
I ∗ ( Q13 ) = 2
Z3 \ Z4
)
−1 ⋅ 5
X \ Z1
.
Proof: By definition of the given semilattice D we have Q13θ XI = {{Z 7 , Z 6 , Z 4 , Z 3 , Z1}} .
If the following equality is hold D1′ = {Z7 , Z6 , Z4 , Z3 , Z1} , then I ∗ ( Q13 ) = I ( D1′ ) . From
this equality and statement (13) of Lemma 2.1 we obtain validity of Lemma 3.3.
The Lemma is proved.
Lemma 3.4: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q14 ) can be calculated by the formula
(
I ∗ ( Q14 ) = 2
Proof:
{{
By
Z3 \ Z 4
)(
−1 ⋅ 6
⌣
D \ Z1
definition
−5
of
}}
⌣
D \ Z1
)⋅6
⌣
X \D
the
.
given semilattice
D
following
equality
⌣
Q14θ XI = Z 7 , Z 6 , Z 4 , Z 3 , Z1 , D .
If
the
⌣
D1′ = {Z7 , Z6 , Z4 , Z3 , Z1 , D} , then I ∗ ( Q14 ) = I ( D1′ )
we
is
have
hold
. From this equality and statement (14)
of Lemma 2.1 we obtain validity of Lemma 3.4.
The Lemma is proved.
Lemma 3.5: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q15 ) can be calculated by the formula
(
)(
)
I ∗ ( Q15 ) = 2 Z3 \ Z 2 − 1 ⋅ 4 Z 2 \ Z1 − 3 Z 2 \ Z1 ⋅ 7
⌣
X \D
.
Proof: By definition of the given semilattice D we have
Q15θ XI =
{{Z , Z , Z , Z , Z , Z , D}} .
⌣
7
6
4
3
2
1
⌣
If the following equality is hold D1′ = {Z7 , Z6 , Z4 , Z3 , Z2 , Z1 , D} , then I ∗ ( Q15 ) = I ( D1′ ) .
From this equality and statement (15) of Lemma 2.1 we obtain validity of Lemma
3.5.
Idempotent Elements of the Semigroups BX(D)...
29
The Lemma is proved.
Let us assume that
k3 = I ∗ ( Q9 ) + I ∗ ( Q10 ) + I ∗ ( Q11 ) + I ∗ ( Q12 ) + I ∗ ( Q13 ) + I ∗ ( Q14 ) + I ∗ ( Q15 ) =
=3
X \ Z1
(
+ (4
+ (2
+ 4
+3
Z1 \ Z 4
Z1 \ Z 4
Z3 \ Z 4
X \ Z4
−3
−3
(
+ 4
Z1 \ Z 4
Z1 \ Z 4
)
−1 ⋅ 5
⌣
D \ Z1
)⋅4
) ⋅ (5
X \ Z1
−3
⌣
D \ Z1
(
(
+ 4
X \ Z1
+ 2
⌣
D \ Z1
−4
Z3 \ Z 4
)
⌣
D \ Z4
⋅4
⌣
X \D
−3
)⋅5
− 1) ⋅ ( 6
⌣
D \ Z1
(
+ 4
⌣
D \ Z4
⌣
X \D
⌣
D \ Z1
)
Z2 \ Z4
⋅4
+3
⌣
X \D
−5
)⋅4
Z2 \ Z4
+
X \ Z2
) ⋅ (5
)
⋅(4
−3
)⋅(4
) ⋅ 6 + ( 2 − 1) ⋅ ( 4
( Z 2 ∩ Z1
⌣
D \ Z1
−3
(
+ 4
Z 2 \ Z4
\ Z4
−3
Z2 \ Z4
Z1 \ Z 2
⌣
X \D
⌣
D \ Z2
Z1 \ Z 2
Z3 \ Z 2
−4
Z 2 \ Z1
Z 2 \ Z1
⌣
D \ Z2
−3
−3
)
⋅5
Z 2 \ Z1
Z 2 \ Z1
⌣
X \D
)⋅6
)⋅7
+
⌣
X \D
⌣
X \D
Theorem 3.2: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 ≠ ∅ . If X is a
finite set and I D is the set of all idempotent elements of the semigroup BX ( D ) , then
ID = k1 + k3 .
Proof: This Theorem immediately follows from the Theorem 3.1.
The Theorem is proved.
Theorem 4.1: Let D ∈ Σ2 ( X ,8) , Z7 ∩ Z6 = ∅, Z6 ∩ Z5 = ∅, Z7 ∩ Z3 ≠ ∅ and α ∈ BX ( D ) .
The binary relation α is an idempotent relation of the semmigroup BX ( D ) iff
binary relation α satisfies only one condition of the Theorem 2.1 and only one
following conditions:
13) α = (Y7α × Z 7 ) ∪ (Y6α × Z 6 ) ∪ (Y5α × Z 5 ) ∪ (Y4α × Z 4 ) ∪ (Y2α × Z 2 ) ,
where Y7α , Y6α , Y5α ∉ {∅}
and satisfies the conditions: Y7α ⊇ Z7 , Y6α ⊇ Z6 , Y7α ∪ Y5α ⊇ Z5 , Y5α ∩ Z5 ≠ ∅;
⌣
14) α = (Y7α × Z7 ) ∪ (Y6α × Z6 ) ∪ (Y5α × Z5 ) ∪ (Y4α × Z4 ) ∪ (Y2α × Z2 ) ∪ (Y0α × D) ,
where Y7α , Y6α , Y5α , Y0α ∉ {∅} and satisfies the conditions:
⌣
Y7α ⊇ Z 7 , Y6α ⊇ Z 6 , Y7α ∪ Y5α ⊇ Z5 , Y5 ∩ Z5 ≠ ∅, Y0α ∩ D ≠ ∅;
⌣
15) α = (Y7α ×Z7 ) ∪(Y6α ×Z6 ) ∪(Y5α ×Z5 ) ∪(Y4α ×Z4 ) ∪(Y2α ×Z2 ) ∪(Y1α ×Z1 ) ∪(Y0α ×D) , where
Y7α , Y6α , Y5α , Y1α ∉ {∅} and satisfies the conditions:
Y7α ⊇ Z 7 , Y6α ⊇ Z 6 , Y7α ∪ Y5α ⊇ Z5 , Y7α ∪ Y6α ∪ Y1α ⊇ Z1 , Y5α ∩ Z5 ≠ ∅, Y1α ∩ Z1 ≠ ∅.
Proof: In this case, when Z7 ∩ Z6 = ∅, Z6 ∩ Z5 = ∅, Z7 ∩ Z3 ≠ ∅ , from the Lemma 2.5 in
[6] it follows that diagrams 1-15 given in fig.1 exhibit all diagrams of XI −
subsemilattices of the semilattices D, a quasinormal representation of idempotent
+
30
Nino Tsinaridze
elements of the semigroup BX ( D ) , which are defined by these XI − semilattices,
may have one of the forms listed above. The statements 13) , 14 ) immediately
follows from the Corollary 13.4.1 in [1], 13.4.1 in [2], the statement 15)
immediately follows from the Theorem 13.10.2 in [1].
The Theorem is proved.
Lemma 4.1: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z6 ∩ Z5 = ∅, Z7 ∩ Z3 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q9 ) can be calculated by the formula
I ∗ ( Q9 ) = 3
Proof:
X \ Z2
By
+3
X \ Z4
.
definition
D
of the given semilattice
If the following equalities
Q9θ XI = {{Z 7 , Z 6 , Z 4 } , {Z 6 , Z 5 , Z 2 }} .
we
are
have
hold
D1′ = {Z7 , Z6 , Z4 } , D2′ = {Z6 , Z5 , Z2 } , then I ∗ ( Q9 ) = I ( D1′ ) + I ( D2′ ) . From this equality and
statement (9) of Lemma 2.1 we obtain validity of Lemma 4.1.
The Lemma is proved.
Lemma 4.2: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z6 ∩ Z5 = ∅, Z7 ∩ Z3 ≠ ∅ . If X is a
finite set, then the number I ∗ ( Q10 ) can be calculated by the formula
(
I ∗ ( Q10 ) = 4
(
+ 4
⌣
D \ Z2
Z1 \ Z 4
−3
−3
⌣
D \ Z2
Z1 \ Z 4
)⋅4
(
⌣
X \D
)⋅4
+ 4
(
Z2 \ Z4
+ 4
X \ Z1
⌣
D \ Z4
−3
−3
Z2 \ Z4
)⋅4
)⋅4
⌣
D \ Z4
X \ Z2
⌣
X \D
+
.
Proof: By definition of the given semilattice D we have
Q10θ XI =
{{Z , Z , Z , Z } ,{Z , Z , Z , Z } ,{Z , Z , Z , D} ,{Z , Z , Z , D}} .
⌣
7
6
4
2
7
6
4
1
7
6
4
⌣
6
5
2
If the following equalities are hold
⌣
⌣
D1′ = {Z7 , Z6 , Z4 , Z2} , D2′ = {Z7 , Z6 , Z4 , Z1} , D3′ = Z7 , Z6 , Z4 , D , D4′ = Z6 , Z5 , Z2 , D
{
}
{
},
then I ∗ ( Q10 ) = I ( D1′ ) + I ( D2′ ) + I ( D3′ ) + I ( D4′ ) . From this equality and statement (10)
of Lemma 2.1 we obtain validity of Lemma 4.2.
The Lemma is proved.
Lemma 4.3: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z6 ∩ Z5 = ∅, Z7 ∩ Z3 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q13 ) can be calculated by the formula
Idempotent Elements of the Semigroups BX(D)...
(
I ∗ ( Q13 ) = 2
)
−1 ⋅ 5
Z5 \ Z 4
X \ Z2
31
.
Proof: By definition of the given semilattice D we have Q13θ XI = {{Z 7 , Z 6 , Z 5 , Z 4 , Z 2 }} .
If the following equality is hold D1′ = {Z7 , Z6 , Z5 , Z4 , Z2 } , then I ∗ ( Q13 ) = I ( D1′ ) . From
this equality and statement (13) of Lemma 2.1 we obtain validity of Lemma 4.3.
The Lemma is proved.
Lemma 4.4: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z6 ∩ Z5 = ∅, Z7 ∩ Z3 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q14 ) can be calculated by the formula
(
I ∗ ( Q14 ) = 2
Proof:
{{
By
)(
−1 ⋅ 6
Z5 \ Z 4
⌣
D \ Z2
definition
−5
⌣
D \ Z2
of
}}
)⋅6
⌣
X \D
the
.
given semilattice
D
following
equality
we
is
⌣
Q14θ XI = Z 7 , Z 6 , Z 5 , Z 4 , Z 2 , D .
If
the
⌣
D1′ = {Z7 , Z6 , Z5 , Z4 , Z2 , D} , then I ∗ ( Q14 ) = I ( D1′ )
have
hold
. From this equality and statement (14)
of Lemma 2.1 we obtain validity of Lemma 4.4.
The Lemma is proved.
Lemma 4.5: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z6 ∩ Z5 = ∅, Z7 ∩ Z3 ≠ ∅ . If X is a finite
set, then the number I ∗ ( Q15 ) can be calculated by the formula
(
)(
)
I ∗ ( Q15 ) = 2 Z5 \ Z1 − 1 ⋅ 4 Z1 \ Z 2 − 3 Z1 \ Z 2 ⋅ 7
Proof:
{{
By
definition
of
⌣
X \D
the
}}
.
given semilattice
D
following
equality
we
is
⌣
Q15θ XI = Z 7 , Z 6 , Z 5 , Z 4 , Z 2 , Z1 , D .
If
the
⌣
D1′ = {Z7 , Z6 , Z5 , Z4 , Z2 , Z1 , D} , then I ∗ ( Q15 ) = I ( D1′ )
have
hold
. From this equality and statement
(15) of Lemma 2.1 we obtain validity of Lemma 4.5.
The Lemma is proved.
Let us assume that
k4 = I ∗ ( Q9 ) + I ∗ ( Q10 ) + I ∗ ( Q11 ) + I ∗ ( Q12 ) + I ∗ ( Q13 ) + I ∗ ( Q14 ) + I ∗ ( Q15 ) =
=3
X \ Z2
(
+ (4
+ (2
+ 4
Z1 \ Z 4
Z1 \ Z 4
Z5 \ Z 4
+3
(
+ 4
X \ Z4
−3
−3
Z1 \ Z 4
Z1 \ Z 4
)
−1 ⋅ 5
⌣
D \ Z2
)⋅4
) ⋅ (5
X \ Z2
X \ Z1
⌣
D \ Z1
(
+ 2
−3
⌣
D \ Z2
(
+ 4
−4
Z5 \ Z 4
)⋅4
⌣
D \ Z4
⌣
D \ Z1
⌣
X \D
−3
)
⋅5
)(
−1 ⋅ 6
(
+ 4
⌣
D \ Z4
⌣
X \D
⌣
D \ Z2
)
Z2 \ Z 4
⋅4
+3
⌣
X \D
−3
(
+ 4
( Z 2 ∩ Z1 ) \ Z 4
−5
⌣
D \ Z2
)⋅6
)⋅4
Z2 \ Z4
Z 2 \ Z4
(
⋅ 4
⌣
X \D
X \ Z2
−3
Z1 \ Z 2
(
+ 2
Z2 \ Z4
−3
+
) ⋅ (5
)⋅(4
− 1) ⋅ ( 4
Z1 \ Z 2
Z5 \ Z1
⌣
D \ Z2
−4
Z 2 \ Z1
Z1 \ Z 2
⌣
D \ Z2
−3
−3
)
⋅5
Z 2 \ Z1
Z1 \ Z 2
⌣
X \D
)⋅6
)⋅7
+
⌣
X \D
⌣
X \D
+
32
Nino Tsinaridze
Theorem 4.2: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z6 ∩ Z5 = ∅, Z7 ∩ Z3 ≠ ∅ . If X is a
finite set and I D is the set of all idempotent elements of the semigroup BX ( D ) , then
ID = k1 + k4 .
Proof: This Theorem immediately follows from the Theorem 4.1.
The Theorem is proved.
Theorem 5.1: Let D ∈ Σ2 ( X ,8) , Z7 ∩ Z6 = ∅, Z 6 ∩ Z5 = ∅, Z7 ∩ Z3 = ∅, Z5 ∩ Z3 ≠ ∅ and
α ∈ BX ( D) . The binary relation α is an idempotent relation of the semmigroup
BX ( D ) iff binary relation α satisfies only one conditions of the Theorem 3.1 and
only one conditions of the Theorem 4.1.
Proof: In this case, when Z7 ∩ Z6 = ∅, Z 6 ∩ Z5 = ∅, Z7 ∩ Z3 = ∅, Z5 ∩ Z3 ≠ ∅ , from the
Lemma 2.6 in [6] it follows that diagrams 1-15 given in fig.1 exhibit all diagrams
of XI − subsemilattices of the semilattices D , a quasinormal representation of
idempotent elements of the semigroup BX ( D ) , which are defined by these XI −
semilattices, may have one of the forms listed above. This Theorem immediately
follows from the Theorems 3.1 and 4.1.
The Theorem is proved.
Let us assume that
k5 = I ∗ ( Q9 ) + I ∗ ( Q10 ) + I ∗ ( Q11 ) + I ∗ ( Q12 ) + I ∗ ( Q13 ) + I ∗ ( Q14 ) + I ∗ ( Q15 ) =
=3
X \ Z1
(
+ (4
+ (2
+ (2
+ 4
+3
Z1 \ Z4
X \ Z2
−3
Z1 \ Z4
Z1 \ Z4
−3
Z5 \ Z4
−1 ⋅ 5
Z3 \ Z2
+3
Z1 \ Z4
)
−1) ⋅ ( 4
X \ Z4
)⋅4
) ⋅ (5
X \ Z1
X \ Z2
(
+ 4
⌣
D \ Z1
(
⌣
D \ Z1
(
+ 4
−4
−3
⌣
D \ Z4
⌣
D \ Z1
⌣
D \ Z1
−3
) ⋅5
)
+ 2
Z3 \ Z4
−1 ⋅ 5
−3
Z2 \ Z1
) ⋅7
Z2 \ Z1
⌣
X \D
⌣
D \ Z4
⌣
X \D
X \ Z1
⌣
X \D
) ⋅4 + (4
) ⋅4 + (4
(
⌣
D \ Z2
⌣
X \D
+3
(
( Z2 ∩Z1 ) \ Z4
+ 2
+ 2
Z5 \ Z4
Z5 \ Z1
−3
Z2 \ Z4
(
⋅ 4
)(
)(
−3
Z1 \ Z2
−1 ⋅ 6
−1 ⋅ 4
⌣
D \ Z2
⌣
D \ Z2
Z1 \ Z2
)⋅4
Z2 \ Z4
−3
) ⋅ (5
) ⋅(4
⌣
D \ Z2
Z1 \ Z2
(
+ 4
⌣
D \ Z2
Z1 \ Z2
−5
−3
⌣
X \D
)⋅6
) ⋅7
−4
Z2 \ Z1
⌣
X \D
⌣
X \D
Z2 \ Z4
−3
⌣
D \ Z2
−3
(
+ 2
Z2 \ Z4
)⋅5
Z2 \ Z1
Z3 \ Z4
)⋅4
⌣
X \D
X \ Z2
+
+
) ⋅6 +
−1) ⋅ ( 6
⌣
X \D
⌣
D \ Z1
−5
⌣
D \ Z1
) ⋅6
Theorem 5.2: Let D ∈ Σ2 ( X ,8) and Z7 ∩ Z6 = ∅, Z7 ∩ Z3 = ∅, Z6 ∩ Z5 = ∅, Z5 ∩ Z3 ≠ ∅ . If
X is a finite set and I D is the set of all idempotent elements of the semigroup
BX ( D ) , then ID = k1 + k5 .
Proof: This Theorem immediately follows from the Theorem 5.1.
The Theorem is proved.
⌣
X \D
+
Idempotent Elements of the Semigroups BX(D)...
33
Theorem 6.1: Let D ∈ Σ2 ( X ,8) , Z5 ∩ Z3 = ∅ and α ∈ BX ( D ) . The Binary relation α
is an idempotent relation of the semmigroup BX ( D ) iff binary relation α satisfies
only one conditions of the Theorem 5.1 and only one following condition:
13) α = (YTα ×T ) ∪(YTα′ ×T ′) ∪(YTα∪T′ ×(T ∪T′) ) ∪(YTα′′ ×T ′′) ∪(YZ × Z ) , where
T , T ′, T ′′, Z ∈ D, (T ∪ T ′ ) ⊂ Z , T ′ ⊂ T ′′ ⊂ Z , (T ∪T′) \ T′ ≠∅, T′ \ (T ∪T′) ≠∅, T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅
, YTα , YTα′ , YTα′′ ∉ {∅} and satisfies the conditions: YTα ⊇ T , YTα′ ⊇ T ′ , YTα′ ∪ YTα′′ ⊇ T ′′ ,
YTα′′ ∩ T ′′ ≠ ∅ ;
⌣
16) α = (Y7α × Z7 ) ∪(Y6α × Z6 ) ∪(Y5α × Z5 ) ∪(Y4α × Z4 ) ∪(Y3α × Z3 ) ∪(Y2α × Z2 ) ∪(Y1α × Z1 ) ∪(Y0α × D) ,
where Y7α ,Y6α ,Y5α , Y3α ∉{∅} and satisfies the conditions:
Y7α ⊇ Z 7 , Y6α ⊇ Z 6 , Y7α ∪ Y5α ⊇ Z 5 , Y6α ∪ Y3α ⊇ Z3 , Y5α ∩ Z5 ≠ ∅ , Y3α ∩ Z3 ≠ ∅ .
Proof: In this case, when Z5 ∩ Z3 = ∅ , from the Lemma 2.7 in [6] it follows that
diagrams 1-16 given in fig.1 exhibit all diagrams of XI − subsemilattices of the
semilattices D , a quasinormal representation of idempotent elements of the
semigroup BX ( D ) , which are defined by these XI − semilattices, may have one of
the forms listed above. The statement 13) immediately follows from the Corollary
13.4.1 in [1], 13.4.1 in [2]. Now we will proof the statement 16). It is easy to see,
⌣
that the set D (α ) = {Z7 , Z6 , Z5 , Z4 , Z3 , Z2 , Z1, D} is a generating set of the semilattice D .
Then the following equalities are hold:
ɺɺ (α ) = {Z } , D
ɺɺ (α ) = {Z } , D
ɺɺ (α ) = {Z , Z } , D
ɺɺ (α ) = {Z , Z , Z } ,
D
7
6
7
5
7
6
4
Z7
Z6
Z5
Z4
ɺɺ
ɺɺ
ɺɺ
D (α )Z = {Z6 , Z3} , D (α )Z = {Z7 , Z6 , Z5 , Z4 , Z2 } , D (α )Z = {Z7 , Z6 , Z4 , Z3 , Z1} .
3
2
1
By statement b) of the Theorem 1.2 follows that the following conditions are true:
Y7α ⊇ Z7 , Y6α ⊇ Z6 , Y7α ∪ Y5α ⊇ Z5 , Y7α ∪ Y6α ∪ Y4α ⊇ Z 4 , Y6α ∪ Y3α ⊇ Z3 ,
Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y2α ⊇ Z 2 , Y7α ∪ Y6α ∪ Y4α ∪ Y3α ∪ Y1α ⊇ Z1 ;
Y7α ∪ Y6α ∪ Y4α ⊇ Z7 ∪ Z6 ∪ Y4α = Z 4 ∪ Y4α ⊇ Z 4 ,
Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y2α = (Y7α ∪ Y5α ) ∪ (Y7α ∪ Y6α ∪ Y4α ) ∪ Y2α ⊇
⊇ Z5 ∪ Z 4 ∪ Y2α = Z 2 ∪ Y2α ⊇ Z 2 ,
Y7α ∪ Y6α ∪ Y4α ∪ Y3α ∪ Y1α = (Y7α ∪ Y6α ∪ Y4α ) ∪ (Y6α ∪ Y3α ) ∪ Y1α ⊇
⊇ Z 4 ∪ Z3 ∪ Y1α = Z1 ∪ Y1α ⊇ Z1 ,
i.e., the inclusions
Y7α ∪ Y6α ∪ Y4α ⊇ Z 4 , Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y2α ⊇ Z 2 , Y7α ∪ Y6α ∪ Y4α ∪ Y3α ∪ Y1α ⊇ Z 1
hold. Further, it is to see, that the following conditions are true:
are always
34
Nino Tsinaridze
(
(
(
ɺɺ
l (D
ɺɺ
l (D
ɺɺ
l (D
ɺɺ
l (D
) ( Dɺɺ
) ( Dɺɺ
) ( Dɺɺ
ɺɺ
, Z ) = ∪(D
ɺɺ
, Z ) = ∪(D
ɺɺ
, Z ) = ∪(D
ɺɺ
, Z ) = ∪( D
ɺɺ , Z = ∪
l D
Z7
7
ɺɺ , Z = ∪
l D
Z6
6
ɺɺ , Z = ∪
l D
Z5
Z3
Z4
Z2
Z1
Z7
Z6
5
Z5
3
Z3
4
Z4
2
Z2
1
Z1
)
(
)
)
(
)
)
(
)
ɺɺ
\ { Z } ) = Z , Z \ l ( D , Z ) = Z \ Z ≠ ∅;
ɺɺ , Z ) = Z \ Z = ∅;
\ {Z } ) = Z , Z \ l ( D
ɺɺ , Z ) = Z \ Z = ∅;
\ {Z } ) = Z , Z \ l ( D
ɺɺ , Z ) = Z \ Z = ∅.
\ {Z } ) = Z , Z \ l ( D
ɺɺ , Z = Z \ ∅ ≠ ∅;
\ { Z 7 } = ∅, Z 7 \ l D
Z7
7
7
ɺɺ , Z = Z \ ∅ ≠ ∅;
\ { Z 6 } = ∅, Z 6 \ l D
Z6
6
6
ɺɺ , Z = Z \ Z ≠ ∅;
\ {Z } = Z , Z \ l D
5
7
5
Z5
5
5
7
3
6
3
Z3
3
3
6
4
4
4
Z4
4
4
4
2
2
2
Z2
2
2
2
1
1
1
Z1
1
1
1
We have the elements Z 7 , Z 6 , Z5 , Z3 are nonlimiting elements of the sets Dɺɺ (α )Z ,
7
ɺɺ (α ) , D
ɺɺ (α ) and D
ɺɺ (α ) respectively. By statement c ) of the Theorem 1.2 it
D
Z
Z
Z
6
5
3
follows, that the conditions Y7α ∩ Z7 ≠ ∅ , Y6α ∩ Z6 ≠ ∅, Y5α ∩ Z5 ≠ ∅ and Y3α ∩ Z3 ≠ ∅ are
hold. Since Z 7 ⊂ Z 5 , Z6 ⊂ Z3 , we have Y5α ∩ Z5 ≠ ∅ and Y3α ∩ Z3 ≠ ∅ . Therefore the
following conditions are hold:
Y7α ⊇ Z 7 , Y6α ⊇ Z 6 , Y7α ∪ Y5α ⊇ Z 5 , Y6α ∪ Y3α ⊇ Z 3 ,
Y5α ∩ Z 5 ≠ ∅, Y3α ∩ Z 3 ≠ ∅.
The Theorem is proved.
Lemma 6.1: Let D ∈ Σ2 ( X ,8) and Z5 ∩ Z3 = ∅. If X is a finite set, then the number
I ∗ ( Q9 ) can be calculated by the formula
I ∗ ( Q9 ) = 3
+3
X \ Z4
X \ Z2
+3
X \ Z1
+3
⌣
X \D
.
Proof: By definition of the given semilattice D we have
Q9θ XI =
{{Z , Z , Z } ,{Z , Z , Z } ,{Z , Z , Z } ,{Z , Z , D}} .
⌣
7
6
4
6
5
2
7
3
1
5
3
If the following equalities are hold
⌣
D1′ = {Z7 , Z6 , Z4 } , D2′ = { Z6 , Z5 , Z2 } , D3′ = {Z7 , Z3 , Z1} , D4′ = Z5 , Z3 , D , then
{
}
I ∗ ( Q9 ) = I ( D1′ ) + I ( D2′ ) + I ( D3′ ) + I ( D4′ )
. From this equality and statement (9) of
Lemma 2.1 we obtain validity of Lemma 6.1.
The Lemma is proved.
Lemma 6.2: Let D ∈ Σ2 ( X ,8) and Z5 ∩ Z3 = ∅. If X is a finite set, then the number
I ∗ ( Q16 ) can be calculated by the formula
Idempotent Elements of the Semigroups BX(D)...
(
)(
)
I ∗ ( Q16 ) = 2 Z5 \ Z1 − 1 ⋅ 2 Z3 \ Z 2 − 1 ⋅ 8
Proof:
{{
By
definition
of
⌣
X \D
.
the
}}
35
given
⌣
Q16θ XI = Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D .
If the
⌣
D1′ = {Z7 , Z6 , Z5 , Z4 , Z3 , Z2 , Z1 , D} , then I ∗ ( Q16 ) = I ( D1′ )
semilattice
D
we
following equality is
have
hold
. From this equality and statement
(16) of Lemma 2.1 we obtain validity of Lemma 6.2.
The Lemma is proved.
Let us assume that
k6 = I ∗ ( Q9 ) + I ∗ ( Q10 ) + I ∗ ( Q11 ) + I ∗ ( Q12 ) + I ∗ ( Q13 ) + I ∗ ( Q14 ) + I ∗ ( Q15 ) + I ∗ ( Q16 ) =
=3
X \ Z1
(
+ (4
+ (2
+ (2
+ (2
+ 4
+3
Z1 \ Z4
Z1 \ Z4
Z5 \ Z 4
Z5 \ Z 4
Z3 \ Z 2
X \ Z2
−3
−3
+3
Z1 \ Z4
Z1 \ Z4
)
− 1) ⋅ ( 6
− 1) ⋅ ( 4
−1 ⋅ 5
X \ Z4
)⋅4
) ⋅ (5
+3
⌣
D \ Z1
(
+ 2
⌣
D \ Z2
Z2 \ Z1
−5
(
+ 4
X \ Z1
X \ Z2
⌣
X \D
−4
Z3 \ Z 4
⌣
D \ Z2
−3
Z2 \ Z1
(
+ 4
⌣
D \ Z4
−3
)
⌣
D \ Z1
⌣
D \ Z1
)
⋅5
−1 ⋅ 5
)⋅6
)⋅7
⌣
X \D
⌣
X \D
⌣
D \ Z1
)⋅4
⌣
D \ Z4
X \ Z1
⌣
X \D
−3
+3
(
+ (2
⌣
X \D
⌣
X \D
(
+ 4
( Z2 ∩ Z1 ) \ Z4
(
+ 2
+ 2
)
⋅4
Z3 \ Z 2
Z3 \ Z 4
Z5 \ Z1
(
+ 4
Z2 \ Z4
(
⋅ 4
)
−1 ⋅ 5
)(
− 1) ⋅ ( 4
−1 ⋅ 6
⌣
D \ Z2
⌣
X \D
⌣
D \ Z1
Z1 \ Z2
⌣
D \ Z2
)
⋅4
(
⌣
X \D
+ 4
Z2 \ Z 4
−3
Z2 \ Z4
) ⋅ (5 − 4 ) ⋅ 5 +
−3
) ⋅(4 − 3 ) ⋅ 6
+ (2
− 1) ⋅ 5
+
−3
Z1 \ Z2
−3
−5
⌣
D \ Z2
Z2 \ Z4
Z1 \ Z2
Z2 \ Z1
−3
)⋅6
Z1 \ Z2
⌣
X \D
⌣
X \D
Z2 \ Z1
)⋅4
X \ Z2
+
⌣
X \D
)⋅7
+
⌣
X \D
(
+ 2
Z5 \ Z1
)(
−1 ⋅ 2
Z3 \ Z 2
)
−1 ⋅ 8
⌣
X \D
Theorem 6.2: Let D ∈ Σ2 ( X ,8) and Z5 ∩ Z3 = ∅ . If X is a finite set and I D is the
set of all idempotent elements of the semigroup BX ( D ) , then ID = k1 + k6 .
Proof: This Theorem immediately follows from the Theorem 6.1.
The Theorem is proved.
References
[1]
[2]
[3]
[4]
[5]
+
⌣
X \D
Z5 \ Z1
⌣
D \ Z1
⌣
D \ Z2
Ya. Diasamidze and Sh. Makharadze, Complete Semigroups of Binary
Relations, Monograph, Kriter, Turkey, (2013), 1-520.
Ya. Diasamidze and Sh. Makharadze, Complete Semigroups of Binary
Relations, Monograph, M., Sputnik+, (2010), 657 (Russian).
Ya. I. Diasamidze, Complete semigroups of binary relations, Journal of
Mathematical Sciences, Plenum Publ. Cor., New York, 117(4) (2003),
4271-4319.
Ya. Diasamidze, Sh. Makharadze and N. Rokva, On XI − semilattices of
union, Bull. Georg. Nation. Acad. Sci., 2(1) (2008), 16-24.
Ya. Diasamidze, Sh. Makharadze and I. Diasamidze, Idempotents and
regular elements of complete semigroups of binary relations, Journal of
Mathematical Sciences, Plenum Publ. Cor., New York, 153(4) (2008),
481-499.
36
Nino Tsinaridze
[6]
N. Tsinaridze, Sh. Makharadze and N. Rokva, Idempotent elements of the
semigroup BX ( D ) defined by semilattices of the class Σ2 ( X ,8) , J. of Math.
Sci., New York (To appear).