1. Introduction

The symbols denote sets of real numbers, complex numbers and bicomplex numbers, respectively.

A bicomplex number is defined as (cf.[1,2])

,

where

and .

With usual binary compositions, becomes a commutative algebra with identity. Besides the additive and multiplicative identities 0 and 1, there exist exactly two non-trivial idempotent elements denoted by and defined as and . Note that and .

A bicomplex number can be uniquely expressed as a complex combination of and as (cf. [3])

,

where and.

The complex coefficients and are called the idempotent components and the combination is known as idempotent representation of bicomplex number.

The auxiliary complex spaces and are defined

as follows:

and

The idempotent representation

associates with each point in, the points and in and , respectively and to each pair of points , there corresponds a unique bicomplex point . Some updated details of the theory of Bicomplex Numbers can be found in [5,6].

2. Order Topologies on

Srivastava[3] initiated the topological study of . He defined three topologies on , viz., norm topology , complex topology and idempotent topology and has proved some results on these topological structures.

Throughout, < denotes the ordering of real numbers and denotes the dictionary ordering of the complex numbers. Denote by , the dictionary ordering of bicomplex numbers expressed in the real component form. The order topologies induced by this ordering will be called as Real Order Topology (cf.[4]), denoted by and is generated by the basis comprising of the members of the following families of subsets of :

,

the set denoting the open interval with respect to the ordering .

Throughout the discussion, we shall consider some special types of subsets of the bicomplex space , equipped with real order topology.

A set of the type

and is called an open space segment.

A set of the type {ξ: ξ = a + i₁ x₂ + i₂ x₃ + i₁ i₂ x₄} is called a frame and is denoted as (). A set of the type {ξ: ξ = a + i₁ x₂ + i₂ x₃ + i₁ i₂ x₄; b < x₂ and x₂ < c} is called an open frame segment.

A set of the type {ξ: ξ = a + i₁ b + i₂ x₃ + i₁ i₂ x₄} is called as plane and is denoted as (). A set of the type is called an open plane segment.

A set of the type {ξ: ξ = a + i₁ b + i₂ c + i₁i₂ x₄} is called as a line and is denoted as (). A set of the type { and } is called an open line segment.

Note that is a family of open space segments, is a family of open frame segments, is a family of open plane segments and is a family of open line segments.

Further, we shall consider some special types of subsets of the bicomplex space equipped with the idempotent order topology, (cf.[4]).

Denote by , the dictionary ordering of the bicomplex numbers expressed in the idempotent form. The order topology induced by this ordering is called as Idempotent Order Topology (cf.[4]). Hence, idempotent order topology, is generated by the basis comprising of members of the following families of subsets of :

,

the set denoting the open interval with respect to the ordering .

A set of the type is called an open ID – space segment.

A set of the type is called an ID – frame and is denoted as. A set of the type is called an open ID – frame segment.

A set of the type is called as an ID – plane and is denoted as.

A set of the type is called an open ID – plane segment. A set of the type is called an ID – line and is denoted as .

A set of the type

is called an open ID – line segment.

2.1. Remark

Note that, and can also be described as and , where

In other words,

2.2. Remark

The geometry of the Cartesian idempotent set determined by and , i.e., is entirely different from the geometry of Cartesian set determined by and , i.e., (for definitions see[3]). Obviously, the members of the families , , and are open ID – space segments, open ID – frame segments, open ID – plane segments and open ID – line segments, respectively.

3. Confinements of Bicomplex Nets in the Real Order Topology

3.1. Static and Eventually Static Bicomplex Net

Let D be an arbitrary directed set. Then a bicomplex net can be defined as such that

Further, a bicomplex net is said to be static on if , . It is said to be eventually static on if there exists such that , .

3.2. Real Frame Confinement (RF Confinement)

A bicomplex net

is said to be Real Frame confined (in short, RF confined) to the frame , if it eventually in every member of the family (of open space segments) containing the frame .

3.3. Real Plane Confinement (RP Confinement)

A bicomplex net

is said to be Real Plane confined (in short, RP confined) to the plane, if it is eventually in every member of the family (of open frame segments) containing the plane.

3.4. Real Line Confinement (RL Confinement)

A bicomplex net

is said to be Real Line confined (in short, RL confined) to the line, if it is eventually in every member of the family (of open plane segments) containing the line .

3.5. Real Point Confinement (R – Point Confinement)

A bicomplex net

is said to be Real Point confined (in short, R – Point confined) to the bicomplex point , if it is eventually in every member of the family (of open line segments) containing the point .

3.6. Remark

Note that if a bicomplex net is RF confined to the frame , it will not be eventually in any member of the family (and therefore will not be RP confined to any plane) unless is eventually static on ‘a’ (and in this case, will be RP confined to the plane () provided that ). Similar cases will arise with the other types of bicomplex nets in the real form.

3.7. Remark

The R – Point confinement of a bicomplex net is a necessary but not sufficient condition for the convergence of the net in the classical sense (i.e., in the topology induced by the Euclidean norm). In fact, every eventually static net,, converges and therefore, R – Point confinement of to implies convergence of to in the norm. To verify insufficiency, consider the bicomplex net on the directed set of positive rationals (with usual ordering) defined as follows:

,

where , .

The net converges to the point in the norm but not R – Point confined to . As a matter of fact, no member of belongs to the line segment

.

3.8. Theorem

A bicomplex net

is RF confined to the frame () if and only if the net converges to a.

Proof: Assume that converges to a. Given , let

(3.1)

be the member of containing the frame .

Since, converges to a, an index such that ,,

,

,

and

and

So that the net is eventually in.

Since is arbitrary and every member of contains a (for some ), is RF confined to the frame .

Conversely, let the bicomplex net

be RF confined to the frame .

Therefore, the bicomplex net

is eventually in every member of containing the frame .

In particular, is eventually in () defined by (3.1).

Thus, such that

,

,

.

Hence the theorem.

3.9. Theorem

A bicomplex net

is RP confined to the plane if and only if the net is eventually static on ‘a’ and net converges to ‘b’.

Proof: Let the net be eventually static on ‘a’ and the net converge to ‘b’.

Since is eventually static on a, , such that , .

Given , let

(3.2)

be a member of containing the plane .

As the net converges to b, there exists some such that

,

Since , there exists some such that and .

Therefore, and,.

,

for any , .

So that the bicomplex net

is eventually in .

Since is arbitrary and every member of contains an for some , the bicomplex net is RP confined to the plane .

Conversely, suppose that the bicomplex net

is RP confined to the plane .

Therefore, it is eventually in every member of the type () defined by (3.2), of the family containing the plane .

So that for given , there exists some such that

and ,

Therefore, the net is eventually static on a and the net converges to b.

On the similar lines, the following theorems can be proved.

3.10. Theorem

A bicomplex net

is RL confined to the line if and only if the nets and are eventually static on a and b, respectively and the net converges to c.

3.11. Theorem

A bicomplex net

is R – Point confined to the bicomplex point if and only if the nets , , and are eventually static on a, b and c, respectively and the net is converges to d.

3.12. Theorem

(i) Every R – Point confined bicomplex net is RL confined.

(ii) Every RL confined bicomplex net is RP confined.

(iii) Every RP confined bicomplex net is RF confined.

The converses of these implications are not true, in general.

Proof: In fact, if a bicomplex net is R – Point confined to the bicomplex point , then it is RL confined to the line . Similarly, a bicomplex net which is RL confined to the line is RP confined to the plane and a bicomplex net which is RP confined to the plane is RF confined to the frame( ).

That the converse is not true, in general, is shown with the help of the two examples below.

3.13. Example

Consider the directed set . Define the bicomplex net as follows:

such that

, ,

and , ,

where the net is eventually static on 0.

Therefore, the net is eventually static on ‘a’ and the net converges on ‘0’. So that the bicomplex net is RP confined to the plane. Since, the net is eventually static on ‘a’. Therefore, the bicomplex net is eventually in every member of the family containing the frame . Hence, the bicomplex net is RF confined to the frame .

3.14. Example

Consider the bicomplex net

such that

, ,

and , .

This bicomplex net is RF confined to the frame . The component net converges to ‘a’ but component net is not convergent. Therefore, the bicomplex net is not RP confined to any plane contained in the frame .

4. Confinements of Bicomplex Nets in Idempotent Order Topology

In this section, we assume to be furnished with the idempotent order topology (cf. [4]). Hence, the net

will be treated as the net ,

where

(4.1)

For the sake of brevity, we shall express the numbers the numbers and as and , respectively. Thus denote the frame , whereas denote the net and so on. Under these notations, (4.1) can be rewritten as

(4.2)

Note that for the net to be convergent, either both the nets and are convergent or both nets and are divergent but is convergent. Similarly, for net will be convergent if either both the nets and are convergent or both nets and are divergent but is convergent. The convergence of nets and can be similarly interpreted.

4.1. ID – Frame Confinement (ID – F Confinement)

A bicomplex net is said to be ID – Frame confined (in short, ID – F confined) to the ID – frame if it is eventually in every member of the family (of open ID – space segments) that contains the ID – frame .

4.2. ID – Plane Confinement (ID – P Confinement)

A bicomplex net is said to be ID – Plane confined (in short, ID – P confined) to the ID – plane if it is eventually in every member of the family (of open ID – frame segments) that contains the ID – plane .

4.3. ID – Line Confinement (ID – L Confinement)

A bicomplex net is said to be ID – Line confined (in short, ID – L confined) to the ID – line if it is eventually in every member of the family (of open ID – plane segments) containing the ID – line .

4.4. ID – Point Confinement

A bicomplex net is said to be ID – Point confined to the point if it is eventually in every member of the family (of ID – line segments) containing the point .

4.5. Remark

Note that if a bicomplex net defined by (4.2) is ID – F confined to the frame , it cannot be eventually in any member of the family unless is eventually static on ‘a’. Similar cases will arise with the other types of the confinements of the bicomplex nets with respect to the idempotent order topology.

4.6. Theorem

A bicomplex net is ID – F confined to the ID – frame if and only if the net converges to a.

Proof: Assume that converges to a.

Given , let

(4.3)

be a member of containing the ID – frame .

Since , there exists an index such that

,

Hence, by the definition of , we have and ,

and

i.e,

for any , .

So that the bicomplex net is eventually in .

Since is arbitrary and every member of contains for some , by definition, is ID – F confined to the ID – frame .

Conversely, let the bicomplex net be ID – F confined to the ID – frame .

By definition, it is eventually in every member of (in fact, of ) containing the frame . In particular, it is eventually in , for every ε, defined by (4.3).

Thus, such that ,, i.e., such that and

By definition of and , we infer

,

.

Hence the theorem.

4.7. Theorem

A bicomplex net is ID – P confined to the ID – plane if and only if the net is eventually static on a and the net converges to b.

Proof: Suppose that the net is eventually static at a and the net converges to b.

Since the net is eventually static on a, , such that , .

Given , let

(4.4)

be a member of containing the ID – plane .

Since the net converges to b, there exists some such that , .

Since . Then, there exists some such that and .

Therefore, and ,.

Therefore, the net is eventually in .

Since is arbitrary and every member of contains an for some , by the definition, the bicomplex net is ID – P confined to the ID – plane .

Conversely, suppose that the bicomplex net is ID – P confined to the ID – plane .

Therefore, the bicomplex net is eventually in every member of the family containing the ID – plane . In particular, the net is eventually in every open ID – frame segment , (), defined by (4.4) containing the ID – plane .

Given , there exists some such that

, .

Therefore, by definition of , and , .

Hence the theorem.

On similar lines, the following theorems can be proved.

4.8. Theorem

A bicomplex net is ID – L confined to the ID – line if the nets and are eventually static on a and b, respectively and converges to c.

4.9. Theorem

A bicomplex net is ID – Point confined to the point if the nets, and are eventually static on a, b and c, respectively and the net converges to d.

4.10. Theorem

(i) Every ID – Point confined bicomplex net is ID – L confined.

(ii) Every ID – L confined bicomplex net ID – P confined.

(iii) Every ID – P confined bicomplex net ID – F confined.

The converses of these implications are not true, in general.

Proof: In fact, if a bicomplex net is ID – Point confined to the point , say, then it is ID – L confined to the ID – line . Further, a bicomplex net which is ID – L confined to the ID – line is ID – P confined to the ID – plane . Furthermore, a bicomplex net which is ID – P confined to the ID – plane is ID – F confined to the ID – frame .

To show that that the converse of these implications are not true, in general, we give below, in particular, an example of ID – F confined net which is also ID – P confined and an example of an ID – F confined net which is not ID – P confined.

4.11. Example

Consider the directed set . Define the bicomplex net

,

where is eventually static on 0.

By (4.1), the net is eventually static on 2a and hence converges to 2a, the bicomplex net is eventually in every member of the family containing the ID – frame . Hence the net is ID – F confined to the frame .

Also, the net is eventually static on 2a and the net converges to 2a.

Therefore, bicomplex net is ID – P confined to the ID – plane .

4.12. Example

Consider the bicomplex net

.

By (4.1), the net is eventually static on 2a and the net converges to 0. Therefore, the bicomplex net is ID – F confined to the ID – frame .

Note that although the component net converges to 2a, it is not eventually static on 2a. Therefore, the bicomplex net is not ID – P confined to any ID – plane contained in the ID – frame .

5. Bicomplex Net and its Projection Nets

This section is devoted to the study of relations between confinements of a bicomplex net and the convergence of its projection nets (cf.[7]) in the idempotent product topology (cf.[4]). Recall the definitions of the auxiliary complex spaces and .

5.1. Theorem

A bicomplex net converges to a bicomplex point in the idempotent product topology if and only if the net in is confined to in (k = 1, 2).

Proof: If converges to the bicomplex point , it is eventually in every neighbourhood of with respect to the idempotent product topology.

Note that and the projections are continuous, k = 1, 2. (cf. [5]).

Therefore, the net in is eventually in every neighbourhood of , k = 1, 2.

Hence the net in is confined to in , k = 1, 2.

The converse is straightforward.

5.2. Note

The analogue of the above result is not true for any type of ID – confinement (except ID – Point confinement) of the bicomplex nets with respect to the idempotent order topology on . Further, there is a characteristic difference between the convergence in the idempotent product topology and the confinement in the idempotent order topology in the sense that for any type of confinement (except ID – Point confinement) of a bicomplex net with respect to the idempotent order topology it is not necessary to have all the component nets to be convergent. We prove the following results in this context.

5.3. Theorem

If the bicomplex net is ID – F confined to the ID – frame , the projection net in is confined to the line in .

Proof: Suppose that the bicomplex net defined by (4.2), is ID – F confined to the ID – frame .

Therefore the bicomplex net is eventually in every member of the family (of open ID – space segments) containing the ID – frame .

Now, the projection of every member of on is a plane segment in and therefore is a basis element of the dictionary order topology on . Hence, the projection net is eventually in every basis element of the dictionary order topology on containing the line segment in .

Hence the net is confined to the line segment in .

5.4. Theorem

If the bicomplex net is ID – P confined to the ID – plane , the projection net is confined to the point in .

Proof: Suppose that the bicomplex net defined by (4.2), is ID – P confined to the ID – plane .

Therefore, the bicomplex net is eventually in every member of the family (of open ID – frame segments) containing the ID – plane .

Note that the projection of every member of on the auxiliary space is a basis element (in fact, an open interval) of the dictionary order topology on containing the point .

So the projection net is eventually in every basis element of dictionary order topology on the auxiliary complex space containing the point .

Therefore, the projection net is confined to the point in .

5.5. Theorem

If the bicomplex net is ID – L confined to the ID – line , the projection net converges to the point in and the projection net is confined to the line in .

Proof: Let the bicomplex net defined by (4.2), be

ID – L confined to the ID – line

.

Therefore, the bicomplex net is eventually in every member of the family (of open ID – plane segments) containing the ID – line

.

Since the projection of every member of on is a point .

Therefore, the projection net in is eventually static on in , so it is convergent to the point in .

The projection on of every member of (of open ID – plane segments) is a plane segment in , which is a basis element of the dictionary order topology on .

Therefore, the projection net in is eventually in every basis element of the dictionary order topology on containing the line in .

Hence, the projection net in is confined to the line in .

5.6. Theorem

If the bicomplex net is ID – Point confined to the bicomplex point , the projection net in converge to the point in and the projection net in converge to the point in .

Proof: Let the bicomplex net defined by (4.2), be ID – Point confined to the bicomplex point .

Therefore, it is eventually in every member of the family (of open ID – line segments) containing the point. So that the projection net in is confined to the point in and the projection net in is confined to the point in .

5.7. Example

Consider the bicomplex net of Example 4.11.

The net converges to 2a and the net converges to 2a. However, the nets and are not convergent and hence the projection net in is not confined in . So the bicomplex net does not converge in the idempotent product topology.

But the projection net in is confined to the point in . Hence the bicomplex net is ID – P confined to the ID – plane .

5.8. Example

Consider the directed net . Define a bicomplex net as follows:

and being eventually static on 0.

The projection net in is confined to the point in and the projection net in is confined to the point in . Therefore, the bicomplex net is ID – Point confined to the point .

Since, all the component nets are convergent. Therefore, the bicomplex net converges to the point in the classical sense.