Received 15 January 2016; accepted 24 May 2016; published 27 May 2016
![](//html.scirp.org/file/8-7403039x9.png)
1. Introduction
In the present work our aim is to identify regular elements of thesemigroup
when
and ![](//html.scirp.org/file/8-7403039x16.png)
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![](//html.scirp.org/file/8-7403039x19.png)
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). □