Gen. Math. Notes, Vol. 27, No. 1, March 2015, pp. 69-89
ISSN 2219-7184; Copyright © ICSRS Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Maximal Subgroups of the Semigroup BX (D)
Defined by Semilattices of the Class Ʃ3(X, 8)
Giuli Tavdgiridze
Faculty of Physics, Mathematics and Computer Sciences
Department of Mathematics, Shota Rustaveli Batumi State University
35, Ninoshvili St., Batumi 6010, Georgia
E-mail: g.tavdgiridze@mail.ru
(Received: 19-12-14 / Accepted: 30-1-15)
Abstract
By the symbol Ʃ3(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
⌣
⌣
⌣
⌣
Z7 ⊂ Z5 ⊂ Z3 ⊂ Z1 ⊂ D⌣, Z7 ⊂ Z6 ⊂ Z 4 ⊂ Z 2 ⊂ D, Z7 ⊂ Z5 ⊂ Z 4 ⊂ Z1 ⊂ D, Z7 ⊂ Z5 ⊂ Z 4 ⊂ Z 2 ⊂ D,
Z7 ⊂ Z 6 ⊂ Z 4 ⊂ Z1 ⊂ D,
Zi \ Z j ≠ ∅, ( i, j ) ∈ {( 5,6 ) , ( 6,5) , ( 3,6 ) , ( 6,3) , ( 4,3) , ( 3, 4 ) , ( 2,3) , ( 3, 2 ) , ( 2,1) , (1, 2 )}.
In the given paper we give a full description maximal subgroups of the complete
semigroups of binary elations defined by semilattices of the class Ʃ3(X, 8).
Keywords: Semilattice, Semigroup, Binary Relation, Idempotent Element.
70
1
Giuli Tavdgiridze
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 set-theoretic
operations of unification of elements from D, f be an arbitrary mapping from X
into D. To each such a mapping f there corresponds a binary relation α f on the set
X that satisfies the condition α f = ∪ ({ x} × f ( x ) ) . The set of all such α f (f : X→D)
x∈ X
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, p. 34]).
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 . Further let x, y∈X , Y ⊆ X ,
⌣
α ∈ BX ( D ) , T ∈ D , ∅ ≠ D′ ⊆ D and t ∈ D = ∪ Y . Then by symbols we denote the
Y ∈D
following 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 } , l ( D′, T ) = ∪ ( D′ \ D′ ) .
DT′ = {Z ′ ∈ D′ | T ⊆ Z ′} , D
T
T
(1.1)
Under symbol ˄ (D, Dt) we mean an exact lower bound of the set Dt in the
semilattice D.
Definition 1.1: Let ε ∈ BX ( D ) . If ε ε = ε or α ε = α for any α ∈ BX ( D ) , then ε is
called an idempotent element or called right unit of the semigroup BX (D)
respectively (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)
b)
⌣
∧ ( D, Dt ) ∈ D for any t ∈ D ;
Z = ∪ ∧ ( D, Dt ) for any nonempty element Z of D (see [1, Definition
t∈Z
1.14.2], [2, Definition 1.14.2] or [6]).
Definition 1.3: The one-to-one mapping ϕ between the complete X − semilattices
of unions D′ and D′′ is called a complete isomorphism if the condition
ϕ ( ∪ D1 ) = ∪ ϕ ( T ′ ) is fulfilled for each nonempty subset D1 of the semilattice
T ′∈D1
D′ (see [1, Definition 6.2.3], [2, Definition 6.2.3] or [5]).
Maximal Subgroups of the Semigroup BX (D)…
71
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] or [2, Definition 1.13.1
and 1.13.2]).
Theorem 1.1: 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 ≠ ∅
⌣
Theorem 1.2: Let D = { D, Z1 , Z 2 ,..., Z m −1} be some finite X-semilattice of unions and
C ( D ) = { P0 , P1 , P2 ,..., Pm −1} be the family of sets of pairwise nonintersecting subsets of
the set X. If ϕ is a mapping of the semilattice D on the family of sets C(D) which
⌣
satisfies the condition ϕ ( D ) = P0 and ϕ ( Zi ) = Pi for any i = 1, 2,..., m − 1 and
Dˆ Z = D \ {T ∈ D | Z ⊆ T } ,
then the following equalities are valid:
⌣
D = P0 ∪ P1 ∪ P2 ∪ ... ∪ Pm −1 , Z i = P0 ∪
∪ ϕ (T ).
(1.2)
T ∈Dˆ Zi
In the sequel these equalities will be called formal.
It is proved that if the elements of the semilattice D are represented in the form
(1.2), then among the parameters Pi ( i = 0,1, 2,..., m − 1) there exist such parameters
that cannot be empty sets. Such sets Pi ( 0 < i ≤ m − 1) are called basis sources,
whereas sets Pj ( 0 ≤ j ≤ m − 1) which can be empty sets too are called completeness
sources.
The number the basis sources we denote by symbol δ .
It is proved that under the mapping ϕ the number of covering elements of the preimage of a basis source is always equal to one, while under the mapping ϕ the
number of covering elements of the pre-image of a completeness source either
does not exist or is always greater than one (see [1, Item 11.4], [2, Item 11.4] or
[3]).
72
Giuli Tavdgiridze
Denote by the symbol GX ( D,ε ) a maximal subgroup of the semigroup BX (D)
whose unit is an idempotent binary relation ε of the semigroup BX (D).
Theorem 1.3: For any idempotent element ε ∈ BX ( D ) , the group GX ( D,ε ) is
antiisomorphic to the group of all complete automorphism of the semilattice
V ( D, ε ) (see [1, Theorem 7.4.2], [2, Theorem 7.4.2] or [4]).
2
Results
Let X and Σ3 ( X ,8) be respectively an any nonempty set and a class
intreisomorphic X-semilattices of unions where every element is isomorphic to
⌣
some X-semilattice of unions D = {Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D} , that satisfying the
conditions.
⌣
Z7 ⊂ Z 6 ⊂ Z 4 ⊂ Z 2 ⊂ D
⌣,
Z7 ⊂ Z5 ⊂ Z 4 ⊂ Z 2 ⊂ D
⌣,
Z 7 ⊂ Z 5 ⊂ Z 3 ⊂ Z1 ⊂ D;
Z1 \ Z 2 ≠ ∅, Z 2 \ Z1 ≠ ∅,
Z 3 \ Z 4 ≠ ∅, Z 4 \ Z 3 ≠ ∅,
Z 5 \ Z 6 ≠ ∅, Z 6 \ Z 5 ≠ ∅;
⌣
Z 7 ⊂ Z 6 ⊂ Z 4 ⊂ Z1 ⊂ D
⌣,
Z 7 ⊂ Z5 ⊂ Z 4 ⊂ Z1 ⊂ D,
Z 3 \ Z 2 ≠ ∅, Z 2 \ Z 3 ≠ ∅,
Z 3 \ Z 6 ≠ ∅, Z 6 \ Z 3 ≠ ∅,
(2.1)
The semilattice satisfying the conditions (2.1) is shown in Figure 1.
⌣
D
Z1
Z2
Z4
Z3
Z5
Z6
Z7
Fig. 1
Lemma 2.1: Let D ∈ Σ3 ( X ,8) . Then the following sets exhaust all subsemilattices
⌣
of the semilattice D = {Z7 , Z6 , Z5 , Z4 , Z3 , Z2 , Z1 , D} .
⌣
1){Z 7 } , {Z 6 } , {Z 5 } , {Z 4 } , {Z 3 } , {Z 2 } , {Z1 } , D . (see diagram 1 of the figure 2);
{ }
⌣
2 ){ Z 7 , Z1 } , {Z 7 , Z 2 } , { Z 7 , Z 3 } , { Z 7 , Z 4 } , {Z 7 , Z 5 } , { Z 7 , Z 6 } , Z 7 , D ,
⌣
{Z 6 , Z 4 } , {Z 6 , Z⌣ 2 } , {Z 6 , Z1} , Z 6 , D , {Z 5 , Z⌣ 4 } , {Z 5 , Z 3 } , {Z 5 ,⌣Z 2 } ,
, { Z 4 , Z 2 } , { Z 4 , Z1 } , Z 4 , D , { Z 3 , Z1 } , Z 3 , D ,
{Z 5 , Z⌣1 } , Z 5 , D
⌣
Z 2 , D , Z1 , D .
{
{
}{
}
}
{
}
(see diagram 2 of the figure 2);
{
{
}
{
}
}
Maximal Subgroups of the Semigroup BX (D)…
73
⌣
{Z 7 , Z 6 , Z 4 } , {Z 7 , Z5 , Z 2 } ,{Z 7 , Z 6 , Z1} , {Z 7 , Z 6 , D} , {Z 7 , Z5 , Z 4 } ,
3)
⌣
{Z 7 , Z5 , Z3 } , {Z 7 , Z5 , Z 2 } , {Z 7 , Z 5 , Z1} , {Z 7 , Z5 , D} , {Z 7 , Z 4 , Z 2 } ,
⌣
⌣
⌣
{Z 7 , Z 4 , Z1} ,{Z 7 , Z 4 , D} , {Z 7 , Z 3 , Z1} , {Z 7 , Z 3 , D} , {Z 7 , Z 2 , D} ,
⌣
⌣
⌣
{Z , Z , D} , {Z , Z , Z } , {Z , Z , Z } ,{Z , Z , D} ,{Z , Z , D} ,
⌣
⌣
{Z , Z , D} , {Z , Z , Z } , {Z , Z , Z } , {Z , Z , D} , {Z , Z , Z } ,
⌣
⌣
⌣
⌣
⌣
{Z , Z , D} , {Z , Z , D} , {Z , Z , D} ,{Z , Z , D} , {Z , Z , D} ,
⌣
{Z , Z , D}.
7
1
6
4
2
6
4
1
6
4
6
2
6
1
5
4
2
5
4
1
5
4
5
3
5
3
3
1
5
2
5
1
4
2
4
1
1
(see diagram 3 of the figure 2);
4)
⌣
⌣
{Z7 , Z6 , Z 4 , Z 2 } , {Z7 , Z6 , Z 4 , Z1} ,{Z7 , Z6 , Z 4 , D} , {Z7 , Z 6 , Z2 , D} ,
⌣
⌣
{Z7 , Z6 , Z1 , D}{Z7 , Z5 , Z 4 , Z 2 } ,{Z7 , Z5 , Z4 , Z1} ,{Z7 , Z5 , Z 4 , D} ,
⌣
⌣
⌣
{Z7 , Z5 , Z3 , Z1} , {Z7 , Z5 , Z3 , D} , {Z7 , Z5 , Z2 , D} ,{Z7 , Z5 , Z1 , 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}.
7
4
2
5
3
1
7
5
4
1
4
7
1
3
6
4
1
5
4
2
2
6
4
1
(see diagram 4 of the figure 2);
5)
⌣
⌣
⌣
⌣
{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}.
7
6
4
2
7
5
3
1
7
6
4
1
7
5
4
1
7
5
4
2
(see diagram 5 of the figure 2);
6)
⌣
⌣
{Z 7 , Z 4 , Z 3 , Z⌣1} , {Z 7 , Z 2 , Z1 , D} , {Z 7 , Z 6 , Z 5 , Z⌣4 } , {Z 4 , Z 2 , Z1 , D
},
⌣
{Z 6 , Z 2 , Z1 , D} , {Z 7 , Z 6 , Z 3 , Z1} ,{Z 7 , Z 3 , Z 2 , D} , {Z 5 , Z3 , Z 2 , D}.
(see diagram 6 of the figure 2);
7)
⌣
⌣
{Z , Z , Z , Z , D⌣ } , {Z , Z , Z , Z , Z⌣ } ,{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}.
7
6
2
1
7
5
4
3
7
4
2
1
7
5
2
1
5
4
2
1
1
7
6
5
4
3
2
2
1
(see diagram 7 of the figure 2);
⌣
⌣
8)
{Z , Z , Z , Z , Z , D} ;{Z , Z , Z , Z , Z , D}. (see diagram 8 of the figure 2);
9)
{Z , Z , Z , Z , Z , D}. (see diagram 9 of the figure 2);
7
6
4
2
1
7
5
4
2
1
⌣
7
5
4
3
1
⌣
10 ) {Z 7 , Z 6 , Z 5 , Z 4 , Z 2 } , {Z 7 , Z 6 , Z 5 , Z 4 , Z1 } , Z 7 , Z 6 , Z 5 , Z 4 , D ,
⌣
⌣
⌣
Z 7 , Z 6 , Z 3 , Z1 , D , Z 5 , Z 4 , Z 3 , Z1 , D , Z 7 , Z 4 , Z 3 , Z1 , D ,
{
}{
}{
{
}
}
(see diagram 10 of the figure 2);
⌣
⌣
11) Z 7 , Z 6 , Z 5 , Z 4 , Z 2 , D , Z 7 , Z 6 , Z5 , Z 4 , Z1 , D . (see diagram 11 of the figure 2);
{
}{
}
74
Giuli Tavdgiridze
⌣
⌣
⌣
12 ) {Z 7 , Z6 , Z5 , Z 4 , Z3 , Z1} , Z 7 , Z 6 , Z3 , Z 2 , Z1 , D , Z 7 , Z 4 , Z3 , Z 2 , Z1 , D , Z5 , Z 4 , Z3 , Z 2 , Z1 , D .
{
}{
}{
(see diagram 12 of the figure 2);
⌣
13)
{Z , Z , Z , Z , Z , Z , D}. (see diagram 13 of the figure 2);
14)
{Z , Z , Z , Z , Z , Z , D} ,
15)
{Z , Z , Z , Z , Z , Z , D} , (see diagram 15 of the figure 2);
16)
{Z , Z , Z , Z , Z , Z , Z , D}. (see diagram 16 of the figure 2);
17)
{Z , Z , D} ,{Z , Z , D}.{Z , Z , Z } ,{Z , Z , Z } ,{Z , Z , Z }.
7
5
4
3
2
1
⌣
7
6
5
4
3
(see diagram 14 of the figure 2);
1
⌣
7
6
5
4
2
1
⌣
7
6
5
4
3
2
⌣
3
1
⌣
2
2
1
6
5
4
6
3
1
4
3
1
(see diagram 17 of the figure 2);
18)
⌣
⌣
{Z , Z , Z , D⌣ } , {Z , Z , Z , Z } , {Z , Z , Z , Z } , {Z , Z , Z , D} ,
{Z , Z , Z , D}.
6
5
4
6
3
1
6
5
4
2
6
5
4
1
4
3
1
(see diagram 18 of the figure 2);
⌣
19) Z 3 , Z 2 , Z1 , D , { Z 6 , Z 4 , Z 3 , Z1 } . (see diagram 19 of the figure 2);
{
}
⌣
⌣
20) Z 6 , Z5 , Z 4 , Z 2 , D , Z 6 , Z5 , Z 4 , Z1 , D , (see diagram 20 of the figure 2);
{
}{
}
⌣
21)
{Z , Z , Z , Z , D} , (see diagram 21 of the figure 2);
22)
{Z7 , Z 6 , Z 4 , Z3 , Z1} , {Z5 , Z3 , Z 2 , Z1 , D} , {Z7 , Z3 , Z 2 , Z1 , D} ,
6
4
3
1
⌣
⌣
⌣
⌣
(see diagram 22 of the figure 2);
23)
{Z 6 , Z5 , Z 4 , Z3 , Z1} , {Z 6 , Z3 , Z 2 , Z1 , D} , {Z 4 , Z3 , Z 2 , Z1 , D} ,
(see diagram 23 of the figure 2);
⌣
24)
{Z , Z , Z , Z , Z , D} ,
25)
{Z , Z , Z , Z , Z , D} , (see diagram 25 of the figure 2);
26)
{Z , Z , Z , Z , Z , D} ,
27)
{Z , Z , Z , Z , Z , D} ; (see diagram 27 of the figure 2);
28)
{Z , Z , Z , Z , Z , D} , (see diagram 28 of the figure 2);
6
5
4
3
1
(see diagram 24 of the figure 2);
⌣
6
5
4
2
1
⌣
6
4
3
2
1
(see diagram 26 of the figure 2);
⌣
7
5
3
2
1
⌣
7
6
4
3
1
}
Maximal Subgroups of the Semigroup BX (D)…
75
⌣
29)
{Z , Z , Z , Z , Z , Z , D} , (see diagram 29 of the figure 2);
30)
{Z , Z , Z , Z , Z , Z , D} ,
6
5
4
3
2
1
⌣
7
6
4
3
2
1
(see diagram 30 of the figure 2);
⌣
Proof: It is ease to see that, the sets {Z7} ,{Z6} ,{Z5} ,{Z4} ,{Z3} ,{Z2} ,{Z1} ,{D} are
subsemilattices of the semilattice D .
The number all subsets of the semilattise D, every set of which contains two
elements, is equal to C82 = 28 . In this case X- subsemilattices of the semilattice D
are the following sets:
⌣
⌣
{Z7 , Z⌣1} ,{Z7 , Z⌣ 2} ,{Z7 , Z3} ,{Z7 , Z4} ,{Z7 , Z5} ,{Z7 , Z6} ,{Z7 , D} ,{Z5, D} ,{Z6 , Z⌣ 4} ,{Z6 , Z2} ,{Z6 ,⌣Z1} , ⌣
{Z6, D} ,{Z3, D} ,{Z5, Z4} ,{Z5, Z3} ,{Z5, Z2} ,{Z5, Z1} ,{Z4 , Z2} ,{Z4 , Z1} ,{Z4, D} ,{Z3, Z1} ,{Z2 , D} ,{Z1, D}
Remainder 5 subsets of the semilattice D, whose every element contains two
elements is not an X-subsemilattice.
The number all subsets of the semilattise D, every set of which contains three
elements, is equal to C83 = 56 . In this case X-subsemilattices of the semilattice D
are the following sets:
⌣
{Z 7 , Z 6 , Z⌣4 } , {Z 7 , Z 5 , Z 2 } , {Z 7 , Z 6 , Z1} ,{Z 7 , Z 6 , ⌣D} , {Z 7 , Z5 , Z 4 } , {Z 7 , Z5 ,⌣Z3 } , {Z 7 , Z5 ,⌣Z 2 } , {Z 7 , Z 5⌣, Z1} ,
{Z7 , Z5 , D} , {Z 7 , Z 4 , Z 2 } , {Z 7 , Z 4 , Z1} , {Z 7 , Z 4 , D⌣} , {Z7 , Z3 , Z1} , {Z7 , Z3 , D⌣} , {Z7 , Z 2 , D⌣} , {Z7 , Z1 , D} ,
{Z 6 , Z5 , Z 4 } , {Z 6 , Z 4 , Z⌣ 2 } ,{Z 6 , Z 4 , Z1} , {Z 6 , Z 4 ,⌣D} ,{Z 6 , Z 3 ,⌣Z1} , {Z 6 , Z 2⌣, D} ,{Z 6 , Z1 , D} , {Z 5 , Z 4 ,⌣Z 2 } ,
{Z 5 , Z 4 , Z⌣1} , {Z5 , Z 4 , ⌣D} , {Z 5 , Z 3 , ⌣Z1} , {Z5 , Z3 ,⌣D} , {Z5 , Z 2 , D} , {Z 5 , Z1 , D} , {Z 4 , Z3 , Z1} , {Z 4 , Z 2 , D} ,
{Z , Z , D} , {Z , Z , D} , {Z , Z , D} ,{Z , Z , D}
4
1
3
2
3
1
2
1
Remainder 20 subsets of the semilattice D, whose every element contains three
elements is not an X- subsemilattice.
The number all subsets of the semilattise D, every set of which contains four
elements, is equal to C84 = 70 . In this case X- subsemilattices of the semilattice D
are the following sets:
⌣
⌣
{Z7 , Z6 , Z5 , Z4 } ,{Z7 , Z6 , Z4 , Z⌣ 2 } ,{Z7 , Z6 , Z4 ,⌣Z1} ,{Z7 , Z5 , Z4 , Z2 } ,{Z7 , Z6 , Z4⌣, D} ,{Z7 , Z6 , Z3⌣, D} ,{Z7 , Z6 , Z3⌣, Z1} ,
{Z7 , Z5 , Z3 , Z⌣1} ,{Z7 , Z6 , Z2 , D} ,{Z7 , Z6 , Z1 , D⌣} ,{Z7 , Z5 , Z4 , Z⌣1} ,{Z7 , Z5 , Z4 , ⌣D} ,{Z7 , Z5 , Z3 , ⌣D} ,{Z7 , Z5 , Z2 ,⌣D} ,
{Z7 , Z5 , Z1, D⌣} ,{Z7 , Z4 , Z3 , Z1} ,{Z7 , Z4 , Z2 , D⌣ } ,{Z7 , Z4 , Z1, D} ,{Z7 , Z3 , Z2 , D⌣} ,{Z7 , Z3 , Z1 , D} ,{Z7 , Z2 , Z1 , D} ,
{Z6 , Z2 , Z1, D} ,{Z5 , Z4 , Z3 , Z⌣1} ,{Z5 , Z4 , Z2 , D⌣} ,{Z6 , Z5 , Z4 , Z⌣2 } ,{Z6 , Z5 , Z4 ,⌣D} ,{Z6 , Z5 , Z4 ,⌣Z1} ,{Z6 , Z4 , Z3⌣, Z1} ,
{Z6 , Z5 , Z3 , Z⌣1} ,{Z6 , Z4 , Z2 ,⌣D} ,{Z6 , Z4 , Z1 , D} ,{Z6 , Z3 , Z1 , D} ,{Z4 , Z3 , Z1 , D} ,{Z4 , Z2 , Z1 , D} ,{Z5 , Z3 , Z2 , D} ,
{Z , Z , Z , D} ,{Z , Z , Z , D} ,
5
4
1
5
3
1
Remainder 33 subsets of the semilattice D, whose every element contains four
elements is not an X- subsemilattice.
76
Giuli Tavdgiridze
The number all subsets of the semilattise D, every set of which contains five
elements, is equal to C85 = 56 . In this case X- subsemilattices of the semilattice D
are the following sets:
⌣
⌣
⌣
⌣
{Z7 , Z6 , Z5 , Z4 , Z⌣2 } ,{Z7 , Z6 , Z5 , Z4 , Z1} ,{Z7 , Z6 , Z5 , Z4 ,⌣D} ,{Z7 , Z6 , Z4 , Z1 ,⌣D} ,{Z7 , Z6 , Z4 , Z2 ,⌣D} ,{Z7 , Z6 , Z3 , Z1 ,⌣D} ,
{Z7 , Z6 , Z2 , Z1, D⌣} ,{Z7 , Z5 , Z4 , Z3 , Z⌣1} ,{Z7 , Z5 , Z4 , Z2 ,⌣D} ,{Z7 , Z5 , Z4 , Z1,⌣D} ,{Z7 , Z5 , Z3 , Z2 , D} ,{Z7 , Z5 , Z3 , Z1, D⌣ } ,
{Z7 , Z5 , Z2 , Z1, D⌣} ,{Z7 , Z4 , Z3 , Z1, D⌣} ,{Z7 , Z4 , Z2 , Z1, D⌣} ,{Z7 , Z3 , Z2 , Z1, D⌣ } ,{Z6 , Z5 , Z4 , Z3 , Z⌣1} ,{Z6 , Z5 , Z4 , Z2 ,⌣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} ,{Z , Z , Z , Z , D} ,{Z , Z , Z , Z , Z } ,
6
5
4
1
6
4
3
1
6
4
2
1
5
3
2
1
4
3
2
1
7
6
4
3
6
3
2
1
5
4
3
1
5
4
2
1
1
Remainder 29 subsets of the semilattice D, whose every element contains five
elements is not an X- subsemilattice.
The number all subsets of the semilattise D, every set of which contains six
elements, is equal to C86 = 28 . In this case X- subsemilattices of the semilattice D
are the following sets:
⌣
⌣
⌣
⌣
{Z7 , Z6 , Z5 , Z4 , Z3 , Z⌣1} ,{Z7 , Z6 , Z5 , Z4 , Z2 ,⌣D} ,{Z5 , Z4 , Z3 , Z2 , Z1 , ⌣D}{Z7 , Z6 , Z5 , Z4 , Z1 , D⌣} ,{Z7 , Z5 , Z 4 , Z2 , Z1 , 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} ,{Z , Z , Z , Z , Z , D} ,{Z , Z , Z , Z , Z , D} ,{Z , Z , Z , Z , Z , D} ,
6
4
3
2
1
7
6
4
3
1
7
6
4
2
1
7
6
3
2
1
7
5
4
3
1
7
5
3
2
1
7
4
3
2
1
6
5
4
3
1
6
5
4
2
1
7
6
5
4
3
Remainder 13 subsets of the semilattice D , whose every element contains six
elements is not an X - subsemilattice.
The number all subsets of the semilattise D , every set of which contains seven
elements, is equal to C87 = 8 . In this case X - subsemilattices of the semilattice
D are the following sets:
⌣
⌣
⌣
{Z , Z , Z , Z , Z , Z , D⌣} , {Z , Z , Z , Z , Z , Z , D⌣ } ,{Z , Z , Z , Z , Z , Z , D} ,
{Z , Z , Z , Z , Z , Z , D} , {Z , Z , Z , Z , Z , Z , D} ,
7
6
5
4
3
1
7
6
5
4
2
1
7
5
4
3
2
1
6
5
4
3
2
1
7
6
4
3
2
1
Remainder 3 subsets of the semilattice D , whose every element contains seven
elements is not an X - subsemilattice.
The number all subsets of the semilattise D , every set of which contains eight
⌣
elements, is equal to C88 = 1 . This set is {Z7 , Z6 , Z5 , Z 4 , Z3 , Z2 , Z1 , D} .
The Lemma is proved.
From the proven lemma it follows that diagrams shown in fig. 2, exhaust all
diagrams of subsemilattices of the semilattice D .
Maximal Subgroups of the Semigroup BX (D)…
•
1 2 3 4 5 6
7
17 18
21
19 20
8
9 10
22 23
77
11
12
13
24 25 26
14
15
27 28
16
29
30
Figure 2
Lemma 2.2: Let D ∈ Σ3 ( X ,8) and Z7 ≠ ∅ . Then any subsemilattices of the
semilattice D having diagram 17 − 30 of the figure 2 are never XI − semilattice.
Proof: Remark, that the all subsemilattices of semilattice D which has diagrams
of form 17 − 30 are never XI − semilattices. For example we consider the
semilattice which has the diagram of the form 30 of the figure 3 (see diagram
figure 30).
⌣
Let D′ = {Z 7 , Z 6 , Z 4 , Z 3 , Z 2 , Z1 , D} and C ( D′) = { P0 , P1 , P2 , P3 , P4 , P5 , P6 } is a family sets,
where
P0 , P1 , P2 , P3 , P4 , P5 , P6
are pairwise disjoint subsets
of
the set X and
⌣


ϕ =  D Z1 Z 2 Z3 Z 4 Z 6 Z 7  is a mapping of the semilattice D ′ onto the family
 P0 P1 P2 P3 P4 P5 P6 
sets C ( D′) . Then for the formal equalities of the semilattice D ′ we have a form:
⌣
D
Z1
Z3
Z2
Z4
Z6
Z7
Figure 3
⌣
D = P0 ∪ P1 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ,
Z1 = P0 ∪ P2 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ,
Z2 = P0 ∪ P1 ∪ P3 ∪ P4 ∪ P6 ∪ P7 ,
Z3 = P0 ∪ P2 ∪ P4 ∪ P6 ∪ P7 ,
Z4 = P0 ∪ P3 ∪ P6 ∪ P7 ,
Z6 = P0 ∪ P3 ∪ P7
Z7 = P0
Here the elements P1 , P2 , P3 , P6 are basis sources, the element P0 , P4 , P7 is sources of
completenes of the semilattice D ′ . Therefore X ≥ 3 and, that
78
Giuli Tavdgiridze
 D ′, if t ∈ P ,
0
⌣

 Z 2 , D ,⌣if t ∈ P1
 Z 3 , Z1 , D , if t ∈ P2 ,
⌣

Dt′ =  Z 6 , Z 4 , Z 2 , Z1 , D , if t ∈ P3 ,
⌣
 Z , Z , Z , D , if t ∈ P ,
3
2
1
4
⌣

 Z 4 , Z 3 , Z 2 , Z1 , D , ⌣if t ∈ P5 ,
 Z 6 , Z 4 , Z 3 , Z 2 , Z1 , D , if t ∈ P6 ,

{
{
{
{
{
{
}
}
}
}
}
}
Z7 ,
Z2 ,
Z ,
 3
Λ ( D ′, Dt′ ) =  Z 6 ,
Z7 ,
Z7 ,
Z ,
 7
t ∈ P0
t ∈ P1
t ∈ P2
t ∈ P3
t ∈ P4
t ∈ P5
t ∈ P6
if
if
if
if
if
if
if
⌣
We have D′∧ = {Z7 , Z6 , Z3 , Z2 } and Λ( D′, Dt′) ∈ D′ for all t ∈ D . But element Z 4 is not
union of some elements of the set D′∧ . So, from the Definition 1.2 follows that
semilattice D ′ which has diagram 41 of the figure 3 never is XI − semilattice.
In the same manner it can be proved that any subsemilattice of the semilattice D
having diagrams 17-30 are never an XI − semilattice.
Lemma is proved.
Lemma 2.3: Let D ∈ Σ3 ( X ,8) and Z7 ≠ ∅ . Then the following sets are all XIsubsemilattices of the given semilattice D:
⌣
1)
{Z 7 } ,{Z 6 } , {Z5 } ,{Z 4 } , {Z3 } , {Z 2 } ,{Z1} , {D} ; (see diagram 1 of the figure 2);
2)
{Z7 , Z6 } , {Z7 , Z5 } , {Z7 , Z 4 } ,{Z7 , Z3 } ,{Z7 , Z 2 } , {Z7 , Z1} ,{Z7 , D} ,{Z6 , Z 4 } , {Z6 , Z2 } ,
⌣
⌣
{Z6 , Z1} , {Z6 , D} , {Z5 , Z 4 } ,{Z5 , Z3 } , {Z5 , Z 2 } , {Z5 , Z1} , {Z5 , D} , {Z 4 , Z 2 } , {Z 4 , Z1} ,
⌣
⌣
⌣
⌣
{Z4 , D} ,{Z3 , Z1} ,{Z3 , D} ,{Z 2 , D} ,{Z1 , D} ;
⌣
(see diagram 2 of the figure 2);
3)
⌣
{Z7 , Z6 , Z4 } ,{Z7 , Z6 , Z2 } ,{Z7 , Z6 , Z1} ,{Z7 , Z6 , D} ,{Z7 , Z5 , Z4 } ,{Z7 , Z5 , Z3} ,{Z7 , Z5 , Z2 } ,{Z7 , Z5 , Z1} ,
⌣
⌣
⌣
⌣
{Z7 , Z5 , D} ,{Z7 , Z4 , Z2} ,{Z7 , Z4 , Z⌣1} ,{Z7 , Z4 , D⌣} ,{Z7 , Z3 , Z⌣1} ,{Z7 , Z3 , D} ,{Z7 , Z2 , D} ,{Z7 , Z1, D⌣} ,
{Z6 , Z4 , Z2 } ,{Z6 , Z4 , Z1} ,{Z6 , Z4 , D} ,{Z6 , Z2 , D} ,{Z6 , Z1, D} ,{Z5 , Z4 , Z2 } ,{Z5 , Z4 , Z1} ,{Z5 , Z4 , D} ,
⌣
⌣
⌣
⌣
⌣
⌣
{Z5 , Z3 , Z1} ,{Z5 , Z3 , D} ,{Z5 , Z2 , D} ,{Z5 , Z1, D} ,{Z4 , Z2 , D} ,{Z4 , Z1, D} ,{Z3 , Z1, D} ;
(see diagram 3 of the figure 2);
4)
⌣
⌣
{Z 7 , Z 6 , Z 4 , Z 2 } , {Z 7 , Z 6 , Z 4 , Z1} , {Z 7 , Z6 , Z 4 , D} , {Z7 , Z 6 , Z 2 , D} , {Z 7 , Z 6 , Z1 , D} ,
⌣
⌣
{Z 7 , Z5 , Z 4 , Z 2 } , {Z7 , Z5 , Z 4 , Z1} , {Z 7 , Z5 , Z 4 , D} {Z 7 , Z5 , Z3 , Z1} , {Z7 , Z5 , Z3 , D} ,
⌣
⌣
⌣
⌣
{Z 7 , Z5 , Z 2 , D} , {Z 7 , Z5 , Z1 , D} , {Z 7 , Z 4 , Z 2 , D} , {Z 7 , Z 4 , Z1 , D} , {Z 7 , Z3 , Z1 , D}
⌣
⌣
⌣
⌣
⌣
{Z 6 , Z 4 , Z 2 , D} , {Z 6 , Z 4 , Z1 , D} ,{Z5 , Z 4 , Z 2 , D} , {Z5 , Z 4 , Z1 , D} , {Z5 , Z3 , Z1 , D} ;
(see diagram 4 of the figure 2);
⌣
⌣
⌣
⌣
⌣
5 ) Z 7 , Z 6 , Z 4 , Z 2 , D , Z 7 , Z 6 , Z 4 , Z1 , D , Z 7 , Z 5 , Z 4 , Z 2 , D , Z 7 , Z 5 , Z 4 , Z1 , D , Z 7 , Z 5 , Z 3 , Z1 , D ,
{
}{
(see diagram 5 of the figure 2);
}{
}{
}{
}
Maximal Subgroups of the Semigroup BX (D)…
79
⌣
⌣
{Z 7 , Z6 , Z5 , Z 4 } , {Z7 , Z 6 , Z3 , Z1} ,{Z 7 , Z 4 , Z3 , Z1} , {Z 7 , Z3 , Z 2 , D} , {Z 7 , Z 2 , Z1 , D} ,
⌣
⌣
⌣
⌣
{Z6 , Z 2 , Z1, D} , {Z5 , Z 4 , Z3 , Z1} , {Z5 , Z3 , Z 2 , D} ,{Z5 , Z 2 , Z1 , D} , {Z 4 , Z 2 , Z1 , D} ;
6)
(see diagram 6 of the figure 2);
⌣
⌣
⌣
{Z7 , Z6 , Z2 , Z1, D⌣} , {Z7 , Z5 , Z 4 , Z3 , Z⌣1 ,} ,{Z7 , Z5 , Z3 , Z 2 ⌣, D} ,{Z7 , Z5 , Z2 , Z1 , D} ,
{Z7 , Z4 , Z 2 , Z1 , D} ,{Z6 , Z 4 , Z 2 , Z1 , D}{Z5 , Z 4 , Z 2 , Z1 , D} ;
7)
(see diagram 7 of the figure 2);
⌣
⌣
8 ) Z 7 , Z 6 , Z 4 , Z 2 , Z1 , D , Z 7 , Z 5 , Z 4 , Z 2 , Z1 , D ;
{
}{
⌣
9 ) Z 7 , Z 5 , Z 4 , Z 3 , Z1 , D ;
{
10 )
}
}
(see diagram 8 of the figure 2);
(see diagram 9 of the figure 2);
⌣
⌣
{Z 7 , Z 6 , Z5 , Z 4 , Z 2 } ,{Z 7 , Z 6 , Z5 , Z 4 , Z1} , {Z 7 , Z 6 , Z5 , Z 4 , D} , {Z7 , Z 4 , Z3 , Z1 , D} ,
⌣
⌣
{Z7 , Z6 , Z3 , Z1, D} , {Z5 , Z4 , Z3 , Z1 , D} ;
(see diagram 10 of the figure 2);
⌣
⌣
11)
{Z7 , Z6 , Z5 , Z 4 , Z 2 , D} , {Z7 , Z6 , Z5 , Z 4 , Z1 , D} ; (see diagram 11 of the figure 2);
12 )
{Z7 , Z 6 , Z5 , Z 4 , Z3 , Z1} ,{Z 7 , Z 6 , Z3 , Z 2 , Z1 , D} , {Z 7 , Z 4 , Z3 , Z 2 , Z1 , D} ,{Z5 , Z 4 , Z3 , Z 2 , Z1 , D} ;
⌣
⌣
⌣
(see diagram 12 of the figure 2);
⌣
13) Z 7 , Z 5 , Z 4 , Z3 , Z 2 , Z1 , D ;
{
}
(see diagram 13 of the figure 2);
⌣
14 )
{Z7 , Z6 , Z5 , Z4 , Z3 , Z1 , D} ; (see diagram 14 of the figure 2);
15)
{Z7 , Z6 , Z5 , Z 4 , Z 2 , Z1 , D} ;
16 )
{Z7 , Z6 , Z5 , Z4 , Z3 , Z 2 , Z1 , D} ;
⌣
⌣
T T
⌣
D
T ′′
Z7
11
T′
T
3
1 2
T ′′′
T ′′
T′
T′
T′
Z4
T
(see diagram 15 of the figure 2);
T
4
T ′′
T′ T′
T
Z7
5
T ′′
T
12
T ′′′
T ′′
T ′′′ T ′
T′
T ′′
T
6
T
7
Z1
Z3
⌣
DZ
Z4
Z5
2
Z4
Z6
Z7
13
Z1
T ′′
Z4
T
T ′′′
Z1
T ′ ∪ T ′′
Z4 ′
T
T ′′
Z5
Z3
Z7
8
Z7
9
⌣
D
⌣
D
Z3
Z1
Z5 Z5
Z7
14
Figure 4
⌣
D
⌣
D
T ′′ ∪ T ′′′
⌣ T ′ ∪ T ′′
D
T ′ ∪ T ′′ ∪ T ′′′
T ′ ∪ T ′′
T′ T′
(see diagram 16 of the figure 2);
Z2
Z4
Z6
Z7
15
T
10
Z1
Z3
Z5
⌣
D
Z2
Z4
Z6
Z7
16
80
Giuli Tavdgiridze
Proof: The statements 1) , 2 ) , 3) , 4 ) 5) immediately follows from the Theorems
[1] 11.6.1, [2] 11.6.1, the statements 6 ) , 7 ) ,8 ) 9 ) 10 ) 11) immediately follows
from the Theorems [1] 11.6.3, [2] 11.6.3 the statement 12 ) immediately follows
from the Theorems [1] 11.7.2 and the statements 15 ) immediately follows from
the Theorems [1] 13.11.1.
⌣
Now we will proof the statement 16 ) . Let D = {Z 7 , Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D} where
⌣
⌣
Z7 ⊂ Z5 ⊂ Z3 ⊂ Z1 ⊂ D⌣, Z7 ⊂ Z6 ⊂ Z4 ⊂ Z2 ⊂ D
⌣,
Z7 ⊂ Z5 ⊂ Z4 ⊂ Z1 ⊂ D
⌣, Z7 ⊂ Z5 ⊂ Z4 ⊂ Z2 ⊂ D,
Z7 ⊂ Z6 ⊂ Z4 ⊂ Z1 ⊂ D, Z5 \ Z6 ≠ ∅, Z6 \ Z5 ≠ ∅,
Z4 \ Z3 ≠ ∅, Z3 \ Z4 ≠ ∅, Z2 \ Z1 ≠ ∅, Z⌣1 \ Z2 ≠ ∅,
Z6 ∪ Z5 = Z4, Z4 ∪ Z3 = Z1, Z2 ∪ Z1 = D
⌣
D
Z1
Z3
Z5
Z2
Z4
Z6
Z7
Let C ( D ) = { P0 , P1 , P2 , P3 , P4 , P5 , P6 , P7 } is a family sets, where P0 , P1 , P2 , P3 , P4 , P5 , P6 , P7
⌣
are pairwise disjoint subsets of the set X and ϕ =  D Z1 Z 2 Z3 Z 4 Z 5 Z 6 Z 7  is a
P P P P P P P P

0
1
2
3
4
5
6
7

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 = 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 ∪ P6 ∪ P7 ,
Z 6 = P0 ∪ P3 ∪ P5 ∪ P7 ,
Z 7 = P0 .
(2.5)
Of the formal equalities (2.5) follows, that
⌣
 D , if ⌣ t ∈ P0 ,
Z7 ,
 Z , D , if t ∈ P ,
2
1
Z2 ,
⌣

Z
,
Z
,
D
,
if
t
∈
P
,
Z ,
 3 1
2
⌣
 Z 3 ,
 Z 6 , Z 4 , Z 2 , Z 1 , D , if t ∈ P3 ,
⌣
∧ ( D, Dt ) =  6
Dt′ = 
Z ,
Z 3 , Z 2 , Z 1 , D , if t ∈ P4 ,
 5
⌣

Z7 ,
 Z 6 , Z 4 , Z 3 , Z 2 , Z 1 , D⌣ , if t ∈ P5 ,
 Z , Z , Z , Z , Z , D , if t ∈ P ,
Z5 ,
6
⌣
 5 4 3 2 1
 Z 7 ,
 Z 6 , Z 5 , Z 4 , Z 3 , Z 2 , Z 1 , D , if t ∈ P7 ,
{
{
{
{
{
{
{
}
}
}
}
}
}
}
if
if
if
if
if
if
if
if
t ∈ P0 ,
t ∈ P1 ,
t ∈ P2 ,
t ∈ P3 ,
t ∈ P4 ,
t ∈ P5 ,
t ∈ P6 ,
t ∈ P7 ,
We have D∧ = {Z7 , Z6 , Z5 , Z3 , Z2 } , ∧ ( D, Dt ) ∈ D for all t and Z 4 = Z 6 ∪ Z5 , Z1 = Z 6 ∪ Z 3 ,
⌣
D = Z3 ∪ Z 2 . So, from the Definition 1.2 follows that semilattice D which has
diagram of the figure 4 is XI − semilattice.
Maximal Subgroups of the Semigroup BX (D)…
81
In the same manner it can be proved that any subsemilattice of the semilattice D
having diagrams 13 and 14 are an XI − semilattice.
Lemma is proved
Corollary 2.1: Let D ∈ Σ3 ( X ,8 ) and Z 7 = ∅ . Then the following sets are all
XI − subsemilattices of the given semilattice D :
1)
{∅} (see diagram 1 of the figure 2);
2)
{∅, Z 6 } ,{∅, Z5 } , {∅, Z 4 } ,{∅, Z3 } , {∅, Z 2 } ,{∅, Z1} , {∅, D}
⌣
(see diagram 2 of the figure 2);
3)
⌣
{∅, Z6 , Z4 } ,{∅, Z6 , Z2 } , {∅, Z6 , Z1} , {∅, Z6 , D} ,{∅, Z5 , Z4 } , {∅, Z5 , Z3 } ,{∅, Z5 , Z 2 } ,{∅, Z5 , Z1} ,
⌣
⌣
⌣
⌣
{∅, Z5 , D} ,{∅, Z4 , Z2 } ,{∅, Z4 , Z1} ,{∅, Z4 , D} ,{∅, Z3 , Z1} ,{∅, Z3 , D} ,{∅, Z2 , D} ,{∅, Z1, D} ,
(see diagram 3 of the figure 2);
⌣
⌣
4 ) {∅, Z 6 , Z 4 , Z 2 } , {∅, Z 6 , Z 4 , Z1} , ∅, Z 6 , Z 4 , D , {∅, Z 6 , Z 2 , D} , ∅, Z 6 , Z1 , D ,
⌣
⌣
{∅, Z5 , Z 4 , Z 2 } , {∅, Z5 , Z 4 , Z1} , ∅, Z5 , Z 4 , D {∅, Z5 , Z3 , Z1} , ∅, Z5 , Z3 , D ,
⌣
⌣
⌣
⌣
∅, Z5 , Z 2 , D , ∅, Z5 , Z1 , D , {∅, Z 4 , Z 2 , D} , ∅, Z 4 , Z1 , D , ∅, Z3 , Z1 , D
{
}{
}
{
{
}
}
{
}{
{
{
}
}
}
(see diagram 4 of the figure 2);
⌣
⌣
⌣
⌣
⌣
5) ∅, Z 6 , Z 4 , Z 2 , D , ∅, Z 6 , Z 4 , Z1 , D , ∅, Z5 , Z 4 , Z 2 , D , ∅, Z5 , Z 4 , Z1 , D , ∅, Z5 , Z3 , Z1 , D ,
{
}{
}{
}{
}{
}
(see diagram 5 of the figure 2);
6)
⌣
⌣
{∅, Z 6 , Z5 , Z 4 } ,{∅, Z 6 , Z3 , Z1} , {∅, Z 4 , Z3 , Z1} , {∅, Z3 , Z 2 , D} , {∅, Z 2 , Z1 , D} ,
(see diagram 6 of the figure 2);
7)
⌣
⌣
⌣
⌣
{∅, Z6 , Z2 , Z1 , D} ,{∅, Z5 , Z 4 , Z3 , Z1 ,} ,{∅, Z5 , Z3 , Z 2 , D} ,{∅, Z5 , Z 2 , Z1 , D} ,{∅, Z 4 , Z2 , Z1 , D}
(see diagram 7 of the figure 2);
⌣
⌣
8)
{∅, Z6 , Z 4 , Z2 , Z1, D} ,{∅, Z5 , Z 4 , Z2 , Z1, D} ;
9)
{∅, Z5 , Z 4 , Z3 , Z1 , D} ;
⌣
(see diagram 8 of the figure 2);
(see diagram 9 of the figure 2);
⌣
⌣
⌣
10 ) {∅, Z6 , Z5 , Z 4 , Z 2 } , {∅, Z 6 , Z5 , Z 4 , Z1} , ∅, Z6 , Z5 , Z 4 , D , ∅, Z 4 , Z3 , Z1 , D , ∅, Z 6 , Z3 , Z1 , D
{
}{
}{
(see diagram 10 of the figure 2);
⌣
⌣
11) ∅, Z 6 , Z5 , Z 4 , Z 2 , D , ∅, Z 6 , Z5 , Z 4 , Z1 , D ; (see diagram 11 of the figure 2);
{
}{
}
⌣
⌣
12 ) {∅, Z6 , Z5 , Z 4 , Z3 , Z1} , ∅, Z 6 , Z3 , Z 2 , Z1 , D , ∅, Z 4 , Z3 , Z 2 , Z1 , D
{
}{
}
}
82
Giuli Tavdgiridze
(see diagram 12 of the figure 2);
⌣
13) ∅, Z 5 , Z 4 , Z 3 , Z 2 , Z1 , D ;
{
14 )
}
⌣
{∅, Z6 , Z5 , Z 4 , Z3 , Z1 , D} ; (see diagram 14 of the figure 2);
⌣
15 ) ∅, Z 6 , Z 5 , Z 4 , Z 2 , Z1 , D ;
{
16 )
(see diagram 13 of the figure 2);
}
(see diagram 15 of the figure 2);
⌣
{∅, Z6 , Z5 , Z4 , Z3 , Z 2 , Z1 , D} ;
(see diagram 16 of the figure 2);
Proof: This corollary immediately follows from the lemmas 2.2.
The corollary is proved.
Theorem 2.1: Let D ∈ Σ3 ( X ,8) , Z7 ≠ ∅ and α ∈ BX ( D) . Binary relation α is an
idempotent relation of the semigroup BX ( D) iff binary relation α satisfies only
one conditions of the following conditions:
1) α = X × T , where T ∈ D ;
2 ) α = (YTα × T ) ∪ (YTα′ × T ′ ) , where T , T ′ ∈ D , T ⊂ T ′ , YTα ,YTα′ ∉{∅} , and satisfies the
conditions: YTα ⊇ T , YTα′ ∩ T ′ ≠ ∅ ;
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 ′′) ∪ (YTα′′′ ×T ′′′) ,
YTα , YTα′ , YTα′′ , YTα′′′ ∉{∅} ,
and
satisfies
where
the
T,T′,T′,T′′ ∈D,
conditions:
YTα ⊇ T ,
T ⊂ T ′ ⊂ T ′′ ⊂ T ′′′ ,
YTα ∪ YTα′ ⊇ T ′ ,
YTα ∪ YTα′ ∪ YTα′′ ⊇ T ′′ , YTα′ ∩ T ′ ≠ ∅ , YTα′ ∩ T ′′ ≠ ∅ , YTα′′′ ∩ T ′′′ ≠ ∅ ;
⌣
5) α = (Y7α ×Z7 ) ∪(YTα ×T ) ∪(YTα′ ×T′) ∪(YTα′′ ×T′′) ∪(Y0α × D) , where
⌣
Z7 ⊂ T ⊂ T ′ ⊂ T ′′ ⊂ D , Y7α , YTα , YTα′ , YTα′′ , Y0α ∉{∅} , and satisfies the conditions: Y7α ⊇ Z7 ,
Y7α ∪YTα ⊇ T , Y7α ∪ YTα ∪ YTα′ ⊇ T ′ , Y7α ∪ YTα ∪ YTα′ ∪ YTα′′ ⊇ T ′′ , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ,
⌣
YTα′ ∩ T ′′ ≠ ∅ , Y0α ∩ D ≠ ∅ ;
6 ) α = (YTα ×T ) ∪ (YTα′ ×T ′) ∪ (YTα′′ ×T ′′) ∪(YTα′∪T ′′ × (T ′ ∪T ′′) ) , where
T , T ′, T ′′ ∈ D , T ⊂ T ′ , T ⊂ T ′′ , T ′ \ T ′′ ≠ ∅ , T ′′ \ T ′ ≠ ∅ , YTα ,YTα′ ,YTα′ ∉{∅} and satisfies the
conditions: YTα ∪YTα′ ⊇ T ′ , YTα ∪YTα′′ ⊇ T ′′ , YTα′ ∩T′ ≠ ∅ , YTα′ ∩T ′′ ≠ ∅ ;
Maximal Subgroups of the Semigroup BX (D)…
83
7 ) α = (YTα ×T) ∪(YTα′ ×T′) ∪(YTα′ ×T′ ) ∪(YTα′′′ ×T′′) ∪(YTα′′∪T′′′ ×(T′ ∪T′′) ) , where, T ⊂ T ′ ⊂ T ′′ , T ⊂ T ′ ⊂ T ′′′ ,
YTα ,YTα′ ,YTα′ ,YTα′′ ∉{∅}
T ′′ \ T ′′′ ≠ ∅ ,
T ′′′ \ T ′′ ≠ ∅
YTα ∪YTα′ ⊇T′ ,
YTα ∪ YTα′ ∪ YTα′′ ⊇ T ′′ ,
and satisfies the conditions: YTα ⊇T
YTα ∪ YTα′ ∪ YTα′′′ ⊇ T ′′′
YTα′ ∩ T ′ ≠ ∅ ,
YTα′ ∩ T ′′ ≠ ∅ ,
YTα′′′ ∩ T ′′′ ≠ ∅ .
⌣
8) α = (Y7α × Z7 ) ∪ (YTα ×T ) ∪ (Y4α × Z4 ) ∪ (Y2α × Z2 ) ∪ (Y1α × Z1 ) ∪ (Y0α × D) ,
Y7 ,YT ,Y4 ,Y2 ,Y1 ∉{∅}
α
α
α
α
α
where
T ∈{Z6, Z5} ,
the conditions: Y7α ⊇ Z7 , Y7α ∪YTα ⊇ T ,
Y7α ∪ YTα ∪ Y4α ⊇ Z4 , Y7α ∪ YTα ∪ Y4α ∪ Y2α ⊇ Z 2 , Y7α ∪YTα ∪Y4α ∪Y1α ⊇ Z1 , YTα ∩ T ≠ ∅ , Y4α ∩Z4 ≠∅
Y2α ∩Z2 ≠∅, Y1α ∩ Z1 ≠ ∅ ;
and
satisfies
⌣
9 ) α = (Y7α × Z7 ) ∪ (Y5α × Z5 ) ∪ (Y4α × Z4 ) ∪ (Y3α × Z3 ) ∪ (Y1α × Z1 ) ∪ (Y0α × D) , where Z 7 ⊂ Z 5 ⊂ Z 3 ,
, Z 4 \ Z 3 ≠ ∅ Y7α ,Y5α , Y4α ,Y3α ,Y1α , Y0α ∉{∅} and satisfies the
conditions: Y7α ⊇ Z7 , Y7α ∪Y5α ⊇ Z5 , Y7α ∪Y5α ∪Y3α ⊇Z3 Y7α ∪Y5α ∪Y4α ⊇Z4 , Y5α ∩Z5 ≠∅,
⌣
Y3α ∩ Z3 ≠ ∅ , Y4α ∩ Z4 ≠ ∅ , Y0α ∩ D ≠ ∅ ;
Z7 ⊂ Z5 ⊂ Z 4 ,
Z3 \ Z 4 ≠ ∅
10 ) α = (YTα ×T ) ∪ (YTα′ × T ′) ∪ (YTα′′ × T ′′) ∪ (YTα′∪T ′′ × (T ′ ∪ T ′′) ) ∪ (YTα′′′ × T ′′′) , where T ⊂ T ′ , T ⊂ T ′′ ,
T ′ \ T ′′ ≠ ∅
T ′′ \ T ′ ≠ ∅ ,
,
T ′ ∪ T ′′ ⊂ T ′′′
YTα ,YTα′ ,YTα′′ ,YTα′′′ ∉{∅}
and satisfies the
conditions: YT ∪YT′ ⊇T′ , YT ∪YT′ ⊇T′′ , YT′ ∩T′ ≠∅ , YT ′′ ∩ T ′′ ≠ ∅ , YT′′′ ∩T ′′′ ≠ ∅ ;
α
α
α
α
α
α
α
⌣
11) α = (Y7α × Z7 ) ∪ (Y6α × Z6 ) ∪ (Y5α × Z5 ) ∪ (Y4α × Z4 ) ∪ (YTα ×T ) ∪ (Y0α × D) , where
T ∈{Z2 , Z1} , Y7α ,Y6α ,Y5α ,YTα ,Y0α ∉{∅} and satisfies the conditions:, Y7α ∪Y6α ⊇ Z6 , Y7α ∪ Y5α ⊇ Z5
⌣
Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ YTα ⊇ T , Y6α ∩ Z6 ≠ ∅ , Y5α ∩Z5 ≠∅ YTα ∩T ≠ ∅, Y0α ∩ D ≠ ∅ ;
12 ) α = (YTα ×T ) ∪ (YTα′ ×T ′) ∪ (YTα′′ ×T ′′) ∪ (YTα′∪T ′′ × (T ′ ∪T ′′) ) ∪ (YTα′′′ ×T ′′′) ∪ (YTα′∪T ′′∪T ′′′ × (T ′ ∪T ′′ ∪T ′′′) ) ,
, T ⊂ T ′′ , T ′ \ T ′′ ≠ ∅ , T ′′ \ T ′ ≠ ∅ , T ′′ ⊂ T ′′′ ,
(T′ ∪T′′) \ T′′′ ≠ ∅,
α α α α α
T′′′ \ (T′ ∪T′′) ≠ ∅ , , YT ,YT′ ,YT′′ ,YT′′′ ,YT′∪T′ ∪T′′′ ∉{∅} and satisfies the conditions:, YTα ∪ YTα′ ⊇ T ′ ,
T ⊂ T′
where
YTα ∪ YTα′′ ⊇ T ′′ YTα ∪ YTα′ ∪ YTα′′′ ⊇ T ′′′ , YTα′ ∩ T ′ ≠ ∅ , YTα′ ∩T′ ≠∅ YTα′′ ∩T′′′ ≠ ∅ ;
⌣
13 ) α = (Y7α ×Z7 ) ∪(Y5α ×Z5 ) ∪(Y4α ×Z4 ) ∪(Y3α ×Z3 ) ∪(Y2α ×Z2 ) ∪(Y1α ×Z1 ) ∪(Y0α ×D) , where
Z5 ⊂ Z3 ,
, Z4 \ Z3 ≠ ∅ , Z4 ⊂ Z2 ,
Z1 \ Z2 ≠ ∅ , Z2 \ Z1 ≠ ∅ ,
Y7 ,Y5 ,Y4 ,Y3 ,Y2 ,Y1 ,Y0 ∉{∅} and satisfies the conditions: Y7α ⊇ Z7 , Y7α ∪Y5α ⊇ Z5
Y7α ∪Y5α ∪Y3α ⊇ Z3 , Y7α ∪Y5α ∪Y4α ⊇ Z4 , Y7α ∪Y5α ∪Y4α ∪Y1α ⊇ Z1 , Y5α ∩Z5 ≠∅, Y3α ∩ Z3 ≠ ∅ ,
Y4α ∩Z4 ≠∅ Y1α ∩Z1 ≠ ∅;
α
α
α
α
Z5 ⊂ Z 4 ,
α
α
Z3 \ Z 4 ≠ ∅
α
⌣
14 ) α = (Y7α ×Z7 ) ∪(Y6α ×Z6 ) ∪(Y5α ×Z5 ) ∪(Y4α ×Z4 ) ∪(Y3α ×Z3 ) ∪(Y1α ×Z1 ) ∪(Y0α ×D) , where
α α α α α α α
Z 6 ⊂ Z 4 , Y7 ,Y6 ,Y5 ,Y4 ,Y3 ,Y1 ,Y0 ∉{∅}
and satisfies the conditions: Y7α ∪Y5α ⊇ Z5 , Y7α ∪Y6α ⊇ Z6 ,
⌣
Y7α ∪Y5α ∪Y3α ⊇ Z3 , Y5α ∩ Z5 ≠ ∅ , Y6α ∩ Z6 ≠ ∅ , Y3α ∩ Z3 ≠ ∅ Y0α ∩ D ≠ ∅ ;
84
Giuli Tavdgiridze
⌣
15 ) α =(Y7α ×Z7 ) ∪(Y6α ×Z6 ) ∪(Y5α ×Z5 ) ∪(Y4α ×Z4 ) ∪(Y2α ×Z2 ) ∪(Y1α ×Z1 ) ∪(Y0α ×D) , where
Y7α ,Y6α ,Y5α ,Y4α ,Y2α ,Y1α ∉{∅}
Y7α ∪Y6α ⊇ Z6 ,
and satisfies the conditions:
Y7α ∪Y6α ∪Y5α ∪Y4α ∪Y2α ⊇ Z2 ,
Y7α ∪Y6α ∪Y5α ∪Y4α ∪Y1α ⊇ Z1
Y6α ∩ Z6 ≠ ∅ ,
Y7α ∪Y5α ⊇ Z5 ,
Y5α ∩ Z5 ≠ ∅ ,
Y2α ∩ Z 2 ≠ ∅ Y1α ∩Z1 ≠ ∅;
⌣
16 ) α =(Y7α ×Z7 ) ∪(Y6α ×Z6 ) ∪(Y5α ×Z ) ∪(Y4α ×Z4 ) ∪(Y3α ×Z3 ) ∪(Y2α ×Z2 ) ∪(Y1α ×Z1) ∪(Y0α ×D) , where,
Y7α ,Y6α ,Y5α,Y4α,Y3α ,Y2α ,Y1α ,Y0α ∉{∅} and satisfies the conditions: Y7α ⊇ Z7 Y7α ∪Y5α ⊇ Z5 , Y7α ∪Y6α ⊇ Z6 ,
Y7α ∪Y5α ∪Y3α ⊇ Z3 , Y7α ∪Y5α ∪Y6α ∪Y4α ∪Y2α ⊇ Z2 , Y5α ∩ Z5 ≠ ∅ ,
Y6α ∩ Z6 ≠ ∅ , Y3α ∩ Z3 ≠ ∅ ,
Y2α ∩ Z 2 ≠ ∅ ;
Proof: The statements 1), 2), 3), 4) and 5) immediately follows from the
Corollary [1] 13.1.1, [2] 13.1.1, the statements 6) -11) immediately follows from
the Corollary [1] 13.3.1, [2] 13.3.1, the statement 15 ) immediately follows from
the Theorem [1] 13.7.2, [2] 13.7.2.
⌣
Now we will prove statement 16. D = {Z7 , Z6 , Z5 , Z4 , Z3 , Z2 , Z1 , D} to begin with, we
note that D is an XI − semilattice of unions(see lemma 2.2) then It is easy to
see, that the set D(α ) = {Z7 , Z6 , Z5 , Z4 , Z3 , Z2 , Z1} is a generating set of the semilattice D .
Then the following equalities are hold:
ɺɺ (α ) = {Z } , D
ɺɺ (α ) = {Z , Z } , D
ɺɺ (α ) = {Z , Z } , D
ɺɺ (α ) = {Z , Z , Z , Z } ,
D
7
7
6
7
5
7
6
5
4
Z7
Z6
Z5
Z4
ɺɺ
ɺɺ
ɺɺ
D (α ) Z = {Z7 , Z5 , Z3} , D (α ) Z = {Z7 , Z6 , Z5 , Z4 , Z2 } , D (α ) Z = {Z7 , Z6 , Z5 , Z4 , Z3 , Z1} .
3
2
1
By statement b ) of the Theorem 1.3 follows that the following conditions are
true:
Y7α ⊇ Z7 , Y7α ∪ Y6α ⊇ Z6 , Y7α ∪ Y5α ⊇ Z5 , Y7α ∪ Y6α ∪ Y5α ∪ Y4α ⊇ Z4
Y7α ∪ Y5α ∪ Y3α ⊇ Z3 , Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y2α ⊇ Z2 ,
Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y3α ∪ Y1α ⊇ Z1 ,
For last conditions we have:
Y7α ∪ Y6α ∪ Y5α ∪ Y4α = (Y7α ∪ Y6α ) ∪ (Y7α ∪ Y5α ) ∪ Y4α ⊇
⊇ T6 ∪ T5 ∪ Y4α = Z6 ∪ Z5 ∪ Y4α = Z4 ∪ Y4α ⊇ Z4 ,
Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y3α ∪ Y1α = (Y7α ∪ Y5α ∪ Y3α ) ∪ (Y7α ∪ Y6α ) ∪ Y4α ∪ Y1α ⊇
⊇ Z3 ∪ T6 ∪ Y4α ∪ Y1α = Z3 ∪ Z6 ∪ Y4α ∪ Y1α = Z1 ∪ Y4α ∪ Y1α ⊇ Z1.
Since ϕ is isomorphism. Further, it is to see, that the following equality are true:
(
ɺɺ
l (D
ɺɺ
l (D
ɺɺ
l (D
) (
)
(
)
ɺɺ \ {Z } ) = Z , Z \ l ( D
ɺɺ , Z ) = Z \ Z ≠ ∅;
, Z ) = ∪(D
ɺɺ \ { Z } ) = ∪ {Z , Z , Z } = Z , Z \ l ( D
ɺɺ , Z ) = Z \ Z
, Z ) = ∪(D
ɺɺ \ {Z } ) = ∪{Z , Z } = Z , Z \ l ( D
ɺɺ , Z ) = Z \ Z ≠ ∅;
, Z ) = ∪( D
ɺɺ , Z = ∪ D
ɺɺ \ {Z } = Z , Z \ l D
ɺɺ , Z = Z \ Z ≠ ∅;
l D
Z6
6
Z6
6
7
6
Z6
6
6
7
Z5
Z4
Z3
5
Z5
5
4
Z4
4
3
Z3
3
7
5
Z5
7
7
6
5
5
5
5
5
4
3
7
4
Z3
4
Z4
3
3
4
5
4
= ∅;
Maximal Subgroups of the Semigroup BX (D)…
(
ɺɺ
l (D
85
) (
)
(
)
ɺɺ
ɺɺ
, Z ) = ∪ ( D \ {Z } ) = ∪ {Z , Z , Z , Z , Z } = Z , Z \ l ( D , Z ) = Z
ɺɺ , Z = ∪ D
ɺɺ \ {Z } = ∪{Z , Z , Z , Z } = Z , Z \ l D
ɺɺ , Z = Z \ Z ≠ ∅;
l D
Z2
2
Z2
2
7
6
5
4
4
2
Z2
2
2
4
Z1
1
Z1
1
7
6
5
4
3
1
1
Z1
1
1
\ Z1 = ∅;
We have the elements Z 6 , Z 5 , Z 3 and Z 2 are nonlimiting elements of the sets
ɺɺ (α ) , D
ɺɺ (α ) , D
ɺɺ (α ) and D
ɺɺ (α ) respectively. By statement c ) of the Theorem
D
z
Z
Z
Z
6
5
3
2
1.1.3 it follows, that the conditions
Y2α ∩ Z 2 ≠ ∅ are hold.
Y6α ∩ Z6 ≠ ∅ , Y5α ∩ Z 5 ≠ ∅ , Y3α ∩ Z3 ≠ ∅
and
Therefore the following conditions are hold:
Y7α ⊇ Z 7 , Y7α ∪ Y6α ⊇ Z 6 , Y7α ∪ Y5α ⊇ Z 5 ,
Y7α ∪ Y5α ∪ Y3α ⊇ Z 3 , Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y2α ⊇ Z 2 ,
Y6α ∩ Z 6 ≠ ∅, Y5α ∩ Z 5 ≠ ∅, Y3α ∩ Z 3 ≠ ∅, Y2α ∩ Z 2 ≠ ∅.
(2.11)
In the same manner it can be proved that any subsemilattice of the semilattice D
having diagrams 13 and 14.
Theorem is proved.
Corollary 2.2: Let D ∈ Σ3 ( X ,8 ) , Z 7 = ∅ and α ∈ BX ( D ) . Binary relation α is an
idempotent relation of the semigroup BX ( D ) iff binary relation α satisfies only
one conditions of the following conditions:
1) α = ∅;
2) α = (Y7α × ∅ ) ∪ (YTα × T ) , where ∅ ≠ T ∈ D , YTα ∉ {∅} , and satisfies the conditions:
YTα ⊇ ∅
, YTα′ ∩ T ′ ≠ ∅ ;
3) α = (Y7α × ∅ ) ∪ (YTα × T ) ∪ ( YTα′ × T ′ ) , where ∅ ≠ T ⊂ T ′ ∈ D , YTα , YTα′ ∉ {∅} , and satisfies
the conditions: Y7α ⊇ ∅ , Y7α ∪ YTα ⊇ T , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ;
4 ) α = (Y7α ×∅) ∪ (YTα × T ) ∪ (YTα′ × T ′) ∪ (YTα′′ × T ′′) , where
∅ ≠ T ⊂ T ′ ⊂ T ′′ ∈ D ,
YTα , YTα′ , YTα′′ ∉ {∅} ,
and satisfies the conditions:
α
α
α
α
α
α
Y7 ∪ YT ⊇ T , Y7 ∪ YT ∪ YT ′ ⊇ T ′ , YT ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ ;
Y7α ⊇ ∅
,
⌣
5) α = (Y7α ×∅) ∪(YTα ×T ) ∪(YTα′ ×T ′) ∪(YTα′′ ×T ′′) ∪(Y0α × D) , where
⌣
∅ ≠ T ⊂ T ′ ⊂ T ′′ ⊂ D , YTα , YTα′ , YTα′′ , Y0α ∉ {∅} , and satisfies the conditions: Y7α ⊇ ∅ ,
Y7α ∪ YTα ⊇ T
, Y7α ∪ YTα ∪ YTα′ ⊇ T ′ , Y7α ∪ YTα ∪ YTα′ ∪ YTα′′ ⊇ T ′′ , YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ ,
⌣
α
YTα′′ ∩ T ′′ ≠ ∅ , Y0 ∩ D ≠ ∅ ;
6 ) α = (Y7α ×∅) ∪ (YTα ×T ) ∪(YTα′ ×T ′) ∪(YTα∪T ′ ×(T ∪T ′) ) , where
86
Giuli Tavdgiridze
∅ ≠ T ,T ′ ∈ D ,
α
α
Y7 ∪ YT ⊇ T
T′\T ≠ ∅ ,
α
T \T′ ≠ ∅ ,
α
α
YTα ,YTα′ ∉{∅}
and satisfies the conditions:
α
, Y7 ∪ YT ′ ⊇ T ′ , YT ∩T ≠ ∅ , YT ′ ∩ T ′ ≠ ∅ ;
7 ) α = (Y7α ×∅) ∪(YTα ×T ) ∪(YTα′ ×T′) ∪(YTα′′ ×T′ ) ∪(YTα′∪T′ ×(T′ ∪T′ ) ) , where, ∅ ≠ T ⊂ T ′ , ∅ ≠ T ⊂ T ′′ ,
T ′ \ T ′′ ≠ ∅ , T ′′ \ T ′ ≠ ∅ YTα ,YTα′ ,YTα′′ ∉{∅}
and satisfies the conditions: Y7α ⊇∅ Y7α ∪YTα ⊇T ,
Y7α ∪ YTα ∪ YTα′ ⊇ T ′ , Y7α ∪ YTα ∪ YTα′′ ⊇ T ′′ YTα ∩ T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ .
⌣
8) α = (Y7α × Z7 ) ∪ (YTα ×T ) ∪ (Y4α × Z4 ) ∪ (Y2α × Z2 ) ∪ (Y1α × Z1 ) ∪ (Y0α × D) , where
T ∈{Z6 , Z5} , YTα ,Y4α , Y2α ,Y1α ∉{∅} and satisfies the conditions: Y7α ⊇ ∅ , Y7α ∪ YTα ⊇ T ,
Y7α ∪ YTα ∪ Y4α ⊇ Z 4 , Y7α ∪ YTα ∪ Y4α ∪ Y2α ⊇ Z 2 , Y7α ∪YTα ∪Y4α ∪Y1α ⊇ Z1 , YTα ∩ T ≠ ∅ , Y4α ∩Z4 ≠∅
Y2α ∩ Z2 ≠ ∅ , Y1α ∩ Z1 ≠ ∅ ;
⌣
9 ) α = (Y7α × Z7 ) ∪ (Y5α × Z5 ) ∪ (Y4α × Z4 ) ∪ (Y3α × Z3 ) ∪ (Y1α × Z1 ) ∪ (Y0α × D) , where
∅ ≠ Z5 ⊂ Z4 ,
Z3 \ Z 4 ≠ ∅ ,
α
Y7 ⊇ ∅
conditions:
α
Y7 ∪Y5 ⊇ Z5 ,
,
Y5α ,Y4α ,Y3α , Y1α ,Y0α ∉{∅}
Z4 \ Z3 ≠ ∅
α
⌣
α
Y3α ∩ Z 3 ≠ ∅ , Y4α ∩ Z4 ≠ ∅ , Y0 ∩ D ≠ ∅ ;
∅ ≠ Z5 ⊂ Z3 ,
α
α
α
Y7 ∪Y5 ∪Y3 ⊇ Z3
and satisfies the
Y7 ∪Y5 ∪Y4α ⊇ Z4 , Y5α ∩ Z5 ≠ ∅ ,
α
α
10 ) α = (Y7α ×∅) ∪ (YTα ×T ) ∪ (YTα′ ×T ′) ∪ (YTα∪T ′ × (T ∪ T ′) ) ∪ (YTα′′ ×T ′′) , where,
, T ′ \ T ≠ ∅ , T ∪ T ′ ⊂ T ′′ YTα , YTα′ ,YTα′′ ∉{∅} and satisfies the
conditions: Y7α ∪YTα ⊇ T , Y7α ∪YTα′ ⊇ T ′ , YTα ∩T ≠ ∅ , YTα′ ∩ T ′ ≠ ∅ , YTα′′ ∩ T ′′ ≠ ∅ ;
∅ ≠ T , T ′′ , T \ T ′ ≠ ∅
⌣
11) α = (Y7α ×∅) ∪ (Y6α × Z6 ) ∪ (Y5α × Z5 ) ∪ (Y4α × Z4 ) ∪ (YTα ×T ) ∪ (Y0α × D) , where
T ∈{Z2 , Z1} , Y6α ,Y5α ,YTα ,Y0α ∉{∅} and satisfies the conditions:, Y7α ∪Y6α ⊇ Z6 , Y7α ∪ Y5α ⊇ Z 5
⌣
α
Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ YTα ⊇ T , Y6α ∩ Z 6 ≠ ∅ , Y5α ∩Z5 ≠∅ YTα ∩T ≠ ∅ , Y0 ∩ D ≠ ∅ ;
12 ) α = (Y7α ×∅) ∪ (YTα ×T ) ∪ (YTα′ ×T ′) ∪ (YTα∪T ′ × (T ∪T ′) ) ∪ (YTα′′ ×T ′′) ∪ (YTα∪T ′∪T ′′ × (T ∪T ′ ∪T ′′) ) ,
where, ∅ ≠ T , T ′ , T \ T ′ ≠ ∅ , T ′ \ T ≠ ∅ , T ⊂ T ′′ , (T ∪T′) \ T′′ ≠ ∅ , T′′ \ (T ∪T′) ≠ ∅ , ,
YTα ,YTα′ ,YTα′′ ,YTα∪T′∪T′′ ∉{∅} and satisfies the conditions:, Y7α ∪ YTα ⊇ T , Y7α ∪ YTα′ ⊇ T ′
Y7α ∪ YTα′ ∪ YTα′′ ⊇ T ′′ , YTα ∩ T ≠ ∅ , YTα′ ∩T′ ≠∅ YTα′′ ∩T ′′ ≠ ∅ ;
⌣
13) α = (Y7α ×∅) ∪(Y5α ×Z5 ) ∪(Y4α ×Z4 ) ∪(Y3α ×Z3 ) ∪(Y2α ×Z2 ) ∪(Y1α ×Z1 ) ∪(Y0α ×D) , where
Z5 ⊂ Z3 ,
Z5 ⊂ Z 4 ,
Z3 \ Z 4 ≠ ∅ ,
Y5 ,Y4 ,Y3 ,Y2 ,Y1 ,Y0 ∉{∅}
α
α
α
α
α
α
Z4 \ Z3 ≠ ∅ ,
Z4 ⊂ Z2 ,
Z1 \ Z2 ≠ ∅ ,
α
Z2 \ Z1 ≠ ∅ ,
α
α
,
satisfies the conditions: Y7 ⊇ ∅ , Y7 ∪Y5 ⊇ Z5
Y7 ∪ Y5 ∪ Y3 ⊇ Z3 , Y7 ∪ Y5 ∪ Y4 ⊇ Z4 , Y7α ∪ Y5α ∪ Y4α ∪ Y1α ⊇ Z1 , Y5α ∩Z5 ≠∅, Y3α ∩ Z 3 ≠ ∅ ,
Y4α ∩Z4 ≠∅ Y1α ∩ Z1 ≠ ∅ ;
α
α
α
and
α
α
α
⌣
14 ) α = (Y7α ×∅) ∪(Y6α ×Z6 ) ∪(Y5α ×Z5 ) ∪(Y4α ×Z4 ) ∪(Y3α ×Z3 ) ∪(Y1α ×Z1 ) ∪(Y0α ×D) , where,
Maximal Subgroups of the Semigroup BX (D)…
Z 6 ⊂ Z 4 , Y6α ,Y5α ,Y4α ,Y3α ,Y1α ,Y0α ∉{∅}
87
and satisfies the conditions: Y7α ∪Y5α ⊇ Z5 , Y7α ∪Y6α ⊇ Z6 ,
⌣
α
Y7α ∪ Y5α ∪ Y3α ⊇ Z3 , Y5α ∩ Z 5 ≠ ∅ , Y6α ∩ Z 6 ≠ ∅ , Y3α ∩ Z 3 ≠ ∅ Y0 ∩ D ≠ ∅ ;
⌣
15) α = (Y7α ×∅) ∪(Y6α ×Z6 ) ∪(Y5α ×Z5 ) ∪(Y4α ×Z4 ) ∪(Y2α ×Z2 ) ∪(Y1α ×Z1 ) ∪(Y0α ×D) , where
Y6α ,Y5α ,Y4α ,Y2α , Y1α ∉{∅}
and satisfies the conditions: Y7α ∪Y6α ⊇ Z6 , Y7α ∪Y5α ⊇ Z5 ,
Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y2α ⊇ Z2 ,
Y7α ∪ Y6α ∪ Y5α ∪ Y4α ∪ Y1α ⊇ Z1
Y6α ∩ Z 6 ≠ ∅ ,
Y5α ∩ Z 5 ≠ ∅ ,
Y2α ∩ Z 2 ≠ ∅ Y1α ∩ Z1 ≠ ∅ ;
⌣
16 ) α =(Y7α ×∅) ∪(Y6α ×Z6) ∪(Y5α ×Z ) ∪(Y4α ×Z4) ∪(Y3α ×Z3) ∪(Y2α ×Z2) ∪(Y1α ×Z1) ∪(Y0α ×D) , where,
Z5 ⊂Z4, Y6α ,Y5α ,Y4α ,Y3α ,Y2α ,Y1α ,Y0α ∉{∅} and satisfies the conditions: Y7α ⊇ Z7
Y7α ∪Y6α ⊇ Z6 ,
Y7α ∪Y5α ∪Y3α ⊇ Z3 ,
Y7α ∪Y5α ∪Y6α ∪Y4α ∪Y2α ⊇ Z2 ,
Y5α ∩ Z 5 ≠ ∅ ,
Y7α ∪Y5α ⊇ Z5 ,
Y6α ∩ Z 6 ≠ ∅ ,
Y3α ∩ Z 3 ≠ ∅ , Y2α ∩ Z 2 ≠ ∅ ;
Proof: This corollary immediately follows from the theorem 2.1.
The corollary is proved.
Lemma 3.1: The number of automorphisms of those semilattices, which are
defined by the diagrams 1), 2), 3), 4), 5), 12), 13), 14) and 16) in fig. 2 is equal to
1, those semilattices which are defined diagrams 6), 7), 8), 9), 10) and 11) in fig.
2 is equal to 2 and that semilattice which is defined by the diagram 15) in fig. 2 is
equal to 4.
Proof: Let us prove the given lemma in case of the semilattice which is defined
by the diagram 16) in fig. 2. The proofs of the other cases are almost identical of
the ongoing one.
T0
T1
T2
T4
T3
T5
T6
T7
Fig. 5
Suppose Q = {T7 , T6 , T5 , T4 , T3 , T2 , T1 , T0 } (see fig. 5). Our purpose is to prove that the
number of automorphisms of the given semilattice Q is equal to 1. Indeed, if
ɺɺ (see
Ti ( ni , mi ) denote the element Ti of the semilattice Q such that ni = QTi , mi = Q
Ti
88
Giuli Tavdgiridze
(1.1)) and ϕ is arbitrary automorphism of the semilattice Q , then ϕ (Ti ) = Tj only if
(
)
ni = n j and mi = m j , i.e. ( ni , mi ) = n j , m j .
For the semilattice Q we have:
T0 = T0 (1,8) , T1 = T1 ( 2, 6 ) , T2 = T2 ( 2,5) , T3 = T3 ( 3,3) ,
T4 = T4 ( 4, 4 ) , T5 = T5 ( 6, 2 ) , T6 = T6 ( 5, 2 ) , T7 = T6 ( 8,1) ,
So, we have ϕ (Ti ) = Ti for every i = 0,1, 2,..., 6 , because ( ni , mi ) = ( n j , m j ) equality is
satisfied if and only if when i = j .
Therefore the number of automorphisms of the given semilattice Q is equal to 1.
Lemma is proved.
Denote by the symbol GX ( D,ε ) a maximal subgroup of the semigroup BX ( D)
whose unit is an idempotent binary relation ε of the semigroup BX ( D ) .
Theorem 3.1: For any idempotent binary relation ε of the semigroup BX ( D ) ,
the order a subgroup G X ( D , ε ) of the semigroup BX ( D ) is one, or two, or four.
Proof: Let ε be an arbitrary binary relation of the semigroup BX ( D ) . Now, if we
denote by Φ the group of all complete automorphisms of the semilattice V ( D, ε ) ,
then by virtue of Theorem 1.3. We have that the groups GX ( D, ε ) and Φ are antiisomorphic.
To prove the theorem, we will consider the following cases with regard to the
idempotent binary relation ε :
1)
relation ε
satisfies the conditions
1) , 2 ) , 3) , 4 ) , 5) , 12 ) , 13) , 14 ) and 16 ) of the Theorem 2.1 and theorem
2.2., then the diagram of the semilattice V ( D, ε ) has form 1, 2, 3, 4, 5, 12,
13, 14 and 16 in Fig. 2. Therefore in this case the number of
automorphisms of the semilattice V ( D, ε ) is equal to one (see Lemma 3.1).
Now, taking into account Theorem 1.3, we obtain GX ( D, ε ) = 1 .
2)
The idempotent binary relation ε satisfies the conditions 6 ) , 7 ) , 8 ) , 9), 10)
and 11) of the Theorem 2.1 and theorem 2.2. So that the diagram of the
emilattice V ( D, ε ) has form 6-11 in Fig. 2. Therefore in this case the
number of automorphisms of the semilattice V ( D, ε ) is equal to two (see
The
idempotent
binary
Maximal Subgroups of the Semigroup BX (D)…
89
Lemma 3.1). Now, taking into account Theorem 1.3, we obtain
GX ( D, ε ) = 2 .
3)
The idempotent binary relation is of type 15 ) so that the number of
automorphisms of the semilattice V ( D, ε ) has form 15 in Fig. 2. Clearly, in
this case the number of automorphisns of the semilattice V ( D, ε ) is four
(see Lemma 3.1). Now, taking into account Theorem 1.3, we obtain
GX ( D, ε ) = 4 .
Since the diagrams shown in Fig. 2 exhaust all the diagrams of the
XI − subsemilattices of the semilattice D , the idempotent binary relations of the
semigroup BX ( D ) are exhausted by types 1-16 from Theorem 2.1 and Theorem
2.2. Hence it follows that for any idempotent ε of the semigroup BX ( D ) , the
order a subgroup GX ( D, ε ) of the semigroup BX ( D ) is one, or two, or four.
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[5]
[6]
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