Received 13 February 2015; accepted 27 December 2015; published 30 December 2015
1. Introduction
An element
taken from the semigroup
is called a regular element of
, if in
there exists an element
such that
(see [1] [2] ).
Definition 1.1. We say that a complete X-semilattice of unions D is an XI-semilattice of unions if it satisfies the following two conditions:
1)
for any
;
2)
for any nonempty element Z of D (see ( [1] , Definition 1.14.2), ( [2] , Definition 1.14.2)).
Definition 1.2. The one-to-one mapping
between the complete X-semilattices of unions
and
is called a complete isomorphism if the condition
is fulfilled for each nonempty sub-
set D1 of the semilattice D' (see ( [1] , Definition 6.3.2), ( [2] , Definition 6.3.2) or [3] ).
Definition 1.3. Let
be some binary relation of the semigroup
. We say that the complete isomorphism
between the complete semilattices of unions Q and
is a complete
-isomorphism if
1)
;
2)
for
and
for all
(see ( [1] , Definition 6.3.3), ( [2] , Definition 6.3.3) or [3] ).
Theorem 1.1. Let
be the set of all regular elements of the semigroup
. Then the following statements are true:
1)
for any
and
;
2)
;
3) If X is a finite set, then
(see ( [1] , Theorem 6.3.6) or ( [2] , Theorem 6.3.6) or [3] ).
2. Result
By the symbol
we denote the class of all X-semilattices of unions whose every element is isomorphic to an X-semilattice of form
, where
(see [4] ).
Now assume that
. We introduce the following notation:
1)
, where
(see diagram 1 in Figure 1);
2)
, where
and
(see diagram 2 in Figure 1);
3)
, where
and
(see diagram 3 in Figure 1);
4)
, where
and
(see diagram 4 in Figure 1);
5)
, where
,
,
and
,
(see diagram 5 in Figure 1);
6)
, where
,
,
and
,
(see diagram 6 in Figure 1);
7)
, where
,
,
and
,
(see diagram 7 in Figure 1);
8)
, where
,
,
, ![]()
,
,
and
,
(see diagram 8 in Figure 1);
9)
, where
,
,
and
(see diagram 9 in Figure 1);
10)
, where
,
,
,
and
(see diagram 10 in Figure 1);
11)
, where
and
(see diagram 11 in Figure 1);
12)
, where
(see diagram 12 in Figure 1);
13)
, where
,
,
,
,
and
(see diagram 13 in Figure 1);
14)
, where
,
,
,
,
,
,
and
(see diagram 14 in Figure 1);
15)
, where
,
,
,
,
,
,
,
,
,
and
(see diagram 15 in Figure 1);
16)
, where
(see diagram 16 in Figure 1).
Denote by the symbol
the set of all XI-subsemilattices of the semilattice D isomorphic to
. Assume that
and denote by the symbol
the set of all regular elements
of the semigroup
, for which the semilattices
and
are mutually
isomorphic and
.
Definition 1.4. Let the symbol
denote the set of all XI-subsemilattices of the semilattice D.
Let, further,
and
. It is assumed that
if and only if there exists some complete isomorphism
between the semilattices D and
. One can easily verify that the binary relation
is an equivalence relation on the set
.
Let the symbol
denote the
-class of equivalence of the set
, where every element is isomorphic to the X-semilattice
and
(see ( [1] , Definition 6.3.5), ( [2] , Definition 6.3.5) or [5] ).Lemma 1.1. If X be a finite set and
, then the following equalities are true:1)
;2)
;3)
;
Figure 1. Diagrams of Qi, (i = 1, 2, 3, ∙∙∙, 16).
4)
;
5)
;
6)
;
7)
;
8)
;
9)
;
10)
;
11)
;
12)
;
13)
;
14)
;
15)
;
16)
.
Proof. The statements 1)-4) immediately follows from the Theorem 13.1.2 in [1] , Theorem 13.1.2 in [2] ; the statements 5)-7) immediately follows from the Theorem 13.3.2 in [1] , Theorem 13.3.2 in [2] ; the statement 8) immediately follows from the Theorem 13.7.5 in [1] , Theorem 13.7.5 in [2] ; the statements 9)-11) immediately follows from the Theorem 13.2.2 in [1] , Theorem 13.2.2 in [2] ; the statement 12) immediately follows from the Theorem 13.5.2 in [1] , Theorem 13.5.2 in [2] ; the statements 13), 14) immediately follows from the Theorem 13.4.2 in [1] , Theorem 13.4.2 in [2] , the statement 15) immediately follows from the Corollary 13.10.2 in [1] and the statement 16) immediately follows from the Theorem 2.2 in [4] .
The lemma is proved.
Lemma 1.2. Let
and
Then the following sets exhibit all XI-subsemilattices of the given semilattice D:
1)
, (see diagram 1 of the Figure 1);
2)
,
(see diagram 2 of the Figure 1);
3)
,
(see diagram 3 of the Figure 1);
4)
,
(see diagram 4 of the Figure 1);
5)
, (see diagram 5 of the Figure 1);
6)
(see diagram 6 of the Figure 1);
7)
(see diagram 7 of the Figure 1);
8)
(see diagram 8 of the Figure 1);
Proof. The statements 1)-4) immediately follows from the Theorems 11.6.1 in [1] , 11.6.1 in [2] or in [5] , the statements 5)-7) immediately follows from the Theorems 11.6.3 in [1] , 11.6.3 in [2] or in [5] and the statement 8) immediately follows from the Theorems 11.7.2 in [1] .
The lemma is proved.
Theorem 2.1. Let
and
Then a binary relation
of the semigroup
that has a quasinormal representation of the form to be given below is a regular element of this semigroup iff there exist a complete
-isomorphism
of the semilattice
on some subsemilattice D' of the semilattice D that satisfies at least one of the following conditions:
1)
, where
;
2)
, where
,
,
and satisfies the conditions:
,
;
3)
, where
,
,
and satisfies the conditions:
,
,
,
;
4)
, where
,
,
and satisfies the conditions:
,
,
,
,
,
;
5)
, where
,
,
, ![]()
and satisfies the conditions:
,
,
,
;
6)
, where
,
,
,
,
,
and satisfies the conditions
,
,
,
,
,
,
;
7)
, where
,
,
,
,
,
and satisfies the conditions
,
;
8)
, where
,
,
, ![]()
,
,
,
,
,
and satisfies the conditions
,
, ![]()
![]()
,
.
Proof. In this case, when
, from the Lemma 1.2 it follows that diagrams 1-8 given in Figure 1 exhibit all diagrams of XI-subsemilattices of the semilattices D, a quasinormal representation of regular elements of the semigroup
, which are defined by these XI-semilattices, may have one of the forms listed above. Then the validity of the statements 1)-4) immediately follows from the Theorem 13.1.1 in [1] , Theorem 13.1.1 in [2] , the statements 5)-7) immediately follows from the Theorem 13.3.1 in [1] , Theorem 13.3.1 in [2] and the statement 8) immediately follows from the Theorem 13.7.1 in [1] , Theorem 13.7.1 in [2] .
The theorem is proved.
1) Lemm 2.1. Let
and
If by
denoted all regular elements of the semigroup
satisfying the condition 1) of the Theorem 2.1, then
.
Proof. According to the definition of the semilattice D we have
.
Assume that
.
Then from Theorem 1.1 we obtain
.
From this and by the statement 1) of Lemma 1.1 we obtain
The lemma is proved.
2) Now let binary relation
of the semigroup
satisfying the condition 2) of the Theorem 2.1. In this case we have
, where
and
. By definition of the semilattice D follows that
If the equalities
Then from Theorem 1.1 we obtain:
. (2.1)
Lemma 2.2. Let
and
If
is a finite set, then
.
Proof. Let
, then
and
. If
then quasinormal representation of a binary relation
has form
for some
,
and by statement 2) of the Theorem 2.1 satisfies the conditions
and
. Since
and
are minimal elements of the semilattice D, we have
or
.
On the other hand,
is maximal elements of the semilattice D, therefore
. Hence, in the considered case, only one of the following two conditions is fulfilled:
and
or
and
.
i.e.,
or
. Hence, using equality (2.1), we obtain
(2.2)
Now, let
then
(2.3)
Of this we have that
, i.e.
and
.
Of the other hand if
, then
and the condition (2.3) is hold. Of this follows
that
, i.e.
. Therefore the equality
(2.4)
is fulfilled. Now of the equalities (2.2) and (2.4) follows the following equality
The lemma is proved.
Lemma 2.3. Let
and
. If X is a finite set, then
.
Proof: It is easy to see
and
, then by statement 2) of the Lemma 1.1 and by Lemma 2.2 we obtain the validity of Lemma 2.3.
The lemma is proved.
3) Let binary relation
of the semigroup
satisfying the condition 3) of the Theorem 2.1. In this case we have
, where
and
. By definition of the semilattice D follows that
Now if
Then from Theorem 1.1 we obtain:
. (3.1)
Lemma 3.1. Let
and
. If X is a finite set, then
Proof. Let ![]()
be arbitrary element of the set
and
. Then quasinormal representation of a binary relation
has form
for some
,
and by statement 3) of the Theorem 2.1 satisfies the conditions
,
,
and
. By definition of the semilattice D we have
or
and
. Of this and by the conditions
,
,
,
we have:
or
i.e.
or
, where
and
. Hence, using equality (3.1), we obtain
. (3.2)
Now we show that the following equalities are true:
(3.3)
For this we consider the following case.
a) If
, then
It follows that
and
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equalities are hold:
,
,
,
,
,
,
,
.
b) If
, then
It follows that
and
. But the inequality
contradic- tion of the condition that representation of binary relation
is quazinormal. So, the equality
is true.
The similar way we can show that the following equalities are hold:
,
,
,
,
.
c) If
, then
(3.4)
It follows that
(3.5)
i.e.,
. So, the inclusion
is hold.
Of the other hand, if
, then the conditions (3.4) and (3.5) are fulfilled, i.e.
and
. Therefore, the equality
is true.
The similar way we can show that the following equality is hold:
.
d) If
, then
(3.6)
It follows that
(3.7)
i.e.,
. So, the inclusion
is hold.
Of the other hand, if
, then the conditions (3.6) and (3.7) are fulfilled, i.e.,
and
. Therefore, the equality
is true.
The similar way we can show that the following equalities are hold:
,
,
,
.
We have that all equalities of (3.3) are true. Now, by the equalities of (3.2) and (3.3) we obtain the validity of Lemma 3.1.
The lemma is proved.
Lemma 3.2. Let
,
, where
and
. If quasinormal representation of binary relation
of the semigroup
has a form
for some
,
and
, then
iff
.
Proof. If
, then by statement 3) of the Theorem 2.1 we have
(3.8)
Of the last condition we have
, (3.9)
since
and
by assumption.
Of the other hand, if the conditions of (3.9) are hold, then also hold the conditions of (3.8), i.e.
.
The lemma is proved.
Lemma 3.3. Let
and
. If X is a finite set, then the following equalities are hold:
Proof. Let
, where
and
. Assume that
and a quasinormal representation of a regular binary relation
has the form
for some
,
and
. Then by state- ment c) of the Theorem 3.1.1, we have
(3.10)
Let
is a mapping of the set X in the semilattice D satisfying the conditions
for all
.
,
,
and
are the restrictions of the mapping
on the sets
,
,
,
respectively. It is clear, that the intersection disjoint elements of the set
is empty set, and
.
We are going to find properties of the maps
,
,
,
.
1)
. Then by the properties (3.10) we have
, i.e.,
and
by definition of the set
. Therefore
for all
.
2)
. Then by the properties (3.10) we have
, i.e.,
and
by definition of the sets
and
. Therefore
for all
.
Preposition we have that
, i.e.
for some
. If
, then
. Therefore
. That is contradict of the equality
, while
by definition of the semilattice D. Therefore
for some
.
3)
. Then by properties (3.10) we have
, i.e.,
and
by definition of the sets
,
and
. Therefore
for all
.
Preposition we have that
, i.e.
for some
. If
. Then
. Therefore
by definition of the set
and
. We have contradict of the equality
. Therefore
for some
.
4)
. Then by definition quasinormal representation binary relation
and by property (3.10) we have
, i.e.
by definition of the sets
and
. Therefore
for all
.
Therefore for every binary relation
exist ordered system
. It is obvious that for disjoint binary relations exist disjoint ordered systems.
Now, let
,
,
,
are such mappings, which satisfying the conditions:
5)
for all
;
6)
for all
and
for some
;
7)
for all
and
for some
;
8)
for all
.
Now we define a map f of a set X in the semilattice D, which satisfies the condition:
Let
,
,
and
. Then binary relation
can be representation by form
and satisfying the conditions:
(By suppose
for some
and
for some
), i.e., by lemma 2.5 we have that
.
Therefore for every binary relation
and ordered system
exist one to one mapping.
By ( [1] , Theorem 1.18.2) the number of the mappings
are respectively:
1,
,
,
.
Note that the number
does not depend on choice of chains
![]()
of the semilattice D. Sins the number of such different chains of the semilattice D is equal to 18, for arbitrary
where
, the number of regular elements of the set
is equal to
.
Note that the number
does not depend on choice of chains
![]()
of the semilattice D. Since the number of such different chains of the semilattice D is equal to 18, for arbitrary
where
, the number of regular elements of the set
is equal to
. Therefore, we obtain the validity of Lemma 3.3.
The lemma is proved.
Lemma 3.4. Let X be a finite set,
and
. If by
denoted all regular elements of the semigroup
satisfying the condition 3) of the Theorem 2.1, then
Proof: It is easy to see
and
, then by statement 3) of the Lemma 1.1, by Lemma 3.1 and by Lemma 3.3 we obtain the validity of Lemma 3.4.
The lemma is proved.
4) Now let binary relation
of the semigroup
satisfying the condition 4) of the Theorem 2.1. In this case we have
where
and
. By definition of the semilattice D follows that
Now if
Then from Theorem 1.1 we obtain
. (4.1)
Lemma 4.1. Let
be a finite set,
and
. If by
denoted all regular elements of the semigroup
satisfying the condition 4) of the Theorem 2.1, then
Proof. First we show that the following equalities are hold:
(4.2)
For this we consider the following case.
a) Let
. If a quasinormal representation of a regular binary relation
has the form
for some
,
and
.
Then by statement 4) of the Theorem 2.1, we have
It follows that
and
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equalities are hold:
b) Let
and a quasinormal representation of a regular binary relation
has the form
for some
,
and
.
Then by statement 4) of the Theorem 2.1, we have
It follows that
and
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equalities are hold:
By equalities (4.1) and (4.2) follows, that
.
It is easy to see
and
, of the last equalities and by statement 4) of the Lemma 1.1 we obtain the validity of Lemma 4.1.
The lemma is proved.
5) Now let binary relation
of the semigroup
satisfying the condition 5) of the Theorem 2.1. In this case we have
where
and
,
,
and
. By definition of the semilattice D follows that
Now if
Then from Theorem 1.1 we obtain
. (5.1)
Lemma 5.1. Let X be a finite set,
and
. If by
denoted all regular elements of the semigroup
satisfying the condition 5) of the Theorem 2.1, then
Proof. Let
be arbitrary element of the set
and
. Then quasinormal representation binary relation
of the semigroup
has a form
,
where
,
,
,
,
and by statement 5) of the Theorem 2.1 satisfies the following conditions:
,
,
and
.
Of this we have that the inclusions
,
are fulfilled. Therefore, of the equality (5.1) follows, that
. (5.2)
Now we show that the following equalities are hold:
(5.3)
a) Let
. Then quasinormal representation binary relation
of the semigroup
has a form
, where
,
,
,
,
and by statement 5) of the Theorem 2.1 satisfies the following conditions:
Of this conditions follows that
, then
.
But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equalities are hold:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
b) Let
. Then quasinormal representation binary relation
of the semigroup
has a form
, where
,
,
,
,
and by statement 5) of the Theorem 2.1 satisfies the following conditions:
Of this conditions follows that
, then
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equalities are hold:
,
,
,
,
,
,
,
,
,
.
c) If
, then
(5.4)
It follows that
(5.5)
i.e.,
. So, the inclusion
is hold.
Of the other hand, if
, then the conditions (5.4) and (5.5) are fulfilled, i.e.,
. Therefore, the equality
is true.
The similar way we can show that the following equalities are hold:
Now by equalities (5.2) and (5.3) we obtain the validity of Lemma 5.1.
The lemma is proved.
Lemma 5.2. Let
and
are arbitrary elements of the set
, where
,
and
. If quasinormal representation of binary relation
of the semigroup
has a form
,
for some
,
,
,
,
and
, then
iff
.
Proof. If
, then we have
(5.6)
Of the last condition we have
, (5.7)
since
and
by supposition.
Of the other hand, if the conditions of (5.7) are hold, then, also hold the conditions of (5.6) i.e.
.
The lemma is proved.
Lemma 5.3. Let X be a finite set,
and
are arbitrary elements of the set
, where
,
and
. Then the following equalities are hold:
Proof. Let
and
are arbitrary elements of the set
, where
,
and
. If
, then quasinormal representation of a binary relation
of semigroup
has a form
for some
,
,
,
,
,
, then by statement 5) of the Theorem 2.1, we have
(5.8)
Let fα is a mapping of the set X in the semilattice D satisfying the conditions
for all
. f0α, f1α, f2α and f3α are the restrictions of the mapping fα on the sets
,
,
,
respectively. It is clear, that the intersection disjoint elements of the set
is empty set and
.
We are going to find properties of the maps f0α, f1α, f2α and f3α.
1)
. Then by the properties (5.8) we have
, since ![]()
and
. i.e.,
and
by definition of the set
. Therefore
for all
.
2)
. Then by the properties (5.8) we have
, i.e.,
and
by definition of the set
and
. Therefore
for all
.
Preposition we have that
, i.e.
for some
. Then
sense
. If
, then
. Therefore
. That is contradiction of the equality
, while
and
by definition of the semilattice D.
Therefore
for some
.
3)
. Then by the properties (5.8) we have
, i.e.,
and
by definition of the set
and
. Therefore
for all
.
Preposition we have that
, i.e.
for some
. Then
sense
. If
then
. Therefore
. That is contradiction of the equality
, while
and
by definition of the semilattice D. Therefore
for some
.
4)
. Then by definition quasinormal representation binary relation
and by property (5.8) we have
, i.e.
by definition of the sets
,
,
and
. Therefore
for all
.
Therefore for every binary relation
exist ordered system
. It is obvious that for disjoint binary relations exist disjoint ordered systems.
Now let
,
,
,
are such mappings, which satisfying the conditions:
5)
for all
;
6)
for all
and
for some
;
7)
for all
and
for some
;
8)
for all
.
Now we define a map f of a set X in the semilattice D, which satisfies the condition:
Now let
,
,
,
and
. Then binary relation
can be representation by form
and satisfying the conditions:
(By suppose
for some
and
for some
), i.e., by Lemma 2.10 we have that
.
Therefore for every binary relation
and ordered system
exist one to one mapping.
By ( [1] , Theorem 1.18.2) the number of the mappings
,
,
and
are respectively:
1,
,
,
.
Note that the number
does not depend on choice of
elements
of the semilattice D, where
,
,
and
. Since the number of such different elements of the semilattice D are equal to 7, the number of regular elements of the
set
is equal to
.
The lemma is proved.
Lemma 5.4. Let X be a finite set,
and
. If by
denoted all regular elements of the semigroup
satisfying the condition 5) of the Theorem 2.1, then
Proof. It is easy to see
and
, then by statement 5) of the Lemma 1.1, by Lemma 5.1 and by Lemma 5.3 we obtain the validity of Lemma 5.4.
The lemma is proved.
6) Let binary relation
of the semigroup
satisfying the condition 6) of the Theorem 2.1. In this case we have
, where
,
,
,
and
. By definition of the semilattice
follows that
.
If
,
,
,
, then from Theorem 1.1 we obtain
. (6.1)
Lemma 6.1. Let X be a finite set,
and
. If by
denoted all regular elements of the semigroup
satisfying the condition 6) of the Theorem 2.1, then
Proof. First we show that the following equalities are hold:
(6.2)
For this we consider the following case.
a) Let
. If a quasinormal representation of a regular binary relation
has the form
for some
,
,
,
and
. Then by statement 6) of the Theorem 2.1, we have
It follows that
and
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equality is hold:
b) Let
and a quasinormal representation of a regular binary relation
has the form
for some
and
,
,
and
. Then by statement 6) of the Theorem 2.1, we have
It follows that
and
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equalities are hold:
By equalities (6.1) and (6.2) follows that
.
It is easy to see
and
, then by statement 6) of the Lemma 1.1 we obtain validity of Lemma 6.1.
The lemma is proved.
7) Let binary relation
of the semigroup
satisfying the condition 7) of the Theorem 2.1. In this case we have
, where
,
,
,
,
. By definition of the semilattice D follows that
.
If
,
,
,
, then from the Theorem 1.1 we obtain
. (7.1)
Lemma 7.1. Let X be a finite set,
and
. If by
denoted all regular elements of the semigroup
satisfying the condition 7) of the Theorem 2.1, then
Proof. First we show that the following equalities are hold:
(7.2)
For this we consider the following case.
a) Let
. If a quasinormal representation of a regular binary relation
has the form
for some
and
,
,
,
and
. Then by statement 7) of the theo- rem 2.1, we have
It follows that
and
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equality is hold:
b) Let
. If a quasinormal representation of a regular binary relation
has the form
for some
and
,
,
,
and
. Then by statement 7) of the theo- rem 2.1, we have
It follows that
and
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
The similar way we can show that the following equalities are hold:
By equalities (7.1) and (7.2) follows that
.
It is easy to see
and
then by statement 7) of the Lemma 1.1 we obtain validity of Lemma 7.1.
The lemma is proved.
8) Let binary relation
of the semigroup
satisfying the condition 8) of the Theorem 2.1. In this case we have
. By definition of the semilattice D follows that
.
If
, then from Theorem 1.1 we obtain
(8.1)
Lemma 8.1. Let X be a finite set,
and
. If by
denoted all regular elements of the semigroup
satisfying the condition 8) of the Theorem 2.1, then
Proof. First we show that the following equalities are hold:
(8.2)
Let
. If a quasinormal representation of a regular binary relation
has the form
,
where
, ![]()
![]()
,
,
,
,
and
. Then by statement 8) of the Theorem 2.1, we have
It follows that
and
. But the inequality
contradiction of the condition that representation of binary relation
is quazinormal. So, the equality
is hold.
By equalities (8.1) and (8.2) follows that
.
It is easy to see
and
, then by statement 8) of the Lemma 1.1 we obtain validity of Lemma 8.1.
The lemma is proved.
Let X be a finite set and
and us assume that
.
Theorem 2.2. Let X is a finite set,
and
. If
is a set of all regular elements of the semigroup
, then
.
Proof. This Theorem immediately follows from the Theorem 2.1.
The theorem is proved.
I was seen in ( [6] , Theorem 2) that if
and
are regular elements of
then
is an XI-subsemilattice of D. Therefore
is regular element of
.
Theorem 2.3. Let
and
. The set of all regular elements is a subsemigroup of the semigroup
which is defined by semilattices of the class
.
Proof. This Theorem immediately follows from the Theorem 2 in [6] .
The theorem is proved.