Fibonacci sequences in groupoids

Transcription

1 Han et al. Advances in Difference Equations 22, 22:9 RESEARCH Open Access Fibonacci sequences in groupoids Jeong Soon Han, Hee Sik Kim 2* and Joseph Neggers 3 * Correspondence: heekim@hanyang.ac.kr 2 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 33-79, Korea Full list of author information is available at the end of the article Abstract In this article, we consider several properties of Fibonacci sequences in arbitrary groupoids (i.e., binary systems). Such sequences can be defined in a left-hand way and a right-hand way. Thus, it becomes a question of interest to decide when these two ways are equivalent, i.e., when they produce the same sequence for the same inputs. The problem has a simple solution when the groupoid is flexible. The Fibonacci sequences for several groupoids and for the class of groups as special cases are also discussed. 2 Mathematics Subject Classification: 2N2; B39. Keywords: fibonacci sequences, groupoid, flexible, wrapped around Introduction In this article, we consider several properties of Fibonacci sequences in arbitrary groupoids (i.e., binary systems). Such sequences can be defined in a left-hand way and a righthand way. Thus, it becomes a question of interest to decide when these two ways are equivalent, i.e., when they produce the same sequence for the same inputs. The problem has a simple solution when the groupoid is flexible. In order to construct sufficiently large classes of flexible groupoids to make the results interesting, the notion of a groupoid (X, *) wrapping around a groupoid (X, ) is employed to construct flexible groupoids (X, ). Among other examples this leads to solving the problem indicated for the selective groupoids associated with certain digraphs, providing many flexible groupoids and indicating that the general groupoid problem of deciding when a groupoid (X, *)is wrapped around a groupoid (X, ) is of independent interest as well. Given the usual Fibonacci-sequences [,2] and other sequences of this type, one is naturally interested in considering what may happen in more general circumstances. Thus, one may consider what happens if one replaces the (positive) integers by the modulo an integer n or what happens in even more general circumstances. The most generalcircumstanceweshalldealwithisthesituationwhere(x, *) is actually a groupoid, i.e., the product operation * is a binary operation, where we assume no restrictions a priori. 2 Fibonacci sequences in groupoids Given a sequence <,,..., n,... > of elements of X, itisaleft-*-fibonacci sequence if n+2 = n+ * n for n, and a right-*-fibonacci sequence if n+2 = n * n+ for n. Unless (X, *) is commutative, i.e., x * y = y * x for all x, y Î X, there is no reason to assume that left-*-fibonacci sequences are right-*-fibonacci sequences and 22 Han et al.; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Han et al. Advances in Difference Equations 22, 22:9 Page 2 of 7 conversely. We shall begin with a collection of examples to note what if anything can be concluded about such sequences. Example 2.. Let(X, *) be a left-zero-semigroup, i.e., x * y := x for any x, y Î X. Then 2 = * =, 3 = 2 * = 2 =, 4 = 3 * 2 = 3 =,... for any, Î X. It follows that < n > L =<,,,... >. Similarly, 2 = * =, 3 = * 2 =, 4 = 2 * 3 = 2 =,...forany, Î X. It follows that < n > R =<,,,,,,... >. In particular, if we let :=, :=, then < n > L = <,,,,,... > and < n > R = <,,,,,,... >. A groupoid (X, *) is said to be a leftoid if x * y = l(x), a function of x in X, for all x, y Î X. We denote it by (X, *,l). Proposition 2.2. Let (X, *,l) be a leftoid and let < n >be a left-*-fibonacci sequence on X. Then < n >=<,, l( ), l (2) ( ),..., l (n) ( ),...>, where l (n+) (x) =l(l n (x)). Proof. If< n > is a left-*-fibonacci sequence on X, then 2 = * = l( )and 3 = 2 * = l( 2 )=l (2) ( ). It follows that k+ = k * k- =( k )=l (k) ( ). A groupoid (X, *) is said to be a rightoid if x * y = r(y), a function of y in X, for all x, y Î X. We denote it by (X, *,r). Proposition 2.2. Let (X, *,r) be a rightoid and let < n >be a right-*-fibonacci sequence on X. Then < n >=<,, r ( ), r( ), r (2) ( ), r (2) ( ), r (3) ( ), r(3)( ),..., r (n) ( ), r (n) ( ),... where r (n+) (x) =r(r n (x)). In particular, if l (r, resp.) is a constant map in Proposition 2.2 (or Proposition 2.2, resp.), say l( )=l( )=a for some x Î X, then < n >=<,, a, a,... >. Theorem 2.3. Let < n > L and < n > R be the left-*-fibonacci and the right-*-fibonacci sequences generated by and. Then < n > L =< n > R if and only if n *( n- * n ) =( n * n- )* n for any n. Proof. If< n > L =< n > R,then * = 2 = * and hence ( * )* = 2 * = 3 = * 2 = *( * ). Similarly ( 2 * )* 2 = 3 * 2 = 4 = 2 * 3 = 2 *( * 2 ). By induction on n, we obtain n *( n- * n )=( n * n- )* n for any n. If we assume that n *( n- * n )=( n * n- )* n for any n, then n * n+ = n *( n- * n )=( n * n- )* n = n+ * n for any n. A groupoid (X, *) is said to be flexible if (x * y) *x = x *(y * x) for any x, y Î X. Proposition 2.4. Let X := R be the set of all real numbers and let A, B Î R. Then any groupoid (X, *)of the types x*y := A + B(x + y) or x*y := Bx +(-B)y for any x, y Î X is flexible. Proof. Define a binary operation * on X by x * y := A + Bx + Cy for any x, y Î X, where A, B, C Î X. Assume that (X, *)isflexible.then(x * y) *x = x *(y * x) for any x, y Î X. It follows that A + B(x + y) +Cx = A + Bx + C(y * x). It follows that AB + B(B )x = AC + C(C )x () for any x Î X. Ifweletx := in (), then we obtain AB = AC. IfA, then B = C and x * y = A + B(x + y). If A =, then it follows from () that B(B )x = C(C )x (2) for any x Î X. If we let x := in (2), then we obtain B(B -)=C(C - ) and hence C = ± +4B(B ) 2 = { B B,

3 Han et al. Advances in Difference Equations 22, 22:9 Page 3 of 7 i.e., x * y = Bx + By or x * y = Bx +(-B)y. This proves the proposition. Proposition 2.5. Let (X, *)be a flexible groupoid. Then (X, *)is commutative if and only if <,,... > L =<,,... > R for any, Î X. Proof. Given, Î X, * = * = 2 since (X, *) is commutative. Since (X, *) is flexible, we obtain ϕ 2 ϕ =(ϕ ϕ ) ϕ = ϕ (ϕ ϕ ) = ϕ ϕ 2 = ϕ 3 By induction on n, we obtain ϕ n ϕ n =(ϕ n ϕ n 2 ) ϕ n = ϕ n (ϕ n 2 ϕ n ) = ϕ n ϕ n Hence <,,... > L =<,,... > R. The converse is trivial, we omit the proof. Example 2.6. LetX := R be the set of all real numbers and let x * y := -(x + y) for any x, y Î X. Consider aright-*-fibonacci sequence < n > R, where, Î X. Since 2 = * =-( + ), 3 = * 2 =-[ + 2 ]=-[ -( + )] =, 4 = 2 * 3 =,..., we obtain < n > R =<,,-( + ),,,-( + ),,,-( + ),,,... > and n+3 = n (n =,, 2,...). Since (X, *) is commutative and flexible, by Proposition 2.5, < n > L =< n > R. Proposition 2.7. Let (X, *) be a groupoid satisfying the following condition: (x y) x = x (y x) =y (3) for any x, y Î X. Then < n > L =< n > R if * = *. Proof. Straightforward. Proposition 2.8. The linear groupoid (R, *),with x * y := A -(x + y), x, y Î R, where A Î R, is the only linear groupoid satisfying the condition (3). Proof. By Proposition 2.4, we consider two cases: x * y := A + B(x + y) orx * y := Bx +(-B)y where A, B Î R. Assume that x * y := A + B(x + y). Since y =(x * y) *x, we have y =(x y) x = A + B(x y + x) = A + B(A + B(x + y)+x) = A(B +)+B(B +)x + B 2 y It follows that B 2 =,B(B +)=,A(B + ) =. If B =, then = B(B + ) = 2, a contradiction. If B = -, then A is arbitrary. Hence x * y = A -(x + y). Assume that x * y := Bx +(-B)y. Since y =(x * y) *x, we have y =(x y) x =[Bx +( B)y] x = B[Bx +( B)y]+( B)x =(B 2 B +)x + B( B)y

4 Han et al. Advances in Difference Equations 22, 22:9 Page 4 of 7 It follows that B 2 - B +=,B( - B) = which leads to B = ± 3i R, a 2 contradiction. This proves the proposition. 3 Flexibility in Bin(X) Given groupoids (X, *) and (X, ), we consider (X, *)tobewrapped around (X, ) if for all x, y, z Î X, (x y) *z = z *(y x). If (X, *) and (X, ) are both commutative groupoids, then (x y) *z = z *(x y) =z *(y x) and(x * y) z = z (x * y) for all x, y, z Î X, i.e., (X, *) and (X, ) are wrapped around each other. Example 3.. Let X := R be the set of all real numbers and let x * y := x 2 y 2, x y := x-y for all x, y Î X. Then(x y) *z =(x - y) *z =(x - y) 2 z 2 = z *(y x) for all x, y, z Î X, i.e.,(x, *) is wrapped around (X, ). On the other hand, (x * y) z = x 2 y 2 - z and z (y * x) =z- y 2 x 2 so that (X, ) is not wrapped around (X, *). The notion of the semigroup (Bin(X), ) was introduced by Kim and Neggers [3]. They showed that (Bin(X), ) is a semigroup, i.e., the operation as defined in general is associative. Furthermore, the left-zero semigroup is an identity for this operation. Proposition 3.2. Let (X, *)be wrapped around (X, ). If we define (X, ) :=(X, *) (X, ), i.e., x y := (x * y) (y * x) for all x, y Î X, then (X, ) is flexible. Proof. Givenx, y, z Î X, since(x, *)iswrappedaround(x, ), we obtain (X, *)be wrapped around (X, ) (x y) x =[(x y) x] [x (x y)] =[{(x y) (y x)} x] [x {(x y) (y x)}] =[x {(y x) (x y)}] [{(y x) (x y)} x] =[x (y x)] [(y x) x] = x (y x), proving the proposition. Example 3.3. Note that in the situation of Example 3., we have x y =(x * y) (y * x) =x 2 y 2 - y 2 x 2 =forallx, y Î X, i.e.,(x, ) =(X, *) (X, ) isatrivialgroupoid(x,, t) where t = and x y = for all x, y Î X. Example 3.4. In Example 3., if we define (X, ) :=(X, ) (X, *), i.e., x y := (x y) * (y x) for all x, y Î X, then x y =(x-y) 4 and hence (x y)δx =((x - y) 4 - x) 4 =(x -(y - x) 4 ) 4 = x (y x). Hence (X, ) is a flexible groupoid. Note that (X, ) isnotasemigroup, since ( z) =z 6 z 4 =( ) z. Obviously,x y = y x for all x, y Î X. By applying Proposition 2.5, we obtain <,,... > L =<,,... > R for any, Î (X, ). We obtain a Fibonacci- -sequence in the groupoid (X, ) discussed in Example 3.4 as follows: Example 3.5. Consider a groupoid (X, ) in Example 3.4. Since x y =(x - y) 4,given, Î X, we have 2 = = =( - ) 4, and 3 = 2 =( 2 - ) 4 = [( - ) 4 - ] 4. In this fashion, we have 4 = [[( - ) 4 - ] 4 -( - ) 4 ] 4. In particular, if we let = =, then 2 =, 3 = 4 =, 5 =, 6 = 7 =, 8 =,... Hence <,,,,,,,,,... > is a Fibonacci- -sequence in (X, ).

5 Han et al. Advances in Difference Equations 22, 22:9 Page 5 of 7 4 Limits of *-Fibonacci sequences In this section, we discuss the limit of left(right)-*-fibonacci sequences in a real groupoid (R, *). Proposition 4.. Define a binary operation * on R by x y := (x + y)for any x, y Î 2 R. If < n >is a *-Fibonacci sequence on (R, *), then lim n ϕ n = 3 (ϕ +2ϕ ). Proof. Since x y = 2 (x + y) =y x for any x, y Î R, < n > L =< n > R for any, Î R. It can be seen that ϕ 2 = 2 (ϕ + ϕ ), ϕ 3 = 2 2 (ϕ +3ϕ ). We let ϕ 3 = 2 2 (A 3ϕ + B 3 ϕ ). Since ϕ 4 = ϕ 2 ϕ 3 = [ ϕ + ϕ 2 2 = 2 3 [3ϕ +5ϕ ], + ] 4 (ϕ +3ϕ ) we let it by ϕ 4 = 2 3 [A 4ϕ + B 4 ϕ ]. In this fashion, if we let ϕ n+2 := 2 n+ [A n+2ϕ + B n+2 ϕ ], we have ϕ n+2 = 2 [ϕ n + ϕ n+ ] = [ An ϕ + B n ϕ 2 2 n + A ] n+ϕ + B n+ ϕ 2 n = 2 n+ [(2A n + A n+ )ϕ +(2B n + B n+ )ϕ ] It follows that A n+2 =2A n + A n+, A 3 =,A 4 =3 B n+2 =2B n + B n+, B 3 =3,B 4 =5 so that the evolution for A k is <, 3, 5,, 2, 43, 85,... > = < A 3,A 4,A 5,... > and the evolution for B k is < 3, 5,, 2, 43, 85,... > so that B k = A k+. If we wish to solve explicitly for A k, we note that the corresponding characteristic equation is r 2 - r -2= with roots r =2orr = -, i.e., A k+3 = a2 k + b(-) k, a + b = when k =,and2a - b = 3 when k = so that α = 4 3, β = 3. Hence A k+3 = 3 [2k+2 +( ) k+ ] and B k+3 = A k+4 = 3 [2k+3 +( ) k+2 ]. It follows that ϕ n+3 = [ 2 k+2 3 (2k+2 +( ) k+ )ϕ + ] 3 (2k+3 +( ) k+2 )ϕ [{ } { } ] = + ( )k+ 3 2 k+2 ϕ + 2+ ( )k+2 2 k+2 ϕ This shows that lim n ϕ n = 3 (ϕ +2ϕ ), proving the proposition.

6 Han et al. Advances in Difference Equations 22, 22:9 Page 6 of 7 Proposition 4.2. Define a binary operation * on R by x * y := Ax +(-A)y, <A <, for any x, y Î R. If < n > L is a left-*-fibonacci sequence on (R, *), then lim n ϕ n = 2 A [( A)ϕ + ϕ ]. Proof. Given, Î R, weconsider< n > L.Since 2 = * = A +(-A) and 3 = 2 * =(A 2 -A +) + A(-A),...,ifweassumethat n := A n + B n (n 2), then ϕ n+2 = ϕ n+ ϕ n =(A n+ ϕ + B n+ ϕ ) (A n ϕ + B n ϕ ) It follows that = A(A n+ ϕ + B n+ ϕ )+( A)(A n ϕ + B n ϕ ) A n+2 = AA n+ +( A)A n, B n+2 = AB n+ +( A)B n Hence we obtain the characteristic equation r 2 - Ar -(-A) = with roots r =or r = A -. Thus A n+2 = a(a -) n+2 + b for some a, b Î R. Since A = A 2 = a(a -) 2 + b and A 2 - A +=A 3 = a(a -) 3 + b, weobtainα = A 2 and β = so that 2 A (A )n+2 A n+2 = A 2. For B n+2 we obtain the same characteristic equation r 2 - Ar -(- A) = with roots r =orr = A -.HenceB n+2 = g(a -) n+2 + δ for some g, δ Î R. Since - A = B 2 = g(a - ) 2 + b, A( - A) = B 3 = g(a - ) 3 + δ, we obtain γ 2 A, δ = A 2 A so that B n+2 = 2 A [(A )n+2 +( A)]. Since <A <, lim n (A - ) n+2 =. It follows that lim n ϕ n+2 = lim n (A n+2 ϕ +B n+2 ϕ ) = (lim n A n+2 )ϕ +(lim n B n+2 )ϕ = 2 A [ϕ +( A)ϕ ], proving the proposition. Note that if A = 2 in Proposition 4.2, then lim n ϕ n = 3 (ϕ +2ϕ ) as in Proposition 4.. Note that (R, *) in Proposition 4.2 is neither a semigroup nor commutative and we may consider a right-*-fibonacci sequence < n > R on (R, *). 5 Fibonacci sequences in a group In this section, we discuss *-Fibonacci sequence in groups. Example 5.. Suppose that X = S 4 is a symmetric group of order 4 and suppose that = (3), = (2). We wish to determine < n > L. Since 2 = = (2)(3) = (23), 3 = 2 = (23)(3) = (2), 4 = 3 2 = (2)(23),..., we obtain < n > L =<(3), (2), (23), (2), (3), (32), (23), (3), (23), (2), (3),... >, i.e., it is periodic of period 6. Proposition 5.2. Let (X,, e)be a group and let, be elements of X such that =. If < n > L is a left- -Fibonacci sequence in (X,, e) generated by and, then ϕ k+2 = ϕ F k+2 ϕ F k+. In particular, if =, then ϕ k+2 = ϕ F k+3 where F k is the k th Fibonacci number.

7 Han et al. Advances in Difference Equations 22, 22:9 Page 7 of 7 Proof. Let, be elements of X such that =. Since < n > L is a left- -Fibonacci sequence in (X,, e) generated by and, we have ϕ 3 = ϕ 2 ϕ, ϕ 4 = ϕ 3 ϕ2, ϕ 5 = ϕ 5 ϕ5 = ϕf 3 ϕf 4, ϕ 6 = ϕ 8 ϕ5 = ϕf 6 ϕf 4 and ϕ 7 = ϕ F 7 ϕf 6. If we assume that ϕ k = ϕ F k ϕf k and ϕ k+ = ϕ F k+ ϕ F k, then ϕ k+2 = ϕ k+ ϕ k = ϕ k+ ϕ F k ϕf k ϕf k = ϕ F k+2 ϕ F k+. In particular, if =, then ϕ k+2 = ϕ F k+2 ϕ F k+ = ϕ F k+3. Proposition 5.3. Let (X,, e) be a group and let, be elements of X such that =. If < n > L is a full left- -Fibonacci sequence in (X,, e) generated by and, then ϕ (2k) = ϕ F k+ ϕ F 2k and ϕ (2k+) = ϕ F 2(k+) ϕ F 2k+. Proof. Since = -,wehave ϕ = ϕ ϕ. It follows from = - -2 that ϕ 2 = ϕ 2ϕ.Inthisfashion,since =,weobtain ϕ 3 = ϕ F 4 ϕ F 3 and ϕ 4 = ϕ F 5 ϕ F 4. By induction, assume that ϕ (2k) = ϕ F k+ ϕ F 2k and ϕ (2k+) = ϕ F 2(k+) ϕ F 2k+. Then we obtain and ϕ (2k+2) =[ϕ (2k+) ] ϕ 2k = ϕ F 2k+2 ϕ F 2k+ ϕ F 2k+ ϕ F 2k = ϕ F 2k+2+F 2k+ ϕ (F 2k++F 2k ) = ϕ F 2k+3 ϕ F 2k+2, ϕ (2k+3) =[ϕ (2k+2) ] ϕ (2k+) [ ] = ϕ F 2k+3 ϕ F 2k+2 ϕ F 2k+2 ϕ F 2k+ = ϕ F 2k+2+F 2k+ ϕ (F 2k+2+F 2k+3 ) = ϕ F 2k+4 ϕ F 2k+3, proving the proposition. Author details Department of Applied Mathematics, Hanyang University, Ahnsan , Korea 2 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 33-79, Korea 3 Department of Mathematics, University of Alabama, Tuscaloosa, AL , USA Authors contributions All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 3 January 22 Accepted: 27 February 22 Published: 27 February 22 References. Atanassov, K, Atanassova, V, Shannon, A, Turner, J: New Visual Perspectives on Fibonacci Numbers. World Scientific, New Jersey (22) 2. Dunlap, RA: The Golden Ratio and Fibonacci Numbers. World Scientific, New Jersey (997) 3. Kim, HS, Neggers, J: The semigroups of binary systems and some perspectives. Bull Korean Math Soc. 45, (28) doi:.86/ Cite this article as: Han et al.: Fibonacci sequences in groupoids. Advances in Difference Equations 22 22:9.