On Bicomplex Nets and their Confinements

We have initiated the study of nets with bicomplex entries. Due to the multi dimensionality of the bicomplex space there arise different types of tendencies called confinements. The bicomplex space equipped with real order topology as well as idempotent order topology exhibits interesting and challenging behaviour of nets. Different types of confinements have been characterized in terms of convergence of the component nets. In the final section, certain relations between bicomplex nets and their projection nets have been derived.


Introduction
The symbols 0 C , 1 C and 2 C denote sets of real numbers, complex numbers and bicomplex numbers, respectively.
A bicomplex number is defined as (cf. [1,2] With usual binary compositions, 2 C 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 1 e and 2 e defined as . Some updated details of the theory of Bicomplex Numbers can be found in [5,6].

Order Topologies on 2 C
Srivastava [3] initiated the topological study of 2 C . He defined three topologies on 2 C , viz., norm topology 1 τ , complex topology 2 τ and idempotent topology 3 τ 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 R  , 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 4 τ and is generated by the basis comprising of the members of the following families of subsets of 2 C : denoting the open interval with respect to the ordering R  . Throughout the discussion, we shall consider some special types of subsets of the bicomplex space 2 C , equipped with real order topology.
A set of the type A set of the type {ξ: ξ = a + i 1 x 2 + i 2 x 3 + i 1 i 2 x 4 } is called a frame and is denoted as ( a x 1 = ). A set of the type {ξ: ξ = a + i 1 x 2 + i 2 x 3 + i 1 i 2 x 4 ; b < x 2 and x 2 < c} is called an open frame segment.
A set of the type {ξ: ξ = a + i 1 b + i 2 x 3 + i 1 i 2 x 4 } is called as plane and is denoted as is called an open plane segment.
A set of the type {ξ: ξ = a + i 1 b + i 2 c + i 1 i 2 x 4 } is called as a line and is denoted as Note that 1 J is a family of open space segments, 2 J is a family of open frame segments, 3 J is a family of open plane segments and 4 J is a family of open line segments. Further, we shall consider some special types of subsets of the bicomplex space 2 C equipped with the idempotent order topology, (cf. [4]).
Denote by ID  , 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, 6 τ is generated by the basis 6 B comprising of members of the following families of subsets of 2 C : is called an open ID -line segment.

Remark
Note that, 1 L and 2 L can also be described as

Remark
The geometry of the Cartesian idempotent set determined by 1 A and 2 A , i.e.,

Static and Eventually Static Bicomplex Net
Let D be an arbitrary directed set. Then a bicomplex net can be defined as . It is said to be eventually static on

Real Frame Confinement (RF Confinement)
A bicomplex net x is said to be Real Frame confined (in short, RF confined) to the frame ( a x 1 = ), if it eventually in every member of the family 1 J (of open space segments) containing the frame ( a x 1 = ).

Real Plane Confinement (RP Confinement)
A bicomplex net

Real Line Confinement (RL Confinement)
A bicomplex net containing the line ( )

Real Point Confinement (R -Point Confinement)
A bicomplex net it is eventually in every member of the family 4 J (of open line segments) containing the point ξ .

Remark
Note that if a bicomplex net { } α ξ is RF confined to the frame ( ) a x 1 = , it will not be eventually in any member of the family 2 J (and therefore will not be RP confined to x is eventually static on 'a' (and in this case, { } α ξ will be RP confined to the plane . Similar cases will arise with the other types of bicomplex nets in the real form.

Remark
The R -Point confinement of a bicomplex net { } α ξ is a necessary but not sufficient condition for the convergence of the net in the classical sense (i.e., in the topology 1 τ induced by the Euclidean norm). In fact, every eventually

Theorem
A bicomplex net is arbitrary and every member of 1 J contains a ε B (for some 0 > ε Hence the theorem.

Theorem
As the net { } α 2 Since 0 > ε is arbitrary and every member of 2 J contains an ε I for some 0 > ε Therefore, it is eventually in every member of the type is eventually static on a and the net { } α 2 x converges to b. On the similar lines, the following theorems can be proved. x converges to c. x is converges to d.

Theorem
(i) Every R -Point confined bicomplex net is RL confined.
(ii) Every RL confined bicomplex net is RP confined.
Similarly, a bicomplex net which is RL confined to the line and a bicomplex net which is RP con- That the converse is not true, in general, is shown with the help of the two examples below.

Example
Consider the bicomplex net This bicomplex net is RF confined to the frame ( ) is not convergent. Therefore, the bicomplex net is not RP confined to any plane contained in the frame ( ) a x 1 = .

Confinements of Bicomplex Nets in Idempotent Order Topology
In this section, we assume 2 C to be furnished with the idempotent order topology (cf. [4] Im can be similarly interpreted.

Remark
Note that if a bicomplex net { } Re 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.

Theorem
Hence the theorem.

Theorem
Therefore, a Re 1 = ξ and Therefore, by definition of . Hence the theorem. On similar lines, the following theorems can be proved. Re converges to c.

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.
. Furthermore, a bicomplex net which is ID -P confined to the ID -plane is ID -F confined to the IDframe ( ) a Re 1 = ξ . 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.

Example
Consider the directed set ( ) x is eventually static on 0.
By (

Example
Consider the bicomplex net By

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 1 A and 2 A . The converse is straightforward.

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 2 C . 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. Now, the projection of every member of 1 N on 1 A is a plane segment in 1 A and therefore is a basis element of the dictionary order topology on 1 A . Hence, the projection

Theorem
is eventually in every basis element of the dictionary order topology on 1 A containing the line segment a x = in 1 A .
Hence the net { } α ξ 1 is confined to the line segment a x = in 1 A .

Theorem
If the bicomplex net { } α ξ is ID -P confined to the ID -

Theorem
If

Example
Consider the directed net ( )