Received 15 January 2016; accepted 24 May 2016; published 27 May 2016
1. Introduction
In the present work our aim is to identify regular elements of thesemigroup when and
The method used in this part does not differ from the method given in [1] .
2. Regular Elements of the Complete Semigroups of Binary Relations of the Class , When and
We denoted the following semilattices by symbols:
1), where (see diagram 1 of the Figure 1);
2) where (see diagram 2 of the Figure 1);
3) where and (see diagram 3 of the Figure 1);
4) where and (see diagram 4 of the Figure 1);
5) where, , , (see diagram 5 of the Figure 1);
6) where, , , (see diagram 6 of the Figure 1);
7), where, , , , , (see diagram 7 of the Figure 1);
8), where, , , , , , (see diagram 8 of the Figure 1);
Figure 1. Diagram of all XI-subsemilattices of semi lattices of unions D.
Note that the semilattices 1)-8), which are given by diagram 1-8 of the Figure 1 always are XI-semilattices (see [2] , Lemma 1.2.3).
Remark that
Lemma 1. Let be an isomorphism between and semilattices, , and. If X is a finite set and and, then the following equalities are true:
1)
2)
3)
4)
5)
6)
7)
8)
Proof. Let. Then given Lemma immediately follows from ( [1] , Lemma 3). □
Theorem 1. Let and. Then a binary relation
of the semigroup whose quasinormal representation has a form will be a
regular element of this semigroup iff there exist a complete a-isomorphism of the semilattice on some subsemilattice of the semilattice D which satisfies at least one of the following conditions:
・
・ , for some and which satisfies the condition;
・ , for some, , and which satis- fies the conditions:, ,;
・ , for some, and which satisfies the conditions:, , , ,;
・ , where, , , and satisfies the conditions:, , ,;
・ , where, , , , and satisfies the conditions:, , , , ,.
・ , where,
, , , and satisfies the conditions:,
, , ,;
・ , where, , , and satisfies the conditions:, , , , ,.
Proof. Let. Then given Theorem immediately follows from ( [1] , Theorem 2). □
Lemma 2. Let and. Let be set of all
regular elements of the semigroup such that each element satisfies the condition a) of Theorem 1. Then.
Now let a binary relation of the semigroup satisfy the condition b) of Theorem 1 (see diagram 2 of the Figure 1). In this case we have, where and. By definition of the semi- lattice D it follows that
It is easy to see and. If
then
(1)
(see remark page 5 in [1] ).
Lemma 3. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition b) of Theorem 1. Then
Proof. Let, and. Then quasinormal representation of a binary relation has a form for some and by statement b) of Theorem 1 satisfiesthe conditions and. By definition of the semilattice D we have, i.e., and. It follows that. Therefore the inclusion holds. By the Equality(1) we have
(2)
From this equality and by statement b) of Lemma 1 it immediately follows that
□Let binary relation of the semigroup satisfy the condition c) of Theorem 1 (see diagram 3 of the Figure 1). In this case we have, where and. By definition of the
semilattice D it follows that
It is easy to see and. If-1
then
(3)
(see remark page 5 in [1] and Theorem 1).
Lemma 4. Let X be a finite set, and. Let
be set of all regular elements of the semigroup such that each element satisfies the condition c) of Theorem 1. Then
where
Proof. Let be arbitrary element of the set and. Then
quasinormal representation of a binary relation has a form for some
, and by statement c) of Theorem 1 satisfies the conditions
, and. By definition of the semilattice D we have. From
this and by the condition, , we have
i.e., where. It follows that, from the last inclusion and by
definition of the semilattice D we have for all, where
Therefore the following equality holds
(4)
Now, let, and. Then for the binary relation we have
From the last condition it follows that.
1). Then we have, that. But the inequality
contradicts the condition that representation of binary relation is quasinormal. So,
the equality is true. From last equality and by definition of the semilattice D we have
for all, where
2), , , ,
and are true. Then we have
and
respectively, i.e., or if and only if
Therefore, the equality is true. From last equality and by defi-
nition of the semilattice D we have: for all, where
3), ,
, , and are true. Then we have
and
respectively, i.e., and if and only if
Therefore, the equality is true. From last equality and by definition of the semilattice D we have: for all, where
Now, by Equality (2) and by conditions 1), 2) and 3) it follows that the following equality is true
where
□
Lemma 5. Let, , where and. If quasinormal repre- sentation of binary relation of the semigroup has a form for some, and, then iff
Proof. If, then by statement c) of theorem 1 we have
(5)
From the last condition we have
(6)
since by assumption. On the other hand, if the conditions of (6) holds, then the conditions of (5) follow, i.e.. □
Lemma 6. Let, and X be a finite set. Then the following equality holds
Proof. Let, where. Assume that
and a quasinormal representation of a regular binary relation has a form
for some, and. Then according to Lemma 5, we have
(7)
Further, let be a mapping from X to the semilattice D satisfying the conditions for all., and are the restrictions of the mapping on the sets, , respec-
tively. It is clear that the intersection of elements of the set is an empty set, and
. We are going to find properties of the maps, ,.
1). Then by the properties of D we have, i.e., and by
definition of the sets and. Therefore for all. By suppose we have that
, i.e. for some. Therefore for some.
2). Then by properties of D we have, i.e.,
and by definition of the sets, and. Therefore for all. By suppose we have, that, i.e. for some. If. Then
. Therefore by definition of the set and. We have contradiction to
the equality. Therefore for some.
3). Then by definition quasinormal representation binary relation a and by property of D we have
, i.e. by definition of the sets and. Therefore
for all. Therefore for every binary relation there exists
ordered system. It is obvious that for disjoint binary relations there exists disjoint ordered
systems. Further, let
be such mappings, which satisfy the conditions: for all and for some
; for all and for some; for all
. Now we define a map f from X to the semilattice D, which satisfies the condition:
Further, let, , and. Then bi-
nary relation may be represented by
and satisfy the conditions:
(By suppose for some and for some), i.e., by lemma 5 we have
that. Therefore for every binary relation and ordered system
there exists one to one mapping. By Lemma 1 and by Theorem 1 in [1] the number of the mappings are respectively:
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 15, for arbitrary where, the number of regular elements of the set is equal to
□
Therefore, we obtain:
(8)
Lemma 7. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition c) of Theorem 1. Then
Proof. Let. Then the given Lemma immediately follows from Lemma 4 and from the Equalities (3).
□
Now let binary relation of the semigroup satisfy the condition d) of Theorem 1 (see diagram 4 of the Figure 1). In this case we have where and. By de- finition of the semilattice D it follows that
It is easy to see and. If
then
(9)
(see Definition [1] , Definition 4 and [1] , Theorem 2).
Lemma 8. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition d) of Theorem 1. Then
Proof. Let Then the given Lemma immediately follows from ( [1] , Lemma 10). □
Now let binary relation of the semigroup satisfy the condition e) of Theorem 1 (see diagram 5 of the Figure 1). In this case we have where and and . By definition of the semilattice D it follows that
It is easy to see and. If
then
(10)
(see [1] , Definition 4 and [1] , Theorem 1).
Lemma 9. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition e) of Theorem 1. Then
where
Proof. Let. Then the given Lemma immediately follows from ( [1] , Lemma 13). □
Lemma 10. Let and be arbitrary elements of the set, where, and. Then the following equality holds
Proof. Let. Then the given Lemma immediately follows from definition semilattice D and by ( [1] , Lemma 13). □
Lemma 11. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition
e) of Theorem 1. Then, where
and
Proof. Let. Then the given Lemma immediately follows from Lemma 9 and 10. □
Let f be a binary relation of the semigroup satisfy the condition g) of Theorem 1 (see diagram 7
of the Figure 1). In this case we have where, and
. By definition of the semilattice D it follows that
It is easy to see and. If
Then
(11)
(see Definition [1] , Definition 4 and [1] , Theorem 2).
Lemma 12. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition f) of Theorem 1. Then
Proof. Let. Then the given Lemma immediately follows from ( [1] , Lemma 15). □
Now let g be a binary relation of the semigroup satisfy the condition f) of Theorem 1 (see
diagram 6 of the Figure 1). In this case we have, where,
and. By definition of the semilattice D it follows that
It is easy to see and. If
then
(12)
(see [1] , Definition 4 and [1] , Theorem 2).
Lemma 13. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition g) of Theorem 1. Then
Proof. Let. Then the given Lemma immediately follows from ( [1] , Lemma 16). □
Let h be a binary relation of the semigroup satisfy the condition h) of Theorem 1 (see diagram 8 of the Figure 1). In this case we have, Where, . By definition of the semilattice D it follows that
It is easy to see and. If
Then
(13)
(see [1] , Definition 4 and [1] , Theorem 2).
Lemma 14. Let X be a finite set, and. Let be set of all regular elements of the semigroup such that each element satisfies the condition h) of Theorem 1. Then
Proof. Let. Then the given Lemma immediately follows from ( [1] , Lemma 17). □
Let us assume that
Theorem 2. Let,. If X is a finite set and is a set of all regular elements of the semigroup, then.
Proof. This Theorem immediately follows from ( [1] , Theorem 2) and Theorem 1. □
Example 1. Let,
Then, , , , , , , , and.
We have, , , , , , , , ,.
Theorem 3. Let. Then the set of all regular elements of the semigroup is a subsemigroup of this semigroup.
Proof. From ( [1] , Lemma 2), and by definition of the semilattice D it follows that the diagrams of XI- semilattices have the form of one of the diagrams given ( [1] , Figure 2). Now the given Theorem immediately follows from ( [3] , Theorem 2). □