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 D_{1} 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 Q_{i}, (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 f_{α} is a mapping of the set X in the semilattice D satisfying the conditions for all. f_{0α}, f_{1α}, f_{2α} and f_{3α} are the restrictions of the mapping f_{α} 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 f_{0α}, f_{1α}, f_{2α} and f_{3α}.
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.