Stud.Cercet.Stiint., Ser.Mat., 16 (2006), Supplement Proceedings of ICMI 45, Bacau, Sept.18-20, 2006, pp. 263-268
Properties of the Generalized Permutations
N. Shajareh-Poursalavati
Abstract
In this article, we study some properties of the generalized permutations. We introduce a fundamental structure of all generalized permutations and by use this concept, we introduce another generalization of permutation matrices, also we obtain some properties of generalized permutation matrices. Key words and phrases: generalized permutation, directed graph, semi- inverse, generalized permutation matrices. (2000) Mathematics Subject Classification: 20B99
1. INTRODUCTION Suppose that X is a non empty set and let P* (X ) be the set of all non empty subset of X . A generalized permutation, which is a generalization of the concept of ordinary permutation, is defined in [2], as follows. The function σ : X → P* (X ) is called a generalized permutation on X , if σ (x) = X . xU∈X Obviously, if we consider the singleton {f (x)} instead f (x) , for every ordinary permutation f on X , then f is a generalized permutation. The set of all generalized permutation on X is denoted by M X . For f1 , f 2 ∈ M X , we say that f1 is a sub generalized permutation of f 2 , or f 2 contains f1 , and write f1 ⊆ f 2 , if f1 (x) ⊆ f 2 (x) for every x ∈ X . The mapping g for which g(x) = X for all x ∈ X , is called the universal generalized permutation and * contains all the element of M X . Every map f : X → P (X ) which contains a generalized permutation is itself a generalized permutation. The map i : X → P* (X ) wherei(x) = {x }, for all x ∈ X , is called the identity generalized permutation. The composition of two generalized permutations
f1 and f 2 is defined as follows:
263 f f ( x ) = f ( y ) ∀x ∈ X . 1 o 2 U 1 y ∈ f 2 ( x )
It is easy to show that f1 o f 2 is a generalized permutation. By this composition, we have a binary operation on the set of all generalized permutation, i.e., f1 o f 2 ∈ M Ω . Definition 1.1. Let f be a generalized permutation on X . The semi-inverse of f is denoted by f ~1 , define as f ~1 (y) = {x ∈ X : y ∈ f (x )}. It is easy to show that identity element of M X , is a sub generalized permutation of ~1 ~1 ~1 ~1 f o f and f o f , i.e.,x ∈ f o f (x) I f o f (x ) for all ∀x ∈ X .
2. GRAPH OF A GENERALIZED PERMUTATION Let f be a generalized permutation on X . We consider X , the set of vertices and for two vertices "x" and "y", we define a directed edge (arc) from "x" to "y", if y ∈ f (x ) . The graph of a generalized permutation is denoted by G(f). The number of the input or output arcs to a vertex α is called input or output degree in α and denoted by id(α) or od(α) , respectively. In [1], we prove that, there is a one to one correspondence between the set of all generalized permutation on the non empty set X and the set of all directed graphs with vertices in X , where input and output degrees in each vertex be positive. Also we introduced a presentation of a generalized permutation f over a finite set X of order n. Without loss of generality, we consider X as the set {1,2,K,n }. Definition 2.1. Let x be a vertex of G(f) and k be the smallest positive integer k numbers such that x ∈ f (x ) . The k-cycle (x0 x1 x2 K xk −1 ) where x = x0 ,
xi ∈ f (xi−1 ), i = 1,2,K,k −1 and x ∈ f (xk −1 ) , is called a closed orbit of length k. Note that every loop is a 1-cycle or closed orbit of length 1. Definition 2.2. Let x be a vertex of G(f) that lies in a closed orbit or loop. The k-symbol order [ x0 x1 x2 K xk −1 ] where k is a positive integer number,
x = x0 , xi ∈ f (xi−1 ), for all i = 1,2,K,k − 1 and xk −1 lies in a closed orbit, and for all i = 1,2,K,k − 2 , xi does not lie in any closed orbit of G(f), is called a pure chain of length k-1.
264 The set of all strongly-connected component’s generalized permutations on a set X is denoted by SM X . In [1], we show that, every elements of SM X is a collection of ordinary cycle permutations over X . Also, we show that, every elements of M X is a collection of closed orbits and pure chains over X . Example 2.3. Let X = {1,2,3,4,5,6} and let f be generalized permutation defined by: 1 2 3 4 5 6 7 f := . {}{}{}{}{}{}{}1,2 3 1,4 5 6 7 5 We present f as the collection {(1), (1 2 3), (5 6 7), [3 4 5]} where (,1 ) (,1 2 3 ) (5 6 7) are closed orbits of length 1, 3 and 3 respectively and []3 4 5 is a pure chain of length 3.
In [1], we obtain an upper bound for SM X and M X . If X be a non empty set of cardinal n, and n(n −1) n(n −1)(n − 2) m = n + + + + (n −1)!, and 2 3 L l = n + n(n −1) + n(n −1)(n − 2) +L+ n!, m m+l then SM X ≤ 2 and M X ≤ 2 . In the next section we define a new generalization of permutation matrices.
3. GENERALIZED PERMUTATION MATRICES
Let R be a field with identity 1R and characteristic of R be zero. By definition of generalized permutation, we can define another generalization of permutation matrix as follows: Definition 3.1. Let X = {}1,2,K,n and σ be a generalized permutation on
X . The generalized permutation matrix Pσ with n elements is defined as: e σ (1) eσ (2) Pσ = , M e σ (n) n×n
265 with for non empty subset S of X , eS := (x1 x2 L xn )1×n , where
1R if i ∈ S xi :.= 0 if i ∉ S In other words, a generalized permutation matrix is an square matrix which its arrays are 0 or 1R , and for each column and row there are at least one array non zero. Example 3.2. Let X = {1,2,3,4,5,6 } and let σ be a generalized permutation defined as follows: 1 2 3 4 5 6 σ = {2,3} {3} {4} {5} {3} {1,6} i.e., σ (1) = {2,3}, σ (2) = {3}, σ (3) = {4}, σ (4) = {5}, σ (5) = {3}, σ (6) = {1,6}. Then 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 P := σ 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 16×6 The transpose of a matrix A = a is denoted by AT :,= b which ( ij )n×m ( ji )m×n b ji = aij for all 1 ≤ i ≤ n and 1 ≤ j ≤ m .
T n Theorem 3.3. If α = ()α1 α 2 Lα n is a vector in R and Pσ is a generalized permutation matrix over {1,2,K,n} then T P α = α , α , , α . σ ∑ i ∑ i K ∑ i i∈σ (1) i∈σ (2) i∈σ (n)
Proof. It is easy to show that (Pσ α)i1 = eσ (i)α = ∑α j for all 1 ≤ i ≤ n . j∈σ (i) □ Theorem 3.4. If σ and π be two generalized permutation over {1,2,K,n}, then for all 1 ≤ i, j ≤ n , (P P ) = 0 if and only if (P ) = 0 . σ π ij π oσ ij
266 Proof. Note that (P ) = 0 if and only if j ∉π σ (i) . Now we calculate π oσ ij o the (Pσ Pπ )ij , we have:
n (P P ) σ π ij = ∑(eσ (i) )k (eπ (k ) ) j k=1 =|{k ∈σ (i): j ∈π (k)}|
~1 ~1 =||{k ∈σ (i):k ∈π ( j)}|=| σ (i) I π ( j) . Hence,
~1 (Pσ Pπ )ij = 0 ⇔ σ (i) I π ( j) = { } ⇔ {k ∈σ (i): j ∈π (k)} = { } ⇔ j ∉π oσ (i) ⇔ (P ) = 0 π oσ ij □
Remark 3.5. In general, Pσ Pπ is not a generalized permutation matrix. As an example, which Pσ Pπ not be generalized permutation matrix, consider 1 2 1 2 σ := and π := generalized permutations on the set {}{}1,2 2 {}1 {1,2 } 1 1 1 0 2 ⋅1 1 R R R R R {}1,2 . Then Pσ = , Pπ = and Pσ Pπ = not a 0 1R 1R 1R 1R 1R generalized permutation matrix. In each case, if Pσ Pπ is a generalized permutation matrix, in Theorem 3.6, we show that P P = P . σ π π oσ In [1], we defined the directed graph and in Definition 1.1 the semi-inverse of a generalized permutation was introduced. Hence, naturally, we can define the ~1 semi-inverse of a generalized permutation matrix, i.e., P := P ~1 . But in the σ σ following theorem we show that this is not a new definition. ~1 T Theorem 3.6. Let σ be a generalized permutation. Then P := P ~1 = P . σ σ σ Proof. We know that the arrays of a generalized permutation matrix, are 0 or
1R . Now we have:
267 P = 1 ⇔ e = 1 ()σ ij R ( σ ()i )1 j R ⇔ j ∈σ (i)
~1 ⇔ i ∈σ ( j) ⇔ (e ~1 ) = 1R ⇔ (P ~1 ) = 1R . σ ( j) 1i σ ji
Therefore P = P ~1 for all i and j. This completes proof. ()σ ij ( σ )ji □
Theorem 3.7. The generalized permutation matrices P ~1 and P ~1 are σ oσ σ oσ symmetric matrices which the diagonal arrays are 1R .
Proof. For every 1,≤ i, j ≤ n (P ~1 )ij = (e ~1 ) j . Now by use the graph of σ oσ σ oσ (i) ~1 ~1 σ and semi-inverse of σ we show that j ∈σ oσ (i) ⇔ i ∈σ oσ ( j ) . Let ~1 ~1 j ∈σ oσ (i) , hence there is a k in σ (i ) such that j ∈σ (k) and i ∈σ (k) , ~1 then i ∈σ σ ( j) . Therefore (P ~1 )ij = (P ~1 ) ji and similarly o σ oσ σ oσ ~1 ~1 (P ~1 )ij = (P ~1 ) ji . It is clearly that i ∈σ σ (i ) and i ∈σ σ (i ) , σ oσ σ oσ o o therefore (P ~1 )ii = 1R = (P ~1 )ii and the proof complet. σ oσ σ oσ □
REFERENCES [1] Shajareh-Poursalavati, N. On the generalized permutations. Journal of Discrete Mathematical Sciences & Cryptography, 8, 3 (Dec. 2005), 365- 672. [2] Vougiouklis, T. Hyperstructures and their representations. Hadronic Press Monographs in Mathematics, Palm Harbor, FL, 1994.
Department of Mathematics, Shahid Bahonar University of Kerman mail address: [email protected]
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