An example of algebraization of analysis and Fibonacci cobweb poset characterization
aa r X i v : . [ m a t h . C O ] N ov An example of algebraization of analysis andFibonacci cobweb poset characterization
Ewa Krot-SieniawskaInstitute of Computer Science, University in Bia lystokPL-15-887 Bia lystok, ul.Sosnowa 64, POLANDe-mail: [email protected]
Abstract
In [10, 17] inspired by O. V. Viskov [23] it was shown that the ψ -calculus in parts appears to be almost automatic, natural extension ofclassical operator calculus of Rota - Mullin or equivalently - of umbralcalculus of Roman and Rota. At the same time this calculus is an exam-ple of the algebraization of the analysis - here restricted to the algebraof polynomials. The first part of the article is the review of the recentauthor’s contribution [4]. The main definitions and theorems of Finite Fi-bonomial Operator Calculus which is a special case of ψ -extented Rota’sfinite operator calculus [9, 10] are presented there. In the second partthe characterization of Fibonacci Cobweb poset P as DAG and oDAG isgiven. The dim P is constructed. KEY WORDS: Extented umbral calculus, Fibonomial calculus, Fibonaccicobweb poset, DAGAMS 2000 numbers: 05A40, 05C20, 06A11, 11B39, 11C08
In [14] it was shown that: ,, any ψ -representation of finite operator calculus orequivalently - any ψ -representation of GHW algebra makes up an example ofthe algebraization of the analysis - naturally when constrained to the algebra ofpolynomials.(...) Therefore the distinction in-between difference and differenti-ation operators disappears. All linear operators on P ( the algebra of polynomi-als) are both difference and differentiation operators if the degree of differenti-ation or difference operator is unlimited. For example ddx = P k ≥ d k k ! ∆ k where d k = (cid:2) ddx x k (cid:3) x =0 = ( − k − ( k − or ∆ = P n ≥ δ n n ! d n dx n where δ n = [∆ x n ] x =0 =1 . Thus the difference and differential operators and equations are treated on he same footing. ” The authors goal was there ,, to deliver the general schemeof ” ψ -umbral” algebraization of the analysis of general differential operators[19]. ” One may consider a plenty of different special cases of these ψ -extensions.Each of them can be obtained by the special choice of an admissible sequence ψ = { ψ n ( q ) } n ≥ ; ψ n ( q ) = 0; n ≥ ψ ( q ) = 1, or equivalently, by thespecial choice of an sequence n ψ , where ([8, 9, 10, 14]) n ψ ≡ ψ n − ( q ) ψ − n ( q ) , n ≥ , and each of them constitutes a representation of GHW algebra and provides anexample of the algebraization of the analysis.One of most interesting cases is the so called Finite Fibonomial OperatorCalcus (FFOC). Its idea comes from [9] and it was considered by the presentauthor in [4]. It is also the main object of this work. In the first part of it, wepresent some definitions and theorems of FFOC. We take there n ψ = n F = F n ,where the famous Fibonacci sequence { F n } n ≥ ( F n +2 = F n +1 + F n F = 0 , F = 1is attributed and refered to the first edition (lost) of ”Liber Abaci” (1202) byLeonardo Fibonacci (Pisano)(see edition from 1228 reproduced as ”Il LiberAbaci di Leonardo Pisano publicato secondo la lezione Codice Maglibecianoby Baldassarre Boncompagni in Scritti di Leonardo Pisano” , vol. 1, (1857)Rome).In order to formulate main results of FFOC the following objects for thesequence F = { F n } n ≥ are defined:(1) F -factorial: F n ! = F n F n − ...F F , F ! = 1 . (2) F -binomial (Fibonomial ) coefficients [2]: (cid:18) nk (cid:19) F = n kF k F ! = F n F n − . . . F n − k +1 F k F k − . . . F F = F n ! F k ! F n − k ! , (cid:18) n (cid:19) F = 1 . It is known that (cid:0) nk (cid:1) F ∈ N for every n, k ∈ N ∪ ψ -extensions of the umbral calculus (includingFFOC) were presented. As annouced there, the combinatorial interpretationof fibonomial coefficients has been found by A. K. Kwa´sniewski in [15]. It wasdone by the use of the so called Fibonacci cobweb poset [11, 12, 13, 15, 17, 18].In [5, 6] the incidence algebra of the Fibonacci cobweb poset was considered bythe present author. In the second part of this work the characterisation of thisposet P as DAG and oDAG is given. The dim P is constructed. Directed acyclic graphs (DAGs)have many important applications in computer science, including: the parse2ree constructed by a compiler, a reference graph that can be garbage collectedusing simple reference counting, dependency graphs such as those used in in-struction scheduling and makefiles, dependency graphs between classes formedby inheritance relationships in object-oriented programming languages. In the-oretical physics a directed acyclic graph can be used to represent spacetime asa causal set. In bioinformatics, DAGs can be used to find areas of syntenybetween two genomes. They can be also used in abstract process descriptionssuch as workflows and some models of provenance. Let P be the algebra of polynomials over the field K of characteristic zero. Definition 2.1.
The linear operator ∂ F : P → P such that ∂ F x n = F n x n − for n ≥ is named the F -derivative. Definition 2.2.
The F -translation operator is the linear operator E y ( ∂ F ) : P → P of the form: E y ( ∂ F ) = exp F { y∂ F } = X k ≥ y k ∂ kF F k ! , y ∈ KDefinition 2.3. ∀ p ∈ P p ( x + F y ) = E y ( ∂ F ) p ( x ) x, y ∈ KDefinition 2.4.
A linear operator T : P → P is said to be ∂ F -shift invariantiff ∀ y ∈ K [ T, E y ( ∂ F )] = T E y ( ∂ F ) − E y ( ∂ F ) T = 0 We shall denote by Σ F the algebra of F -linear ∂ F -shift invariant operators. Definition 2.5.
Let Q ( ∂ F ) be a formal series in powers of ∂ F and Q ( ∂ F ) : P → P . Q ( ∂ F ) is said to be ∂ F -delta operator iff(a) Q ( ∂ F ) ∈ Σ F (b) Q ( ∂ F )( x ) = const = 0Under quite natural specification the proofs of most statements might bereffered to [8](see also references therein).The particularities of the case considered here are revealed in the sequel.There the scope of new possibilities is initiated by means of unknown beforeexamples. Proposition 2.1.
Let Q ( ∂ F ) be the ∂ F -delta operator. Then ∀ c ∈ K Q ( ∂ F ) c = 0 . roposition 2.2. Every ∂ F -delta operator reduces degree of any polynomial byone. Definition 2.6.
The polynomial sequence { q n ( x ) } n ≥ such that deg q n ( x ) = n and:(1) q ( x ) = 1; (2) q n (0) = 0 , n ≥ (3) Q ( ∂ F ) q n ( x ) = F n q n − ( x ) , n ≥ is called ∂ F -basic polynomial sequence of the ∂ F -delta operator Q ( ∂ F ) . Proposition 2.3.
For every ∂ F -delta operator Q ( ∂ F ) there exists the uniquelydetermined ∂ F -basic polynomial sequence { q n ( x ) } n ≥ . Definition 2.7.
A polynomial sequence { p n ( x ) } n ≥ ( deg p n ( x ) = n ) is of F -binomial (fibonomial) type if it satisfies the condition E y ( ∂ F ) p n ( x ) = p n ( x + F y ) = X k ≥ (cid:18) nk (cid:19) F p k ( x ) p n − k ( y ) ∀ y ∈ K Theorem 2.1.
The polynomial sequence { p n ( x ) } n ≥ is a ∂ F -basic polynomialsequence of some ∂ F -delta operator Q ( ∂ F ) iff it is a sequence of F -binomial type. Theorem 2.2. (First Expansion Theorem)
Let T ∈ Σ F and let Q ( ∂ F ) be a ∂ F -delta operator with ∂ F -basic polynomialsequence { q n } n ≥ . Then T = X n ≥ a n F n ! Q ( ∂ F ) n ; a n = [ T q k ( x )] x =0 . Theorem 2.3. (Isomorphism Theorem)
Let Φ F = K F [[ t ]] be the algebra of formal exp F series in t ∈ K ,i.e.: f F ( t ) ∈ Φ F if f f F ( t ) = X k ≥ a k t k F k ! f or a k ∈ K , and let the Q ( ∂ F ) be a ∂ F -delta operator. Then Σ F ≈ Φ F . The isomorphism φ : Φ F → Σ F is given by the natural correspondence: f F ( t ) = X k ≥ a k t k F k ! into −→ T ∂ F = X k ≥ a k F k ! Q ( ∂ F ) k . Remark 2.1.
In the algebra Φ F the product is given by the fibonomial convo-lution, i.e.: X k ≥ a k F k ! x k X k ≥ b k F k ! x k = X k ≥ c k F k ! x k c k = X l ≥ (cid:18) kl (cid:19) F a l b k − l . Corollary 2.1.
Operator T ∈ Σ F has its inverse T − ∈ Σ ψ iff T = 0 . Remark 2.2.
The F -translation operator E y ( ∂ F ) = exp F { y∂ F } is invertiblein Σ F but it is not a ∂ F -delta operator. No one of ∂ F -delta operators Q ( ∂ F ) isinvertible with respect to the formal series ”F-product”. Corollary 2.2.
Operator R ( ∂ F ) ∈ Σ F is a ∂ F -delta operator iff a = 0 and a = 0 , where R ( ∂ F ) = P n ≥ a n F n ! Q ( ∂ F ) n or equivalently : r (0) = 0 & r ′ (0) = 0 where r ( x ) = P k ≥ a k F k ! x k is the correspondent of R ( ∂ F ) under the IomorphismTheorem. Corollary 2.3.
Every ∂ F -delta operator Q ( ∂ F ) is a function Q ( ∂ F ) accordingto the expansion Q ( ∂ F ) = X n ≥ q n F n ! ∂ nF This F -series will be called the F -indicator of the Q ( ∂ F ) . Remark 2.3. exp F { zx } is the F -exponential generating function for ∂ F -basic polynomial sequence { x n } ∞ n =0 of the ∂ F operator. Corollary 2.4.
The F -exponential generating function for ∂ F -basic polynomialsequence { p n ( x ) } ∞ n =0 of the ∂ F -delta operator Q ( ∂ F ) is given by the followingformula X k ≥ p k ( x ) F k ! z k = exp F { xQ − ( z ) } where Q ◦ Q − = Q − ◦ Q = I = id. Example 2.1.
The following operators are the examples of ∂ F -deltaoperators: (1) ∂ F ; (2) F -difference operator ∆ F = E ( ∂ F ) − I such that(∆ F p )( x ) = p ( x + F − p ( x ) for every p ∈ P ; (3) The operator ∇ F = I − E − ( ∂ F ) defined as follows:( ∇ F p )( x ) = p ( x ) − p ( x − F
1) for every p ∈ P ; (4) F -Abel operator: A ( ∂ F ) = ∂ F E a ( ∂ F ) = P k ≥ a k F k ! ∂ k +1 F ; (5) F -Laguerre operator of the form: L ( ∂ F ) = ∂ F ∂ F − I = P k ≥ ∂ k +1 F .5 efinition 2.8. The ˆ x F -operator is the linear map ˆ x F : P → P such that ˆ x F x n = n +1 F n +1 x n +1 f or n ≥ . ( [ ∂ F , ˆ x F ] = id .) Definition 2.9.
A linear map ’ : Σ F → Σ F such that T ’ = T ˆ x F − ˆ x F T = [ T , ˆ x F ] is called the Graves-Pincherle F -derivative [ ? , ? ] . Example 2.2. (1) ∂ F ’ = I = id ;(2) ( ∂ F ) n ’ = n∂ n − F According to the example above the Graves-Pincherle F -derivative is theformal derivative with respect to ∂ F in Σ F i.e., T ’ ( ∂ F ) ∈ Σ F for any T ∈ Σ F . Corollary 2.5.
Let t ( z ) be the indicator of operator T ∈ Σ F . Then t ′ ( z ) is the indicator of T ’ ∈ Σ F . Due to the isomorphism theorem and the Corollaries above the Leibnitz ruleholds .
Proposition 2.4. ( T S ) ’ = T ’ S + ST ’ ; T , S ∈ Σ F . As an immediate consequence of the Proposition 2.4 we get( S n ) ’ = n S ’ S n − ∀ S ∈ Σ F .From the isomorphism theorem we insert that the following is true. Proposition 2.5. Q ( ∂ F ) is the ∂ F -delta operator iff there exists invertible S ∈ Σ F such that Q ( ∂ F ) = ∂ F S. The Graves-Pincherle F -derivative notion appears very effective while for-mulating expressions for ∂ F -basic polynomial sequences of the given ∂ F -deltaoperator Q ( ∂ F ). Theorem 2.4. ( F -Lagrange and F -Rodrigues formulas) [8, 22, 19] Let { q n } n ≥ be ∂ F -basic sequence of the delta operator Q ( ∂ F ) , Q ( ∂ F ) = ∂ F P ( P ∈ Σ F , invertible). Then for n ≥ : (1) q n ( x ) = Q ( ∂ F ) ’ P − n − x n ; (2) q n ( x ) = P − n x n − F n n ( P − n ) ’ x n − ;(3) q n ( x ) = F n n ˆ x F P − n x n − ; (4) q n ( x ) = F n n ˆ x F ( Q ( ∂ F ) ’ ) − q n − ( x ) ( ← Rodrigues F -formula ) . Corollary 2.6.
Let Q ( ∂ F ) = ∂ F S and R ( ∂ F ) = ∂ F P be the ∂ F -delta operatorswith the ∂ F -basic sequences { q n ( x ) } n ≥ and { r n ( x ) } n ≥ respectively. Then: q n ( x ) = R ’ ( Q ’ ) − S − n − P n +1 r n ( x ) , n ≥ ; (2) q n ( x ) = ˆ x F ( P S − ) n ˆ x − F r n ( x ) , n > . The formulas of the Theorem 2.4 can be used to find ∂ F -basic sequences ofthe ∂ F -delta operators from the Example 2.1. Example 2.3. (1)
The polynomials x n , n ≥ ∂ F -basic for F -derivative ∂ F . (2) Using Rodrigues formula in a straighford way one can find the followingfirst ∂ F -basic polynomials of the operator ∆ F : q ( x ) = 1 q ( x ) = xq ( x ) = x − xq ( x ) = x − x + 3 xq ( x ) = x − x + 24 x − xq ( x ) = x − x + 112 . x − x + 156 . xq ( x ) = x − x + 480 x − x + 4324 x − x. (3) Analogously to the above example we find the following first ∂ F -basicpolynomials of the operator ∇ F : q ( x ) = 1 q ( x ) = xq ( x ) = x + xq ( x ) = x + 4 x + 3 xq ( x ) = x + 9 x + 24 x + 16 xq ( x ) = x + 20 x + 112 . x + 250 x + 156 . xq ( x ) = x + 40 x + 480 x + 2160 x + 4324 x + 2605 x. (4) Using Rodrigues formula in a straighford way one finds the following first ∂ F -basic polynomials of F -Abel operator: A ( a )0 ,F ( x ) = 1 A ( a )1 ,F ( x ) = xA ( a )2 ,F ( x ) = x + axA ( a )3 ,F ( x ) = x − ax + 2 a xA ( a )4 ,F ( x ) = x − ax + 18 a x − a x. (5) In order to find ∂ F -basic polynomials of F -Laguerre operator L ( ∂ F ) we7se formula (3) from Theorem 2.4: L n,F ( x ) = F n n ˆ x F (cid:18) ∂ F − (cid:19) − n x n − = F n n ˆ x F ( ∂ F − n x n − == F n n ˆ x F n X k =0 ( − k (cid:18) nk (cid:19) ∂ n − kF x n − = F n n ˆ x F n X k =0 ( − k (cid:18) nk (cid:19) ( n − n − kF x k − == F n n n X k =1 ( − k (cid:18) nk (cid:19) ( n − n − kF kF k x k . F -polynomials Definition 2.10.
A polynomial sequence { s n } n ≥ is called the sequence of Shef-fer F -polynomials of the ∂ F -delta operator Q ( ∂ F ) iff (1) s ( x ) = const = 0(2) Q ( ∂ F ) s n ( x ) = F n s n − ( x ); n ≥ . Proposition 2.6.
Let Q ( ∂ F ) be ∂ F -delta operator with ∂ F -basic polynomialsequence { q n } n ≥ . Then { s n } n ≥ is the sequence of Sheffer F -polynomials of Q ( ∂ F ) iff there exists an invertible S ∈ Σ F such that s n ( x ) = S − q n ( x ) for n ≥ . We shall refer to a given labeled by ∂ F -shift invariant invertible operator S Sheffer F -polynomial sequence { s n } n ≥ as the sequence of Sheffer F -polynomialsof the ∂ F -delta operator Q ( ∂ F ) relative to S . Theorem 2.5. (Second F - Expansion Theorem) Let Q ( ∂ F ) be the ∂ F -delta operator Q ( ∂ F ) with the ∂ F -basic polynomial se-quence { q n ( x ) } n ≥ . Let S be an invertible ∂ F -shift invariant operator and let { s n ( x ) } n ≥ be its sequence of Sheffer F -polynomials. Let T be any ∂ F -shiftinvariant operator and let p(x) be any polynomial. Then the following identityholds : ∀ y ∈ K ∧ ∀ p ∈ P ( T p ) ( x + F y ) = [ E y ( ∂ F ) p ] ( x ) = T P k ≥ s k ( y ) F k ! Q ( ∂ F ) k S T p ( x ) . Corollary 2.7.
Let s n ( x ) n ≥ be a sequence of Sheffer F -polynomials of a ∂ F -delta operator Q ( ∂ F ) relative to S .Then: S − = X k ≥ s k (0) F k ! Q ( ∂ F ) k . Theorem 2.6. ( The Sheffer F -Binomial Theorem )Let Q ( ∂ F ) , invertible S ∈ Σ F , q n ( x ) n ≥ , s n ( x ) n ≥ be as above. Then: E y ( ∂ F ) s n ( x ) = s n ( x + F y ) = X k ≥ (cid:18) nk (cid:19) F s k ( x ) q n − k ( y ) . orollary 2.8. s n ( x ) = X k ≥ (cid:18) nk (cid:19) F s k (0) q n − k ( x ) Proposition 2.7.
Let Q ( ∂ F ) be a ∂ F -delta operator. Let S be an invertible ∂ F -shift invariant operator. Let { s n ( x ) } n ≥ be a polynomial sequence. Let ∀ a ∈ K ∧ ∀ p ∈ P E a ( ∂ F ) p ( x ) = P k ≥ s k ( a ) F k ! Q ( ∂ F ) k S ∂ F p ( x ) .Then the polynomial sequence { s n ( x ) } n ≥ is the sequence of Sheffer F -polynomialsof the ∂ F -delta operator Q ( ∂ F ) relative to S . Proposition 2.8.
Let Q ( ∂ F ) and S be as above. Let q(t) and s(t) be the in-dicators of Q ( ∂ F ) and S operators. Let q − (t ) be the inverse F -exponentialformal power series inverse to q(t). Then the F -exponential generating functionof Sheffer F -polynomials sequence { s n ( x ) } n ≥ of Q ( ∂ F ) relative to S is givenby X k ≥ s k ( x ) F k ! z k = (cid:0) s (cid:0) q − ( z ) (cid:1)(cid:1) − exp F { xq − ( z ) } . Proposition 2.9.
A sequence { s n ( x ) } n ≥ is the sequence of Sheffer F -polynomialsof the ∂ F -delta operator Q ( ∂ F ) with the ∂ F -basic polynomial sequence { q n ( x ) } n ≥ iff s n ( x + F y ) = X k ≥ (cid:18) nk (cid:19) F s k ( x ) q n − k ( y ) . for all y ∈ KExample 2.4.
Hermite F -polynomials are Sheffer F -polynomials of the ∂ F -delta operator ∂ F relative to invertible S ∈ Σ F of the form S = exp F { a∂ F } . One can get them by formula (see Proposition 2.6 ): H n,F ( x ) = S − x n = X k ≥ ( − a ) k k F k ! n kF x n − k . Example 2.5.
Let S = (1 − ∂ F ) − α − . The Sheffer F -polynomials of ∂ F -delta operator L ( ∂ F ) = ∂ F ∂ F − relative to S are Laguerre F -polynomials oforder α . By Proposition 2.6 we have L ( α ) n,F = (1 − ∂ F ) α +1 L n,F ( x ) , From the above formula and using Graves-Pincherle F -derivative we get L ( α ) n,F ( x ) = X k ≥ F n ! F k ! (cid:18) α + nn − k (cid:19) ( − x ) k for α = − . xample 2.6. Bernoullie’s F -polynomials of order 1 are Sheffer F -polynomialsof ∂ F -delta operator ∂ F related to invertible S = (cid:16) exp F { ∂ F }− I∂ F (cid:17) − . UsingProposition 2.6 one arrives at B n,F ( x ) = S − x n = X k ≥ F k ! ∂ k − F x n = X k ≥ F k (cid:18) nk − (cid:19) F x n − k +1 == X k ≥ F k +1 (cid:18) nk (cid:19) F x n − k Theorem 2.7. (Reccurence relation for Sheffer F -polynomials) Let
Q, S, { s n } n ≥ be as above. Then the following reccurence formula holds: s n +1 ( x ) = F n +1 n + 1 (cid:20) ˆ x F − S ′ S (cid:21) [ Q ( ∂ F ) ′ ] − s n ( x ); n ≥ . Example 2.7.
The reccurence formula for the Hermite F -polynomials is: H n +1 ,F ( x ) = ˆ x F H n,F ( x ) − ˆ a F F n H n − ,F ( x ) Example 2.8.
The reccurence relation for the Laguerre F -polynomials is: L ( α ) n +1 ,F ( x ) = − F n +1 n + 1 [ˆ x F − ( α + 1)(1 − ∂ F ) − ]( ∂ F − L ( α ) n,F ( x )= F n +1 n + 1 [ˆ x F ( ∂ F −
1) + α + 1] L ( α +1) n,F ( x ) . F -polynomials (1) Here are the examples of Laguerre F -polynomials of order α = − L ,F ( x ) = 1 L ,F ( x ) = − xL ,F ( x ) = x − xL ,F ( x ) = − x + 4 x − xL ,F ( x ) = x − x + 18 x − xL ,F ( x ) = − x + 20 x − x + 1280 x − xL ,F ( x ) = x − x + 400 x − x + 1200 x − xL ,F ( x ) = − x + 78 x − x + 10400 x − x + 18720 x − xL ,F ( x ) = x − x + 5733 x − x + 382200 x − x ++ 458640 x − x (2) Here are the examples of Laguerre F -polynomials of order α = 1: L (1)0 ,F ( x ) = 1 L (1)1 ,F ( x ) = − x + 2 L (1)2 ,F ( x ) = x − x + 3 L (1)3 ,F ( x ) = − x + 8 x − x + 8 L (1)4 ,F ( x ) = x − x + 60 x − x + 30 L (1)5 ,F ( x ) = − x + 30 x − x + 600 x − x + 240 L (1)6 ,F ( x ) = x − x + 840 x − x + 8400 x − x + 1680(3) Here we give some examples of the Bernoullie’s F -polynomials of order 1: B ,F ( x ) = 1 B ,F ( x ) = x + 1 B ,F ( x ) = x + x + B ,F ( x ) = x + 2 x + x + B ,F ( x ) = x + 3 x + 3 x + x + B ,F ( x ) = x + 5 x + x + 5 x + x + B ,F ( x ) = x + 8 x + 20 x + 20 x + 8 x + x + B ,F ( x ) = x + 13 x + 52 x + x + 52 x + 13 x + x + B ,F ( x ) = x + 21 x + x + 364 x + 364 x + x + 21 x + x + B ,F ( x ) = x + 34 x + 357 x + 1547 x + x + 1547 x + 357 x ++ 34 x + x + Fibonacci cobweb poset characterization
The Fibonacci cobweb poset P has been invented by A.K.Kwa´sniewski in [15,11, 12] for the purpose of finding combinatorial interpretation of fibonomialcoefficients and eventually their reccurence relation.In [15] A. K. Kwa´sniewski defined cobweb poset P as infinite labeled digraphoriented upwards as follows: Let us label vertices of P by pairs of coordinates: h i, j i ∈ N × N , where the second coordinate is the number of level in whichthe element of P lies (here it is the j -th level) and the first one is the number ofthis element in his level (from left to the right), here i . Following [15] we shallrefer to Φ s as to the set of vertices (elements) of the s -th level, i.e.:Φ s = {h j, s i , ≤ j ≤ F s } , s ∈ N ∪ { } , where { F n } n ≥ stands for Fibonacci sequence.Then P is a labeled graph P = ( V, E ) where V = [ p ≥ Φ p , E = {h h j, p i , h q, p + 1 i i} , ≤ j ≤ F p , ≤ q ≤ F p +1 . We can now define the partial order relation on P as follows: let x = h s, t i , y = h u, v i be elements of cobweb poset P . Then( x ≤ P y ) ⇐⇒ [( t < v ) ∨ ( t = v ∧ s = u )] . −→ oDAG problem In [21] A. D. Plotnikov considered the so called ”DAG −→ oDAG problem”. Hedetermined condition when a digraph G may be presented by the corresponding dim R and he established the algorithm for finding it.Before citing Plotnikov’s results lat us recall (following [21]) some indispens-able definitions.If P and Q are partial orders on the same set A , Q is said to be an extension of P if a ≤ P b implies a ≤ Q b , for all a, b ∈ A . A poset L is a chain , or a linearorder if we have either a ≤ L b or b ≤ L a for any a, b ∈ A . If Q is a linear orderthen it is a linear extension of P .The dimension dim R of R being a partial order is the least positive integer s for which there exists a family F = ( L , L , . . . , L s ) of linear extensions of R such that R = T si =1 L i . A family F = ( L , L , . . . , L s ) of linear orders on A iscalled a realizer of R on A if R = s \ i =1 L i . We denote by D n the set of all acyclic directed n -vertex graphs withoutloops and multiple edges. Each digraph ~G = ( V, ~E ) ∈ D n will be called DAG .12 digraph ~G ∈ D n will be called orderable (oDAG) if there exists are dim ~G .Let ~G ∈ D n be a digraph, which does not contain the arc ( v i , v j ) if thereexists the directed path p ( v i , v j ) from the vertex v i into the vertex v j for any v i , v j ∈ V . Such digraph is called regular . Let D ⊂ D n is the set of all regulargraphs.Let there is a some regular digraph ~G = ( V, E ) ∈ D , and let the chain ~X has three elements x i , x i , x i ∈ X such that i < i < i , and, in thedigraph ~G , there are not paths p ( v i , v i ), p ( v i , v i ) and there exists a path p ( v i , v i ). Such representation of graph vertices by elements of the chain ~X is called the representation in inadmissible form . Otherwise, the chain ~X presets the graph vertices in admissible form .Plotnikov showed that: Lemma 3.1. [21]
A digraph ~G ∈ D n may be represented by a dim poset if:(1) there exist two chains ~X and ~Y , each of which is a linear extension of ~G t ;(2) the chain ~Y is a modification of ~X with inversions, which remove theordered pairs of ~X that there do not exist in ~G . Above lemma results in the algorithm for finding dim
Theorem 3.1. [21]
A digraph ~G = ( V, ~E ) ∈ D n can be represented by dim poset iff it is regular and its vertices can be presented by the chain ~X in admis-sible form. In this section we show that Fibonacci cobweb poset is a DAG and it is orderable(oDAG).Obviously, cobweb poset P = ( V, E ) defined above is a DAG (it is directedacyclic graph without loops and multiple edges). One can also verify that it isregular. For two elements h i, n i , h j, m i ∈ V a directed path p ( h i, n i , h j, m i ) / ∈ E will esist iff n < m + 1 but then ( h i, n i , h j, m i ) / ∈ E i.e. P does not contain theedge ( h i, n i , h j, m i ).It is also possible to verify that vertices of cobweb poset P can be presentedin admissible form by the chain ~X being a linear extension of cobweb P asfollows: ~X = (cid:16) h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , ... (cid:17) , where 13 h s, t i ≤ ~X h u, v i ) ⇐⇒ [( s ≤ u ) ∧ ( t ≤ v )]for 1 ≤ s ≤ F t , ≤ u ≤ F v , t, v ∈ N ∪ { } . Fibonacci cobweb poset P satisfies the conditions of Theorem 3.1 so it isoDAG. To find the chain ~Y being a linear extension of cobweb P one usesLemma 3.1 and arrives at: ~Y = (cid:16) h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , h , i , ... (cid:17) , where ( h s, t i ≤ ~Y h u, v i ) ⇐⇒ [( t < v ) ∨ ( t = v ∧ s ≥ u )]for 1 ≤ s ≤ F t , ≤ u ≤ F v , t, v ∈ N ∪ { } and finally( P, ≤ P ) = ~X ∩ ~Y . Remark 3.1.
For any sequence { a n } of natural numbers one can define corre-sponding cobweb poset as follows [17]:Φ s = {h j, s i , ≤ j ≤ a s } , s ∈ N ∪ { } , and P = ( V, E ) where V = [ p ≥ Φ p , E = {h h j, p i , h q, p + 1 i i} , ≤ j ≤ a p , ≤ q ≤ a p +1 with the partial order relation on P :( x ≤ P y ) ⇐⇒ [( t < v ) ∨ ( t = v ∧ s = u )]for x = h s, t i , y = h u, v i being elements of cobweb poset P . Similary as aboveone can show that the family of cobweb posets consist of DAGs representableby corresponding dim Acknowledgements
I would like to thank Professor A. Krzysztof Kwasniewski for his very helpfulcomments, suggestions, improvements and corrections of this note.
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