Cluster Sets of C2-Nets

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2589–2599 © Research India Publications http://www.ripublication.com/gjpam.htm

Cluster Sets of C2-nets

Sukhdev Singh and Rajesh Kumar Gupta Department of Mathematics, Lovely Professional University, Phagwara, Punjab 144411, India.

Abstract

This paper initiates the study of clustering of C2-nets. We gave results on the cluster sets of C2-nets in the Id(o)-topology and its applications to bicomplex numbers. Clustering on different types of zones in C2 have studied. Clustering in Id(o)- topology and idempotent product topology have been compared. Relation between clustering of C2-nets and the clustering of its component nets have been discussed. Finally, investigations have been made connecting clustering of a C2-net and con- ﬁnement of its C2-subnets.

AMS subject classiﬁcation: Primary 06F30; Secondary 54A20. Keywords: Bicomplex numbers, C2-net, Cluster sets, conﬂuence of C2-nets, Id(o)- topology, C2-subnet.

1. Introduction The bicomplex numbers were introduced by Carrado Segre [8] in 1892. The set of bicomplex numbers is denoted by C2 and sets of real and complex numbers are denoted as C0 and C1, respectively.The set of bicomplex number is deﬁned as (cf. [5], [9])

C2 := {a1 + i1a2 + i2a3 + i1i2a4,ak ∈ C0, 1 ≤ k ≤ 4} := {z1 + i2z2,z1,z2 ∈ C1} 2 = 2 =− = where i1 i2 1, i1i2 i2i1. Study of the bicomplex numbers had been started with the work of the Italian school of Segre [8], Spampinato [17, 18], and Dragoni [3]. Their interest arose from the fact that such numbers offer a commutative alternative to the skew ﬁeld of quaternion and that in many ways they generalize complex numbers more closely and more accurately 2590 Sukhdev Singh and Rajesh Kumar Gupta than quaternion do. Segre used some of the Hamilton’s notation to develop his system of bicomplex numbers. The algebra constructed on the basis {1,i1,i2,i1i2} is then nearly the same as James Cockle’s tessarines [4]. Besides the additive and multiplicative identities 0 and 1, there exist exactly two non-trivial idempotent elements denoted by e1 and e2 and deﬁned as e1 = (1 + i1i2)/2 and e2 = (1 − i1i2)/2. Note that e1 + e2 = 1 and e1.e2 = 0. A number ξ = z1 + i2z2 can be uniquely expressed as complex combination of e1 and e2 [10].

1 2 ξ = z1 + i2z2 = ξe1 + ξe2,

1 2 1 2 where ξ = z1 − i1z2 and ξ = z1 + i1z2. The complex coefﬁcients ξ and ξ are called 1 2 the idempotent components of ξ, and ξe1 + ξe2 is known as idempotent representation of bicomplex number ξ. The auxiliary complex spaces A1 and A2 are deﬁned as follows: 1 2 A1 = ξ : ξ ∈ C2 and A2 = ξ : ξ ∈ C2 .

The idempotent representation

1 2 (z1 − i1z2)e1 + (z1 + i1z2)e2 = ξe1 + ξe2

1 2 associates with each point ξ = z1+i2z2 in C2, the points ξ = z1−i1z2 and ξ = z1+i1z2 in A1 and A2, respectively and to each pair of points (z, w) ∈ A1 ×c A2, there is a unique bicomplex point ξ = ze1 + we2. Srivastava [10] initiated the systematic study of topological aspects of C2. He defined three topologies τ 1, τ 2 and τ 3 on C2. Srivastava and Singh [12] continued the study and defined three order topologies τ 4, τ 5, τ 6 and one product topology τ 7 and studied them. In the present paper, we shall confine ourselves mainly to C2 equipped with τ 6. For the sake of ready reference, we give below relevant literature of τ 6 (for details cf. [12]). Denote by ≺ID, the lexicographic ordering of the bicomplex numbers expressed in the idempotent form. The order topology induced by this ordering is called as Id(o)- topology. The Id(o)-topology, τ 6 is generated by the basis B6 comprising of members of the following families of subsets of C2: 1 1 1. L1 = (ξ, η)Id : ξ ≺ η 1 1 2 2 2. L2 = (ξ, η)Id : ξ = η, ξ ≺ η , the set (ξ, η)ID denoting the open interval with respect to the ordering ≺ID and ≺ : 1 denoting the lexicographic order in A1 and A2. A set of the type ξ a

Also, N1, N2, N3 and N4 are the families of Id-open space segments, Id-open frame segments, Id-open plane segments and Id-open line segments, respectively.

Deﬁnition 1.1. Id(p)-topology [11]: The Id(p)-topology denoted as τ 7 on C2 having 1 1 2 2 1 1 as basis B7, is the collection of all sets of the form ( ξ, η) ×e ( ξ, η), where ( ξ, η) 2 2 and ( ξ, η) are basis elements of the order topologies on auxiliary complex spaces A1 and A2, respectively. Theorem 1.2. [11] The Id(o)-topology is strictly ﬁner than the Id(p)-topology.

2. Clustering of C2-nets

In [12], we have initiated the study of C2-nets and their conﬂuence to various types of zones in C2. In this section, we deﬁne the clustering of a C2-net in different types of zones and investigate the conditions required for the clustering of the C2-net. We start with few basics about C2-nets.

Deﬁnition 2.1. C2-net [12]: Let D be any arbitrary directed set, then a C2-net is deﬁned as : D → C2 such that ∀α ∈ D = = + + + (α) ξ α x1α i1x2α i2x3α i1i2x4α = z1α + i2z2α = 1 + 2 ξ α e1 ξ α e2 C = ∀ ∈ Further, a 2-net ξ α is static on ξ if ξ α ξ, α D.Itiseventually static on ξ ∈ = ∀ ≥ if there exists β D such that ξ α ξ, α β. Throughout the paper, the set D will denote the directed set.

Definition 2.2. Cofinal Set [19]: A subset K of directed set D is said to be cofinal in D, if for each α ∈ D, there exists some γ ∈ K such that γ ≥ α. C Definition 2.3. ID-F Confluence [12]: A 2-net ξ α is said to be ID-F confluence to [ 1 = ] ∈ ∈ ∈ ∀ ≥ Re ξ a if for every β D, there exists U N1 such that ξ α U, α β and [Re 1ξ = a]⊂U.

For other types of ID-conﬂuences of a C2-nets, we refer to [12]. C Deﬁnition 2.4. Let ξ α be a 2-net. Then 2592 Sukhdev Singh and Rajesh Kumar Gupta

[ 1 = ] 1. It clusters on Re ξ α a if it is frequently in every member of N1 containing [ 1 = ] Re ξ α a . 1 1 2. It clusters on [Re ξ = a, Im ξ = b] if it is frequently in every member of N2 containing [Re 1ξ = a, Im 1ξ = b].

3. It clusters on [Re 1ξ = a,Im 1ξ = b, Re 2ξ = c] if it is frequently in every 1 1 2 member of N3 containing [Re ξ = a, Im ξ = b, Re ξ = c].

4. It clusters on a point ξ if it is frequently in every member of the family N4 containing ξ. Deﬁnition 2.5. Subnet [19]: A net ηβ β∈E is said to be a subnet of a net ξ α α∈D if for each tail Tα of D, there is a tail Tβ of E such that : ∈ ⊂ : ∈ ηδ δ Tβ ξ γ γ Tα . C Remark 2.6. If any 2-net ξ α frequently in every member of the family N4 containing ξ, then it is frequently in every member of the basis B6 of the idempotent order topology containing ξ. Theorem 2.7. If ξ is the cluster point of ξ α in τ 6, then it is also cluster point of it in τ 7. C = + + + Proof. Let the 2-net ξ α be cluster at ξ (a i1b)e1 (c i1d)e2 in τ 6. 1 1 2 2 Suppose P = ( ζ, η) ×e ( ζ, η) is an arbitrary basis element of τ 7 containing 1 1 2 2 ξ = (a + i1b)e1 + (c + i1d)e2. Thus, a + i1b ∈ ( ζ, η) and c + i1d ∈ ( ζ, η). 2 2 Since ( ζ, η) is open interval in the usual topology on A2, ∃ >0 such that

2 2 (c + i1d − , c + i1d + ) ⊂ ( ζ, η).

Hence, the set K deﬁned as

K = ((a + i1b)e1 + (c + i1(d − ))e2,(a+ i1b)e1 + (c + i1(d + ))e2)ID is contained in P. Clearly, K is a basis element of τ 6 with K ∈ N4, ξ ∈ K and K ⊂ P . = + + + Now as the net ξ α clusters on the point ξ (a i1b)e1 (c i1d)e2 in τ 6,itis frequently in every member of the family N4 containing the point ξ. In particular, the C2- net ξ α is frequently in every open ID-line segment containing ξ. Hence it is frequently in K. So it is frequently in P. Since P is an arbitrary basis element of τ 7 containing the = + + + C point ξ (a i1b)e1 (c i1d)e2, therefore, the 2-net ξ α is frequently in every basis element of τ 7 containing the point ξ = (a + i1b)e1 + (c + i1d)e2. Hence, the net = + + + ξ α clusters at ξ (a i1b)e1 (c i1d)e2 with respect to τ 7. Remark 2.8. The converse of the above theorem is not true, in general. Cluster Sets of C2-nets 2593

+ Example 2.9. Consider the directed set (Q , ≥). Deﬁne the C2-net = + 2 − 2 + ∀ ∈ Q+ ξ α a i1(1/α ) i2(2/α ) i1i20 , α . 1 2 C Since, the both nets ξ α and ξ α are clustering at the point a. Therefore, the 2-net 1 ξ α clusters on the point awith respect to τ 7. However, although the net Re ξ α is 1 static on a, the net Im ξ α does not attain the value 0, frequently. Therefore, the net ξ α is not frequently in any member of the family N4 containing the point a. Therefore, it does not cluster on the point a with respect to τ 6. C [ 1 = ] Theorem 2.10. The 2-net ξ α clusters on Re ξ a if and only if ‘a’ is a cluster 1 point of Re ξ α . C [ 1 = ] Proof. Let the 2-net ξ α cluster on Re ξ a . Therefore, the C2-net is frequently in every member of the family N1 which contains [Re 1ξ = a]. So that for given >0 and ∀β ∈ D, there exists α ∈ D, α ≥ β such that ∈ − + + + + + ξ α ((a i1x2)e1 (x3 i1x4)e2,(a i1y2)e1 + (y3 + i1y4)e2)ID, ∀ xp,yp ∈ C0, 2 ≤ p ≤ 4. ⇒ − 1 + a 0 be given. Then ∈ ∈ ≥ 1 ∈ − + for any β D, there exists α D, α β such that Re ξ α (a , a ) ⇒ − 1 + a 0 is arbitrary. Therefore, the 2-net ξ α clusters on Re ξ a . Remark 2.11. In view of the above theorem one may intuitively infer its analogue for C [ 1 = 1 = ] clustering on an ID-plane viz., A 2-net ξ α clusters on Re ξ a, Im ξ b if 1 1 Re ξ α attains the value a frequently and b is the cluster point of the net Im ξ α . 2594 Sukhdev Singh and Rajesh Kumar Gupta

However, this result does not hold good in the general setup.

Example 2.12. Let D and D be two infinite and disjoint cofinal subsets of D. Define ξ α as follows: 1 = ∀ ∈ 1. Re ξ α a, α D and nowhere else. ∈ ∈ ≥ 1 ∈ 2. Given >0 and given β D, there exists α D , α β such that Im ξ α (b − , b + ).

Note that D and D are disjoint. In particular, this implies there is no member of D for which both the conditions (i) and (ii) are attained, simultaneously. Therefore, the C [ 1 = 1 = ] 2-net ξ α cannot cluster on Re ξ a, Im ξ b . C [ 1 = 1 = ] Theorem 2.13. The 2-net ξ α clusters on Re ξ a, Im ξ b only if there 1 1 exists a coﬁnal subset D of D such that Re ξ α is static on a in D and Im ξ α α∈D cluster on b. C [ 1 = 1 = ] C Proof. Let the 2-net ξ α clusters on Re ξ a, Im ξ b . Thus, the 2-net ξ α 1 1 is frequently in every member of the family N2 containing [Re ξ = a, Im ξ = b]. Let >0 be given. Consider the member F of N2 deﬁned as 1 1 F = ξ : Re ξ = a, b − 0 and given β D, there exists α D, α β such that ∈ + − + + + + ξ α ((a i1(b ))e1 (x3 i1x4)e2,(a i1(b ))e1 + (y3 + i1y4)e2)ID, ∀x3,x4,y3,y4 ∈ C0. ⇒ 1 = − 1 + Re ξ α a and b

Now deﬁne a subset D of D as follows: = : ∈ 1 = D α α D,Re ξ α a .

Obviously, D is the desired coﬁnal subset of D. = ∀ ∈ If we deﬁne a net ηβ on D as ηβ ξ β, β D , we see that ηβ is a subnet 1 = 1 = of ξ α α∈D.Now,if ξ α α∈D clusters on the ID–plane (Re ξ a, Im ξ b), then 1 1 Re ηβ is static on ‘a’ and the net Im ηβ clusters on ‘b’. Hence, the above theorem can be reworded as: C [ 1 = 1 = ] Theorem 2.14. The 2-net ξ α clusters on Re ξ a, Im ξ b only if there exists 1 1 a subnet ηβ of ξ α such that Re ηβ is static on ‘a’ and the net Im ηβ clusters on ‘b’. Cluster Sets of C2-nets 2595

On the similar lines, we can prove the following theorems: C [ 1 = 1 = 2 = ] Theorem 2.15. The 2-net ξ α clusters on Re ξ a, Im ξ b, Re ξ c only 1 if there exists a net η such that Re η is static on ‘a’ and there exists a subnet β β∈D β 1 2 ζ γ γ ∈D of ηβ β∈D such that Im ζ γ is static on the ‘b’ and Im ζ γ clusters on the point ‘c’. C = + + + Theorem 2.16. The 2-net ξα clusters on ξ (a i1b)e1 (c i1d)e 2 only if there exists a subnet η of ξ , a subnet ζ of the net η and a β β∈D α α∈D γ γ ∈D β β∈D 1 1 2 subnet ψδ δ∈D such that Re ηβ is static on ‘a’, Im ζ γ is static on ‘b’, Re ψδ 2 is static on ‘c’ and Im ψδ clusters on the point ‘d’.

Remark 2.17. If a C2-net is confined to a particular ID-zone (ID-frame, ID-plane, ID- line or ID-point) then it clusters on that ID-zone. The converse of this is not true in the general set up. Q+ ≥ C Example 2.18. Consider the directed set ( , ). Define a 2-net ξ α as follows: = + + + 2 ξ α (a i1xα)e1 (b i1(1/α ))e2 , ∀ ∈ Q+ 1 = { } α , where the net Im ξ α xα attains the value 0, frequently. + Therefore, the net ξ α clusters on the bicomplex point ae1 be2 with respect to the 1 idempotent order topology. But as the net Im ξ α is not eventually static on 0, ξ α is not ID-point confined to the bicomplex point ae1 + be2. C Remark 2.19. Let ξ α be a 2-net. Then we have the following observations: + + + [ 1 = 1 = 1. If ξ α clusters on (a i1b)e1 (c i1d)e2, then it clusters on Re ξ a, Im ξ b, Re 2ξ = c]. [ 1 = 1 = 2 = ] 2. If ξ α clusters on Re ξ a, Im ξ b, Re ξ c , then it clusters on [Re 1ξ = a, Im 1ξ = b]. [ 1 = 1 = ] [ 1 = ] 3. If ξ α clusters on Re ξ a, Im ξ b , then it clusters on Re ξ a .

Converses of these implications are not true, in general, for obvious reasons. C { } ∇ For any 2-net ηα , let L(ηα), (ηα) and (ηα) denote the set { ∈ C : { ∈ : ∈ } } z 2 for each neighbourhood V of z the set k D ηα V is inﬁnite , all { } conﬂuence zones and the set of all cluster zones of ηα , respectively. Obviously, ⊂ ∇ ⊂ (ηα) L(ηα) and (ηα) L(ηα). C { } ⊂∇ Remark 2.20. For any 2-net ηα ,wehave (ηα) (ηα). C { } ∇ Theorem 2.21. For any 2-net ηα , the set (ηα) is closed. 2596 Sukhdev Singh and Rajesh Kumar Gupta

∈ ∇ Proof. Let ξ (ηα). Suppose that U is any neighbourhood of ξ. Then there exists ∈ ∩∇ ∈∇ a point ζ U (ηα). Select a neighbourhood V of ζ and as ζ (ηα). Then { ∈ : ∈ }⊃{ ∈ : ∈ } ∈∇ α D ηα U α D ηα V . Hence ξ (ηα). { } { } C { ∈ N : = } Theorem 2.22. If ξ α and ηα be two 2-nets such that δ( n ξ α ηα ) =0, ∇ =∇ then (ξ α) (ηα).

3. Conﬂuence of C2-nets and their C2-subnets

In this section, we investigate the ID – confluence and clustering of subnets of C2-nets. Note that various types of subnets may be formed depending upon its domain. The domains of the net and subnet may be disjoint directed sets or domain of the subnet may be subset of the domain of the net. In the later case, there are two possibilities. Either domain of the subnet is cofinal subset of domain of the net (in which case it is called a cofinal subnet) or a proper subset of domain of the net. C Theorem 3.1. If a 2-net ξ α is ID-point confluence to ξ, then every cofinal subnet of ξ α is also ID-point confluence to ξ. Proof. Let ξ be a C2-net defined on the directed set (D, ≥) which is ID-point con- α fluence to ξ. Now, suppose that D is a cofinal subset of the directed set D, so that for every α ∈ D there exists some λ ∈ D such that λ ≥ α. Also, D as a subset of D,isa directed set under the same order relation of D.

Define a net η on D . Clearly, η is a cofinal subnet of ξ . Therefore, from λ λ α the definition of subnet, for each Tα of D there is a Tλ of D such that for each γ ∈ Tλ here is some δ ∈ Tα such that : ≥ ⊂ : ≥ ηλ λ γ ξ α α δ (3.1) Now as the net ξ α is ID-point confined to the point ξ, the net ξ α is eventually in every member of the family N4 containing ξ. From (3.1) we have obtained that every tail of points of the net ξ α contains some tail of the points of the subnet ηλ and also D is a cofinal subset of D. Therefore, we conclude that the subnet ηλ lies eventually in every member of the family N4 containing the point ξ. Hence, ηλ is ID – Point confined to the point ξ. Remark 3.2. If domain of a subnet of the given net is not cofinal subset of domain of the net, then the ID – confluence to an ID – confluence zone of the net may or may not imply the ID – confluence of the subnet to the same ID – confluence zone.

This assertion can be established with the help of the following example. + + Example 3.3. Let {(α, β) : α,β ∈ D} ⊆ Q ×c Q . Define an order relation as follows: (α, β) (γ , δ) Cluster Sets of C2-nets 2597 if and only if α ≤ γ and β ≤ δ. Obviously, D is a directed set. Now define a C2-net : D −→ C2 as follows: 1 1 1 i i 1 1 (α, β) = + + 1 2 − , ∀ (α, β) ∈ D, 2 α β 2 α β 1 1 1 1 where and are integral parts of and , respectively. α β α β This net is ID-point confined to 0 + i10 + i20 + i1i20.

Further, deﬁne a subset D of D as + D = (α, 1) : α ∈ Q .

Since subset of a directed set is again a directed set under the same order relation, D is a directed set under the order relation of D. Now, consider δ>1. Then for any (λ, δ) ∈ D, there does not exist any element

(γ , 1) ∈ D such that (λ, δ) (γ , 1). Therefore, D is not a coﬁnal subset of D.

Then define a subnet = ⎪D of the net as follows: 1 1 i i (λ, 1) = + 1 + 1 2 , ∀ (λ, 1) ∈ D . 2 λ 2 1 Since the net Re ξ θ θ∈D is not eventually static on 0. Therefore, subnet of the given net is not ID-point confined to the point 0 + i10 + i20 + i1i20. 1 Also as the net Re ξ θ θ∈D does not attain the value 0, frequently, subnet of the given net does not clustering on the point 0 + i10 + i20 + i1i20. Remark 3.4. If the domain of the subnet of given net is not a cofinal subset of domain of the net, then it is be possible that the subnet does not cluster on the ID-zone even when the net is ID-confined to that ID – zone. C Theorem 3.5. A 2-net ξ α α∈D clusters on a bicomplex point ξ if there exists a subnet of the C2-net which is ID-point confined to the bicomplex point ξ. C Proof. 3.6. et ηβ β∈E be a subnet of the 2-net ξ α α∈D, which is ID-point confined to the bicomplex number ξ = (a + i1b)e1 + (c + i1d)e2. We have to show that the net ξ α α∈D clusters on ξ. Let U be an arbitrary neighbourhood ofξ and α ∈ D be given. Now as ηβ β∈E is a subnet of the net ξ α α∈D, there exists a tail Tβ of E such that : ∈ ⊂ ⊂ : ∈ ⊂ ηδ δ Tβ E ξ γ γ Tα D (3.2) = + + + Since the net ηβ is ID – Point confined to the point ξ (a i1b)e1 (c i1d)e2, ∈ ∀ ≥ ∈ there exists λ E such that δ λ, ηδ U. ∈ ≥ ≥ ∈ Now consider some µ E such that µ β and µ λ. Obviously, ηµ U. 2598 Sukhdev Singh and Rajesh Kumar Gupta

∈ = Due to (3.2), there exists ν Tα such that ξ ν ηµ. Thus there corresponds some ∈ ∈ ν D such that ξ ν U. ∈ C Since α D and U are arbitrary. Therefore, the 2-net ξ α clusters on the point ξ.

Remark 3.7. The converse of the above theorem is not true, in general.

Acknowledgements This paper is dedicated to Prof.(late) Rajiv K. Srivastava his great contribution to the theory of bicomplex numbers.

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