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## Chapter three:

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**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.**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.**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**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.**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;**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.**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 .**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.**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.**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.**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.**Example 3.3**• Consider (Z4,๏)and (Z4, *) such that ๏ and * are two binary operations defined by the following tables**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 ;**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.**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 .**The following theorem proves that any two isomorphic far**rings have the same period profiles.**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.**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)**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),…}**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).**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,**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.**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 .**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**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).**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.**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.**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.**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.**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 .**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).**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).**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.**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.**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)**The following theorem gives some conditions to get**isomorphic far rings**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.**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.**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**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).**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,**then , we can show that (Ψ,ø) preserves 2-profiles from**FRX to FRY .Thus the theorem is proved .**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.**Theorem 3.9:**• The composition of far ring isomorphisms (Ψ,ø): FRX→FRY and (Ψ’,ø’):FRY→FRZ is an isomorphism