Applied Mathematics
Vol.07 No.09(2016), Article ID:66807,27 pages
10.4236/am.2016.79078
Regular Elements of Defined by the Class
Yasha Diasamidze1, Nino Tsinaridze1, Neşet Aydn2, Ali Erdoğan3
1Shota Rustavelli University, Batumi, Georgia
2Çanakkale Onsekiz Mart University, Çanakkale, Turkey
3Hacettepe University, Ankara, Turkey
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 15 January 2016; accepted 24 May 2016; published 27 May 2016
ABSTRACT
In order to obtain some results in the theory of semigroups, the concept of regularity, introduced by J. V. Neumann for elements of rings, is useful. In this work, all regular elements of semigroup defined by semilattices of the class are studied. When X has finitely many elements, we have given the number of regular elements.
Keywords:
Semilattice, Semigroup, Binary Relation
1. Introduction
Let D be a nonempty set of subsets of a given set X, closed under union. Such a set D is called a complete X-semilattice of unions. For any map f from X to D, we define a binary relation.
The set of all, denoted by
, is a subsemigroup of
semigroup of all binary relations on X. (See [1] - [6] .)
All notations, symbol and required definitions used in this work can be found in [7] . Recall the following results.
Lemma 1. [1] , Corollary 1.18.1. Let and
be two sets where
and
. Then the number
of all possible mappings from Y to subsets
of
such that
is given by
.
Theorem 1. [1] , Theorem 1.18.1. Let. Let
be nonempty sets. Then the number
of mappings from X to such that
for some
is equal to
.
2. Results
Let X be a nonempty set, D a X-semilattice of union with the conditions (see Figure 1);
(1)
The class of X-semilattices where each element is isomorphic to D is denoted by.
An element is called regular if
for some
. Our aim in this work is to identify all regular elements of
where D is given above.
Definition 1. The complete X-semilattice of unions is called an XI-semilattice of unions if
and for any nonempty Z in D. Here
is an exact lower bound of
in D where
The following Lemma is well known (see [7] , Lemma 3).
Lemma 2. All semilattices in the form of the diagrams in Figure 2 are XI-semilattices.
Figure 1. Diagram of semilattice of unions D.
Figure 2. Diagram of all XI-subsemilattices of D.
Definition 2. Let and
be two X-semilattices of unions. A one to one map from
to
is said to be a complete isomorphism if
for
Definition 3. [1] , Definition 6.3.3. Let. We say that a complete isomorphism
is a complete a-isomorphism if
a)
b) for
and
for any
.
The following subsemilattices are all XI-semilattices of the X-semilattices of unions D.
a), where
(see diagram 1 of the Figure 3);
b) where
and
(see diagram 2 of the Figure 3);
c) where
and
(see diagram 3 of the Figure 3);
d) where
and
(see diagram 4 of the Figure 3);
e) where
,
,
,
,
, (see diagram 5 of the Figure 3);
f) where
,
,
,
,
,
(see diagram 6 of the Figure 3);
g), where
,
,
,
,
,
(see diagram 7 of the Figure 3);
h), where
,
,
,
,
,
,
(see diagram 8 of the Figure 3);
For each we set
One can see that
Assume that and denote by the symbol
the set of all regular elements a of the semigroup
Figure 3. Diagram of all subsemilattices isomorphic to subsemilattices in Figure 2.
, for which the semilattices
and
are mutually a-isomorphic and
and
(see [1] , Definition 6.3.5).
The following results have the key role in this study.
Theorem 2. Let be the set of all regular elements of the semigroup
. Then the following state- ments are true:
a) for any
and
;
b);
c) if X is a finite set, then (see [1] , Theorem 6.3.6).
Lemma 3. Let be isomorphism between
and
semilattices,
,
and
. If X is a finite set and
, then the following equalities are true:
a)
b)
c)
d)
e)
f)
g)
h)
Proof. The propositions a), b), c) and d) immediately follow from ( [1] , Theorem 6.3.5 and Theorem 13.1.2), while the equalities e), f), g) and h) follow from ( [1] , Theorem 6.3.5, Corolaries 13.3.4-5-6 and 13.7.3). □
3. Regular Elements of the Complete Semigroups of Binary Relations of the Class , When
and
Theorem 3. Let and
. Then a binary relation a
of the semigroup whose quasinormal representation has the 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:
a), for some
;
b), for some
,
and
which satisfies the condi- tions:
,
;
c), for some
,
, and
which satisfies the conditions:
,
,
,
;
d), for some
,
and
which satisfies the conditions:
,
,
,
,
,
;
e), where
,
,
,
,
,
and satisfies the conditions:
,
,
,
;
f), where,
,
,
,
and satisfies the conditions:
,
,
,
,
,
,
;
g), where
,
,
,
, and satisfies the conditions:
,
,
,
,
;
h), where
,
,
,
and satisfies the conditions:
,
,
,
,
,
,
.
Proof. In this case from Lemma 2 it follows that diagrams 1-8 given in Figure 2 exhaust all diagrams of XI-subsemilattices of the semilattice D. A quasinormal representation of regular elements of the semigroup, which are defined by these XI-semilattices, may have one of the form listed above. Then the validity of theorem immediately follows from ( [1] , Theorem 13.1.1, Theorem 13.3.1 and Theorem 13.7.1). □
Lemma 4. Let and
. Let
be set of all regular elements of
such that each element satisfies the condition of a) of Theorem 3. Then
.
Proof. Let binary relation a of the semigroup satisfy the condition a) of Theorem 3. Then quasinormal representation of a binary relation a has a form
for some
. It is easy to see that
for all
, i.e. binary relation a is a regular element of the semigroup
. Therefore
□
Now let binary relation a of the semigroup satisfy the condition b) of Theorem 3 (see diagram 2 of the Figure 3). In this case we have
where
and
. By definition of the semi- lattice D it follows that
It is easy to see that there is only one isomorphism from to itself. That is
and
. If
then
(2)
Lemma 5. Let X be a finite set,
and. Let
be the set of all regular elements of
such that each element satisfies the condition b) of Theorem 3. Then
Proof. Let,
,
and
. Then quasinormal representation of a binary relation a has a form
for some
,
and by state- ment b) of theorem 3 satisfies the conditions
and
. By definition of the semilattice D we have
and
, i.e.,
and
. It follows that
. Therefore we have
(3)
From this equality and by statement b) of Lemma 3 it immediately follows that
□
Let binary relation a of the semigroup satisfy the condition c) of Theorem 3 (see diagram 3 of the Figure 3). In this case we have
, where
and
. By definition of the semilattice D it follows that
It is easy to see and
. If
then
(4)
Lemma 6. Let X be a finite set,
and. Let
be the set of all regular elements of
such that each element satisfies the condition c) of Theorem 3. Then
where
Proof. Let
be arbitrary element of the set
and
. Then quasinormal representation of a binary relation a has a form
for some
,
and by statement c) of Theorem 3 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
(5)
holds. Now, let,
and
. Then
for the binary relation a 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 a is quasinormal. So, the equality
is true. From the 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 the last equality and by definition 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 the last equality and by definition of the semilattice D we have
for all, where
Now, by equality (4) and conditions 1), 2) and 3) it follows that the following equality is true
where
□
Lemma 7. Let,
where
and
. If quasinormal repre-
sentation of binary relation a of the semigroup has a form
for
some,
and
, then
iff
Proof. If, then by statement c) of Theorem 3 we have
(6)
From the last condition we have
(7)
since by assumption.
On the other hand, if the conditions of (7) holds, then (6) immediately follows, i.e.. Lemma is proved. □
Lemma 8. 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 a has a form
for some
,
and
. Then ac- cording to Lemma 7, we have
(8)
Further, let be 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 of elements of the set
is an empty set, and
. We are going to find properties of the maps
,
,
,
.
1). Then by the properties (1) we have
, i.e.,
and
by definition of the set
. Therefore
for all
.
2). Then by the properties (1) 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
. That is contradiction to the equality
, while
by definition of the se- milattice D.
Therefore for some
.
3). Then by properties (1) 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
.
4). Then by definition of a quasinormal representation of a binary relation a and by property (1) we have
, i.e.
by definition of the sets
and
. There- fore
for all
.
We have seen that for every binary relation there exists ordered system
. It is obvious that for disjoint binary relations there exist disjoint ordered systems.
Further, let
be such mappings that satisfy the conditions:
for all
;
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
binary relation may be represented by
and satisfies the conditions
.
(By suppose for some
and
for some
), i.e., by lemma 7 we have that
.
Therefore for every binary relation and ordered system
there exists one to one mapping.
By Lemma 1 and by Theorem 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 22, for arbitrary
where
, the number of regular elements of the set
is equal to
□
Therefore we obtain
(9)
Lemma 9. Let X be a finite set, and
. Let
be set of all regular elements of
such that each element satisfies the condition c) of Theorem 3. Then
Proof. The given Lemma immediately follows from Lemma 6 and from the Equalities (5).
Now let a binary relation a of the semigroup satisfy the condition (d) of Theorem 3 (see diagram 4
of the Figure 3). 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
(10)
Lemma 10. Let X be a finite set,
and. Let
be set of all regular elements of
such that each element satisfies the condition d) of Theorem 3. Then
Proof. Let,
and
. Then
, where
,
and the following inclusions and inequalities are true
From this it follows that
We consider the following cases.
1) or
. Then we have
. But the inequality
contradicts the condition that representation of binary relation a is quasinormal. So,
the equality holds. From the last equality and by definition of the semilattice D we have
for all
, where
(10a)
2) or
Then we have
or
. But the inequality
or
contradicts the condition that representation of binary relation a is quasinormal. So, the equality holds. From the last equality and by definition of the semilattice D we have
for all
, where
(10b)
By conditions (10a) and (10b) it follows that
From the last equality we have that the given Lemma is true. □
Now let a binary relation a of the semigroup satisfy the condition e) of Theorem 3 (see diagram 5 of the Figure 3). 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
(11)
Lemma 11. Let X be a finite set,
and. Let
be set of all regular elements of
such that each element satisfies the condition e) of Theorem 3. Then
where
Proof. Let be arbitrary element of the set
and
. Then quasinormal representation of a binary relation a of the semigroup
has a form
where,
,
,
,
and by statement e) of Theorem 3 satisfies the following conditions
From this we have that the inclusions
are fulfilled. Therefore from the Equality (1) it follows that
(12)
Let and
be such elements of the set
that
and
. Then quasinormal representation of a binary relation a of the semigroup
has a form
where,
,
,
,
and by statement e) of Theorem 3 satisfies the following conditions
,
,
and
.
Then by statement e) of Theorem 3 we have
From this conditions it follows that
For and
we consider the following cases.
1) or
. Then
or
respectively. But the inequalities and
contradict the condition that representation of binary relation a is quasinormal. So, the equality
holds. From the last equality it follows that
for all
, where
2) or
. Then by definition of the semilattice D it
follows that the inequalities,
or
,
are true respectively. But the inequalities
and contradict the condition that representation of binary relation a is quasinormal. So,
the equality holds. From the last equality, by definition of the semilattice D it follows
that for all
, where
3) If, then
Then by definition of the semilattice D it follows that the inequalities
are true. But the inequalities contradict the condition that representation of binary relation a is quasinormal. So, the equality
holds. From the last equality it follows that
, where
By similar way one can prove that for any
.
4),
and
are such elements of the set
that
,
,
,
and
, then by statement e) of theorem 3 satisfies the following conditions:
and
respectively, i.e., or
if and only if
Therefore, the equality is true. From the last equality by de- finition of the semilattice D it follows that
for all
, where
From the equalities
and
given above it follows that
where
□
Lemma 12. Let and
be arbitrary elements of the set
, where
,
and
. If quasinormal representation of binary relation a
of the semigroup has a form
, for some
,
,
and
, then
iff
Proof. If, then by statement e) of Theorem 3 we have
(13)
From the last condition we have
(14)
since and
by supposition.
On the other hand, if the conditions of (14) hold, then the conditions of (13) follow, i.e..
□
Lemma 13. Let and
be arbitrary elements of the set
, where
,
and
. Then the following equality holds:
Proof. Let and
be arbitrary elements of the set
, where
,
and
. If
. Then quasinormal repre- sentation of a binary relation a of semigroup
has a form
for some,
,
,
and by the lemma 12 satisfies the conditions
(15)
Now, 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
respectively. 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,
,
and
.
(1). Then by the properties (1) we have
since i.e.,
and
by definition of the set
. Therefore
for all
.
(2). Then by the properties (1) we have
, i.e.,
and
by definition of the set
and
. Therefore
for all
.
By suppose we have that, i.e.
for some
. Then
since
. If
, then
. Therefore
. That contradicts the equality
, while
and
by definition of the semilattice D.
Therefore for some
.
(3). Then by the properties (1) we have
, i.e.,
and
by definition of the set
and
. Therefore
for all
.
By suppose we have that, i.e.
for some
. Then
since
. If
then
. Therefore
. That contradicts the equality
, while
and
by definition of the semilattice D.
Therefore for some
.
(4). Then by definition quasinormal representation of a binary relation a and by property (1) 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
;
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 binary relation
may be represented by
and satisfies the conditions
(By suppose for some
and
for some
), i.e., by lemma 12 we have that
. Therefore for every binary relation
and ordered system
there exists one to one mapping.
The number of the mappings,
,
and
are respectively
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 form
of the semilattice D are equal to 24, the number of regular elements of the set
is equal to
Lemma 14. Let X be a finite set,
□
and. Let
be set of all regular elements of
such that each element satisfies thecondition e) of Theorem 3. Then
, where
and
Proof. The given Lemma immediately follows from Lemma 11 and Lemma 13.
□Let binary relation a of the semigroup satisfy the condition g) of Theorem 3 (see diagram 7 of the Figure 3). In this case we have
, where
,
and
. By definition of the semilattice D it follows that
It is easy to see and
. If
(see Figure 4).
Figure 4. Diagram of all subsemilattices isomorphic to 7 in Figure 2.
Then
(16)
Lemma 15. Let X be a finite set, and
. Let
be set of all regular elements of
such that each element satisfies the condition f) of Theorem 3. Then
Proof. Let,
and
. Then quasinormal representation of a binary relation a of the semigroup
has a form
where,
and
and by statement f) of theorem 3 satisfies the following conditions
From this conditions it follows that
For and
we consider the following case.
or
. Then
or
But the inequality and
contradict the condition that representation of binary relation a is quasinormal. So, the equality
holds. From the last equality by definition of the semilattice D it follows that
for all
, where
(17)
Now by Equalities (16) and by condition (17) it follows that
By statement f) of Lemma 3 the given Lemma is true.
□Now let binary relation a of the semigroup satisfy the condition f) of Theorem 3 (see diagram 6 of the Figure 3). In this case we have
, where
,
and
. By definition of the semilattice D it follows that
It is easy to see and
. If
(see Figure 5).
Then
Lemma 16. Let X be a finite set, and
. Let
be set of all regular elements of
such that each element satisfies the condition g) of Theorem 3. Then
Proof. Let,
and
. Then quasinormal representation of a binary relation a of the semigroup
has a form
Figure 5. Diagram of all subsemilattices isomorphic to 6 in Figure 2.
where,
and
and by statement g) of Theorem 3 satisfies the following conditions
From this conditions it follows that
For and
we consider the following cases.
1) or
. Then
or
But the inequalities and
contradicts the condition that repre- sentation of a binary relation a is quasinormal. So, the equality
holds. From the last equality by definition of the semilattice D it follows that
for all
, where
2) or
. Then
or
But the inequalities and
contradicts the condition that repre- sentation of a binary relation a is quasinormal. So, the equality
holds. From the last equality by definition of the semilattice D it follows that
for all
, where
3) or
. Then
or
But the inequalities and
contradicts the condition that repre- sentation of a binary relation a is quasinormal. So, the equality
holds. From the last equality by definition of the semilattice D it follows that
for all
, where
Now by conditions 1), 2) and 3) it follows that
By statement (g) of Lemma 3 the given Lemma is true.
□Let binary relation a of the semigroup satisfy the condition h) of Theorem 3 (see diagram 8 of the Figure 3). In this case we have
, where
,
. By definition of the semilattice D it follows that
It is easy to see and
. If
(see Figure 6).
Then
Lemma 17. Let X be a finite set, and
. Let
be set of all regular elements of
such that each element satisfies the condition h) of Theorem 3. Then
Figure 6. Diagram of all subsemilattices isomorphic to 8 in Figure 2.
Proof. Let, where
,
,
and
. Then quasinormal representation of a
binary relation a of the semigroup has a form
where,
,
and by statement g) of Theorem 3 sa- tisfies the following conditions
From this conditions it follows that
For and
we consider the following case.
. Then
. But the inequality
contradicts the condition that representation of binary relation a is quasinormal. So,
the equality holds. From the last equality by definition of the semilattice D it follows that
for all
, where
Therefore we have
By statement h) of Lemma 3 the given Lemma is true. □
Let us assume that
Theorem 4. 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 Theorem 2 and Theorem 3. □
Example 1. Let,
Then,
,
,
,
,
,
,
,
and
.
We have,
,
,
,
,
,
,
,
.
Cite this paper
Yasha Diasamidze,Nino Tsinaridze,Neşet Aydn,Neşet Aydn,Ali Erdoğan, (2016) Regular Elements of Bx(D) Defined by the Class ∑1(X,10)-I. Applied Mathematics,07,867-893. doi: 10.4236/am.2016.79078
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http://dx.doi.org/10.4236/am.2015.62026