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1. Chapter three: • Far rings and some results • In This chapter , we introduce some results about far rings and period profiles, and we prove some properties of far rings which are known for the usual rings and some other results.

2. Section one: • Far rings and period profiles • In this section we introduce the meaning of the period profile and we explain the relation between them . This relation is important in the subject of faster way to get fractal images. • First we give the definition of period profile.

3. Definition 3.1: • Let (ZN,+,๏) be a far ring; andα1,…, αk be in ZN. The period profile of α1,…, αk is a list in descending order of the period lengths of the k-sequences which are obtained by recurrence relation over the far ring: • Sn= α1๏Sn-1, α2๏Sn-2+…+ αk๏Sn-k n≥ k

4. Whatever the initial values S0S1…Sk-1 which are elements of ZN. • To illustrate the concept of period profile, we take the simple case k=2 and we consider the far ring of order 4.

5. From examples 2.17 in chapter2 , there are four possible far rings of order 4. They are FRA, FRB,FRC, and FRD. • Let α1, α2 be in Z4. Recurrence relations of length k=2 over the far ring (Z4,+,๏) are given by Sn= α1๏Sn-1+ α2๏Sn-2 n ≥ 2;

6. Where any two elements S0S1from ZN can be taken as initial values to obtain each of these recurrence relation . • The period of the sequence S0S1 … cannot exceed 42=16 , which is the maximum number of distinct pairs SiSi+1 which can be encountered before repetition.

7. Further, any such pair may be taken as initial so is found in some sequence Thus the period lengths sum to 16. • The period profile of (α1, α2) is a list of period lengths in decending order, and the set of lists for all • (α1, α2) in any order , is the profile of the multiplication table of the far ring itself .

8. For example when the coefficients α1, α2 =0,1 are taken in FRA, then we get the sequence 120322…, 102300 ,133…,and 1…, which are periodic of period 6,6,3,and 1 respectively. • Therefore the period profile for 0,1 is 6631.

9. The following tables give the period profile for all coefficients αi, αj in Z4and for the four far rings (of order four) FRA,FRB,FRC, and FRD.

10. Table 3.1

11. Table 3.2

12. Table 3.3

13. Table 3.4

14. Definition 3.2: • Let each of (G1,๏) and (G2,*) be a quasigroup. An isomorphism Ψfrom G1 into G2 is a bijection or a permutation such that for all x,y in G1 • Ψ(x ๏ y)=Ψ(x) *Ψ(y). • Then (G1,๏)and(G2, *) are called isomorphic.

15. Our aim is to obtain far rings (of the same order) which have the same period profiles. • First the following example shows that the isomorphism of quasigroups does not suffice for equal profiles.

16. Example 3.3 • Consider (Z4,๏)and (Z4, *) such that ๏ and * are two binary operations defined by the following tables

17. Define Ψ from the quasigroup (Z4,๏) into the quasigroup (Z4, *)by • Ψ(0)=3, Ψ(1)=1,Ψ(2)=2, and Ψ(3)=0. • It is easy to show that Ψ is an isomorphism. • But (Z4,๏)=FRA and (Z4, *)=FRC have different period profiles ;

18. see table 3.1 and table 3.3. • We shall see that the following definition of isomorphism of far rings is sufficient to make isomorphic far rings of the same period profiles.

19. Definition 3.4: • An isomorphism of far rings (ZN,+,๏)→(ZN,+, *) is a pair of permutations Ψ , ø of ZN such that Ψ(1)=1 and for each α,β,a,b in ZN, • ø (α ๏ a+ β๏b)= Ψ(α ) * ø(a) + Ψ(β) * ø(b) . • Then (ZN,+,๏) and (ZN,+, *) are called isomorphic far rings .

20. The following theorem proves that any two isomorphic far rings have the same period profiles.

21. Theorem 3.5 • Isomorphic far rings have the same period profiles .In particular for k=2 If(Ψ,ø) is an isomorphism from the far ring( ZN,+, ๏ ) into the far ring (ZN,+, *) then the coefficient pairs α,β and Ψ(α), Ψ(β)in ZN have the same period profile, and similarly for k-tuples with k>2.

22. Proof • First let k=2 and suppose α,β give a sequence {an} of period t.That is , with addition as usual mod N, we have • an=α๏an-1+ β๏an-2. • Using ø(α๏a+β๏b) =Ψ(α )*ø(a)+Ψ (β)+ø(b)for any α,β, a, b in ZN ….…(1)

23. Then the sequence {an}={a0,a1,…}satisfies that at=a0,at+1=a1, and so on. • Thus , the coefficients Ψ(α),Ψ(β) • give the sequence {ø(an)}={ø(a0),ø(a1),…,ø(an),…}

24. Now, we need to prove that {ø(an)}is periodic of period t; we must prove that Ø(at)=ø(a0),ø(at-1)=ø(a1) and there is no positive integer r such that r<t and Ø(ar)=ø(a0),ø(ar+1)=ø(a1).

25. It is clear that ø(at)=ø(a0)and ø(at+1)=ø(a1)…,since {an} is periodic of period t.If there is r<t such that ø(ar)=ø(a0), ø(ar+1)=ø(a1)and since ø is bijection, then we get that there is r<t such that ar=a0,ar+1=a1,

26. but this is a contradiction to the fact that the period of {an} is t and r≠t;and t is the least positive integer making {an} periodic .Therefore we get that the sequence{ø(an)} is periodic of period t; • Which generated by Ψ(α),Ψ(β) with * as multiplication.

27. Finally , we may infer that these two far rings have the same period peofile ;since the bijection Ψ defines a bejiction of pairs (α,β )→(Ψ(α),Ψ(β)). • This complete the proof for the case k=2 .

28. The proof for the case k>2 is by induction . • It suffices to illustrate the proof for the case k=3. Let α,β,γ,a,b,c be in ZN and x= α๏a+β๏b+γ๏c, • then

29. Ø(x) =ø(α๏a+β๏b+γ๏c) • = ø(α๏a+1๏(β๏b+γ๏c)) • Since 1 acts as identify for ๏, • =Ψ(α) *ø(a)+Ψ(1) *ø(β๏b+γ๏c) ; since Ψ(1)=1 is an identity for *. • =Ψ(α) *ø(α)+Ψ(β) *ø(b)+Ψ(γ) *ø(c); since +is associative and by using (1).

30. By using the same notions in case k=2; it is easy to show that the coefficients Ψ(α),Ψ(β),Ψ(γ)generate the sequence {ø(an)}of period t,if the coefficients α,β,and γ generate the sequence {an}of period t.

31. We get the same result for k>3 by the same method.This means that if the coefficients α1,α2,…,αk generate a sequence {an} of period t.

32. under the multiplication๏,then the coefficients Ψ(α1),Ψ(α2),…,Ψ(αk) generate the sequence {ø(an)} of period t under multiplication *.Consequently ,isomorphic far rings have the same period profiles for all k≥2.

33. Note: • The important of theorem 3.5 is the result that if any one of these isomorphic far rings gives an M-sequence,then the others do the same. • Now we search for what is required to prove the converse of theorem 3.5 .We give the following preparatory theorem for k=2.

34. Theorem3.6 • let each of (ZN,+,๏) and (ZN,+, *) be a far ring which are denoted by FRX,FRY respectively . Suppose that Ψ,ø are permutations of ZN,Ψ(1)=1 and ø sends sequences over FRX generated by coefficients α,β to sequences over FRY generated by coefficients Ψ(α), Ψ(β) for any α,βin ZN .

35. Then (Ψ,ø) is an isomorphism from FRX into FRY.

36. Proof: • In order to prove this theorem, we must prove the following : • For any α,β,a,b in ZN; • Ø(α๏a+β๏b)=Ψ(α) *ø(a)+Ψ(β) *ø(b).

37. Now consider α,β,a,b in ZN and take k to be 2 to obtain a k-sequences over FRX particularly when k=2. Then a,b must appear as successive members of some sequence with coefficents α,β (we can take a,b as initial members if necessary).

38. Let c=α๏a+β๏b; c will be in ZN. Then ø sends this sequence with coefficients α,β to one over FRY with coefficients Ψ(α),Ψ(β) which results in • ø ( c) = ø(α๏a+β๏b) • = Ψ(α) *ø(a)+Ψ(β) *ø(b); As required.

39. This is true for any four elements of ZN . And since Ψ(1)=1, we get (Ψ,ø) is An isomorphism and the proof is complete . • Remark 3.7: let Ψ and ø be permutations of ZNand each of FRX,FRY is a far ring of order N.

40. The statement that (Ψ,ø) preserves k-profiles from FRX to FRY means that ø sends a sequence defined over FRX by a recurrence with coefficients αi(1≤i≤k) to the sequence over FRY defined by a recurrence with coefficients ø(αi) (1≤i≤k)

41. The following theorem gives some conditions to get isomorphic far rings

42. Theorem 3.8: • Let Ψ,ø be permutations of ZN with Ψ(1)=1 and ø(0)=0, and let FRX, FRY be far rings of order N. Suppose that for some k>2,(Ψ,ø) preserves k-profiles from FRX to FRY then (Ψ,ø) is a far ring isomorphism from FRX to FRY.

43. Proof: • Let (Ψ,ø) preserves k-profiles from FRX= (ZN,+,๏) to FRY= (ZN,+, *) for k=2 . Then , from Remark 3.7 , that means ø Sends sequences over FRX generated by coefficients α,β to sequences over FRY generated by coefficients Ψ(α),Ψ(β); for any α,β in ZN.

44. And since Ψ(1)=1,then we get that (Ψ,ø) is an isomorphism. It is sufficient to show that if for some k>2 ,(Ψ,ø) preserves k-profiles from FRX to FRY then (Ψ,ø) preserves 2- profiles from FRX to FRY ; and then the theorem is proved by theorem 3.6

45. Let (Ψ,ø) preserves 3-profiles from FRX to FRY. If c=α๏a+β๏b α,β,a,b€ZN • Then we can put that • C=α๏a+β๏b+1๏0 • And since (Ψ,ø) preserves 3-profiles from FRX to FRY, then we get • Ø(c) =Ψ(α) * ø(a)+Ψ(β) * ø(b)+Ψ(1) * ø(0).

46. Since Ψ(1)=1 the identity and ø(0)=0, then • Ø(c) =Ψ(α) * ø(a)+Ψ(β) * ø(b) • This shows that (Ψ,ø) preserves 2- profiles from FRX to FRY. • In a similar method if for any k>3,(Ψ,ø) preserves k-profiles from FRX to FRY,

47. then , we can show that (Ψ,ø) preserves 2-profiles from FRX to FRY .Thus the theorem is proved .

48. Section two • Properties of far rings • In this section ; we shall prove some results for the far rings which are knows for the usual rings . • First we give the following theorem.

49. Theorem 3.9: • The composition of far ring isomorphisms (Ψ,ø): FRX→FRY and (Ψ’,ø’):FRY→FRZ is an isomorphism