Bilal Khan Department of Mathematics and Computer Science, John Jay College of Criminal Justice, City University of New York, New York, NY 10019, USA.

Transcription

1 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 A GRAPHIC GENERALIZATION OF ARITHMETIC Bilal Khan Deparmen of Mahemaic and Compuer Science, John Jay College of Criminal Juice, Ciy Univeriy of New York, New York, NY 10019, USA. bkhan@jjay.cuny.edu Kiran R. Bhuani Deparmen of Mahemaic, The Caholic Univeriy of America, Wahingon DC 20064, USA. bhuani@cua.edu Delaram Kahrobaei New York Ciy College of Technology, Ciy Univeriy of New York, New York, NY 11201, USA. DKahrobaei@CiyTech.Cuny.edu Received: 1/21/05, Revied: 1/17/07, Acceped: 1/31/07, Publihed: 2/28/07 Abrac In hi paper, we inroduce a naural arihmeic on he e of all flow graph, ha i, he e of all finie direced conneced muligraph having a pair of diinguihed verice. The propoed model exhibi he propery ha he naural number appear a a ubmodel, wih he direced pah of lengh n playing he role of he andard ineger n. We inveigae he baic feaure of hi model, including aociaiviy, diribuiviy, and variou ideniie relaing he order relaion o addiion and muliplicaion. 1. Inroducion The language of arihmeic L coni of wo conan 0 and 1, one binary relaion, and wo binary operaion + and. In hi paper, we generalize claical arihmeic defined over he naural number N = {0, 1, 2,...}, o he e F coniing of all flow graph: finie direced conneced muligraph 1 in which a pair of diinguihed verice i deignaed a he ource and arge verex. We give naural inerpreaion for L on he e F. To avoid confuion wih he andard model of arihmeic, he correponding operaion in 1 By muligraph we mean graph in which parallel and loop edge are permied.

2 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 2 F are denoed wih a circumcribed circle. The new model F = F, 0, 1,, +, i a naural exenion of he andard model N = N, 0, 1, +,. Specifically, we exhibi an embedding i : N i F aifying: i(0) = 0, i(1) = 1, x, y N, x y i(x) i(y), x, y N, i(x + y) = i(x)+i(y), x, y N, i(x y) = i(x)i(y). There have been oher aemp o define algebraic and meric rucure on he e of all graph. In [6, 2, 1], he auhor ued graph embedding o define a meric on he e of all imple conneced graph of a given order. Thi work differ from hoe inveigaion in ha i conider an infinie collecion of graph in order o exend he andard model of arihmeic, and in doing o doe no eek o eablih a meric rucure. The claical operaion on graph [9] (including exenive lieraure on graph produc [5]) have yielded many reul and a deep mahemaical heory. There ha alo been coniderable prior work on addiion and muliplicaion of ordinal and parially ordered e [3, 4, 7, 8]. To dae, hee prior inveigaion have no yielded an inerpreaion of he language of arihmeic on graph. Thi paper preen reul and open queion in hi direcion. 2. Flow Graph Definiion 2.1 (Flow graph). We define a flow graph A o be a riple (G A, A, A ), where G A = (V A, E A ) i a finie 2 direced conneced muligraph and E A i a muliube of V A V A. Noe ha hi definiion permi parallel and loop edge 3. Given verice u and v, we denoe (u, v) o be he number of edge from u o v. Individual parallel edge from u o v will be referred o a (u, v) 1, (u, v) 2,..., (u, v) i,..., (u, v) (u,v). However, if he argumen doe no depend on a pecific edge from u o v, he ubcrip will be dropped he expreion (u, v) will be ued o mean any one of (poibly many) parallel edge from u o v. The verice A, A V A are called he ource and he arge verex of A, repecively. The e of all flow graph i denoed F. Definiion 2.2 (Flow graph morphim). Le A = (G A, A, A ) and B = (G B, B, B ) be wo flow graph wih G A = (V A, E A ) and G B = (V B, E B ). A map φ : A B i 2 In hi paper, we focu on finie flow graph, alhough many of our reul coninue o hold in he formulaion which conider infinie flow graph a well. 3 We ay ha wo edge e 1 = (u 1, v 1 ) and e 2 = (u 2, v 2 ) are parallel if u 1 = u 2 and v 1 = v 2. An edge e = (u, v) i called a loop edge if u = v.

3 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 3 called a flow graph morphim if (1) A a map of verex e, φ : V A V B repec edge rucure: e = (u, v) E A φ(e) = (φu, φv) E B (2) Source and arge are preerved, i.e. φ( A ) = B, φ( A ) = B. A flow graph morphim i aid o be a flow graph embedding of A ino B if addiionally φ i injecive on boh V A and E A. Flow graph A and B are conidered iomorphic if here i a flow graph embedding φ : A B for which φ(e A ) = E B. Clearly, flow graph iomorphim define an equivalence relaion on flow graph. In hi paper, we hall only conider properie of flow graph which are invarian wih repec o hi equivalence relaion. Conequenly, when dicuing an equivalence cla of flow graph, we will conduc our analyi by rericing ourelve o an arbirary repreenaive from he cla. Whenever we refer o A flow graph F, we hall inend Any flow graph from he equivalence cla of F, bu we will ue he former phrae for uccincne. Likewie, we wrie A = B for flowgraph o indicae only ha A and B are iomorphic a flow graph. Definiion 2.3 (Trivial flow graph). A flow graph A = (G A, A, A ) i called he rivial flow graph if V [G A ] = 1 and E[G A ] = 0. All oher flow graph are conidered nonrivial. Definiion 2.4. Given any flow graph A, le A be he flow graph obained by wapping he ource and he arge of A. Definiion 2.5 (Reflecive flow graph). A flow graph A = (G A, A, A ) i called an reflecive flow graph if A = A. The e of all reflecive flow graph i denoed H. Definiion 2.6 (Infinieimal flow graph). A flow graph A = (G A, A, A ) i called an infinieimal flow graph if A = A. The e of all infinieimal flow graph i denoed I. Noe ha an infinieimal flow graph i necearily reflecive. The convere i fale a he reflecive example in Figure 1 how. Figure 1: A non-infinieimal flow graph in H. Definiion 2.7. Given any flow graph A, le A be he flow graph obained by revering all he arrow of A.

4 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 4 Definiion 2.8 (Reverible flow graph). A flow graph A = (G A, A, A ) i called a reverible flow graph if A = A. The e of all reverible flow graph i denoed J. Noe ha if for all verice u, v in V A we have (u, v) = (v, u), hen A i necearily reverible. The convere i fale a he reverible example in Figure 2 how. Figure 2: A flow graph in J having a non-ymmeric adjacency marix. Definiion 2.9 (Self-conjugae flow graph). A flow graph A = (G A, A, A ) i called an elf-conjugae 4 flow graph if A = A = A. The e of all elf-conjugae flow graph i denoed K. Noe ha if a flow graph i boh reflecive and reverible, i i necearily elf-conjugae. The convere i fale a he elf-conjugae example in Figure 3 how. Indeed, no wo of he e H, J, and K are conained in each oher. The flow graph in Figure 1 belong o H\(J K). The flow graph in Figure 2 belong o J \(H K). The flow graph in Figure 3 belong o K\(H J ). Definiion The roe wih n peal i defined o be he infinieimal flow graph R n having one verex and n loop edge. Roe R 1, R 2, R 3 are hown in he boom lef panel of Figure 4. Definiion The ar (aniar) wih n edge i defined o be he infinieimal flow graph S n (Sn ) having n + 1 verice v 1, v 2,..., v n and u = =, wih n edge from u o v i (v i o u) for each i = 1,..., n. Sar S 1, S 2 and S 3 are hown in he boom cener panel of Figure 4, while ani-ar S1, S 2 and S 3 are hown on he boom righ panel. 4 The moivaion for he erm elf-conjugae will be clarified laer, in iem 4 of Secion 3.4.

5 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 5 Figure 3: A flow graph in K ha i neiher reflecive nor reverible. Definiion 2.12 (Graphical naural number). We repreen he naural number n a a direced chain of lengh n, having n+1 verice. More formally, le P n be a direced chain of lengh n (having n + 1 verice) where each verex ha in-degree 1 and ou-degree 1. Denoe by n, he unique verex in P n having in-degree 0, and le n be he unique verex in P n having ou-degree 0. The flow graph F n = (P n, n, n ) i referred he graphic naural number n. Define he map i : N F a i : n F n. We denoe F 0 a 0 and F 1 a 1. Graphical naural number F 1, F 2 and F 3 are hown in he op lef panel of Figure 4, while he correponding revere flow graph are hown in he op righ panel. 3. Arihmeic on Flow Graph 3.1. Addiion In Definiion 2.1, we repreened he naural number n by he flow graph F n. I follow ha we inerpre he addiion of wo number n 1 and n 2 inide F a concaenaing F n1 wih F n2. Conider, for example, he addiion of 3 and 2 depiced in Figure 5. To exend hi definiion of + o all of F, we define general addiion of flow graph a follow: Given wo flow graph A and B, define A+B o be he flow graph obained by idenifying A wih B and defining A+B = A and A+B = B. An example of uch an addiion i hown in Figure 6.

6 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 6 F 1 F * 1 F 2 F * 2 F 3 F * 3 R 1 = S 1 = = S* 1 = R 2 S 2 = S* 2 = = R 3 S 3 = S* 3 = Figure 4: Some example of pecial flow graph: he graphical naural number F 1, F 2, F 3, he ani-pah F 1, F 2, F 3, he roe R 1, R 2, R 3, he ar S 1, S 2, S 3, and he ani-ar S 1, S 2, S 3.

7 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 7 F 3 2 F + + F F = F Figure 5: Inerpreing addiion of naural number inide F. A B A A B B + A + B A + B A + B Figure 6: General addiion of flow graph. We begin by defining he following Verex gluing operaion on direced muligraph: Definiion 3.1 (Verex gluing of direced graph). Given direced graph G 1 and G 2, and verice u 1 V [G 1 ], u 2 V [G 2 ], we define G 1 + u 1 u 2 G 2 def = (G 1 G 2 )/(u 1 u 2 ) o be he graph obained by aking dijoin copie of G 1 and G 2 and idenifying verex u 1 in G 1 wih verex u 2 in G 2. Noe he obviou and naural graph embedding σ u + 1 u 2 : G 1 G 1 + G 2 u 1 u 2 (1) τ u + 1 u 2 : G 2 G 1 + G 2.. u 1 u 2 Now we can define addiion of flow graph:

8 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 8 Definiion 3.2. Given wo flow graph A = (G A, A, A ) and B = (G B, B, B ), we define A+B def = (G A + G B, A, B ). A B Since A and B are conneced, i follow ha A+B i conneced. The nex lemma follow immediaely from Definiion 2.12 and 3.2. Lemma 3.3. Le m, n be naural number. Then i(n + m) = i(n)+i(m). We preen ome properie of +. Lemma 3.4. The operaion + i aociaive. Proof. Given flow graph A, B, C, (A+B)+C = (G A + A B G B, A, B )+C = ((G A + A B G B ) + B C G C, A, C ) = (G A + A B (G B + B C G C ), A, C ) = A+(G B + B C G C, B, C ) = A+(B+C). One can check ha 1+R 1 R Thu we obain Lemma 3.5. The operaion + i no commuaive. Definiion 3.6. A flow graph A i called +-reducible if here exi non-rivial flow graph B, C, uch ha A = B+C. Oherwie, A i called +-irreducible. Definiion 3.7 (Scalar muliplicaion of flow graph). Given a flow graph A, and a poiive naural number k in N, we define lef calar muliplicaion inducively a follow: 1A = A ka = (k 1)A+A. Righ calar muliplicaion i defined analogouly. A we have een, + i aociaive, and o he wo noion coincide. We hall ubequenly conider only lef calar muliplicaion by ineger calar. Remark 3.8. Noe ha if A i a flow graph wih p A verice and q A edge, and B i a flow graph wih p B verice and q B edge, hen A+B i a flow graph having p A + p B 1 verice and q A + q B edge.

9 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A Muliplicaion In he previou ecion, we preened an inerpreaion of addiion in F ha i a naural exenion of addiion on he naural number. In hi ecion, we preen an inerpreaion of muliplicaion in F which generalize he muliplicaion of naural number. In doing hi, we mu repec he fac ha for each pair of naural number n 1, n 2, he following ideniy hold in N : n 2 + n n }{{} 2 = n 1 n 2 = n 1 + n n 1. }{{} n 1 ime n 2 ime So, in paricular, he definiion of muliplicaion in F mu aify n 1 F n2 = F n1 F n2 = F n1 n 2. (2) Given ha we repreen he naural number n by he flow graph F n, he produc of wo graphical number F n1 and F n2 (denoed F n1 F n2 ) can be made o aify relaion (2) if we ake muliplicaion o be he ac of replacing each edge of F n2 wih a copy of F n1. For example, he muliplicaion of graphical naural number F 3 and F 2 i illuraed in Figure 7. F 3 F 2 x F 3 F 3 F F = F Figure 7: Sandard muliplicaion of naural number in F (repreened a flow graph). To exend hi definiion of muliplicaion o all of F, we define general muliplicaion of flow graph a follow: Given wo flow graph A and B, define AB o be he flow graph obained by replacing every edge e (from E[G B ]) wih a copy of A a follow: For each edge e = (u, v) in B, we remove e and replace i wih a graph A e iomorphic o A, by idenifying u wih Ae, and v wih Ae. An example of uch a muliplicaion operaion i hown in Figure 8. We now formally define muliplicaion of flow graph:

10 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 10 A B A B B A x B B B B B B A B A B A B Figure 8: General muliplicaion of flow graph. Definiion 3.9. Le A = (G A, A, A ) and B = (G B, B, B ) be any wo flow graph. We define an equivalence relaion R on V A E B, a follow: Given verice u 1, u 2 in V A, and edge e 1 = (v 1, w 1 ) and e 2 = (v 2, w 2 ) in E B, le (u 1, (v 1, w 1 )) R (u 2, (v 2, w 2 )) iff he following hold: whenever u 1 i he ource (arge) and u 2 i he ource (arge) hen (repecively) he ail (head) of e 1 coincide wih he ail (head) of e 2 in B. Then R i an equivalence relaion. We define he flow graph AB = (G AB, AB, AB ) a follow. Le G AB = (V AB, E AB ), where V AB = (V A E B )/ R and ((u 1, e 1 ), (u 2, e 2 )) E AB if (u 1, u 2 ) E A and e 1 = e 2 in E B. Define AB = ( A e)/ R where e = ( B, w) for any w V B and AB = ( A e)/ R where e = (v, B ) for any v V B. Since A and B are conneced, i follow ha AB i conneced. We remark ha here i an obviou ymmeric definiion for muliplicaion in which he role of wo flow graph being muliplied i exchanged. To remain in agreemen wih convenion of ordinal and poe muliplicaion eablihed by Canor [3] and oher ubequenly [4, 7, 8], we choe he definiion above.

11 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 11 We now preen ome properie of muliplicaion. Lemma Le A be a flow graph wih p A verice and q A edge, and B be a flow graph having p B verice and q B edge. Then AB ha q A q B edge. If A i eiher rivial or infinieimal hen AB ha 1+q B (p A 1) verice. If A i non-rivial and non-infinieimal hen AB ha p B + q B (p A 2) verice. Proof. The flow graph AB i obained by replacing each edge e = (u, v) in G B wih a copy of A a follow: remove e from G B and replace i wih a flow graph A e iomorphic o A, idenifying u wih Ae and v wih Ae. Thu each edge of G B produce q A edge in AB and o he by doing he ame operaion wih every edge of G B, we ee ha he number of edge in AB will be q A q B. If A i non-rivial and non-infinieimal, hen each edge e = (u, v) of G B produce in addiion o i end verice u and v, an addiional (p A 2) verice. Thu he number of verice in AB i p B + q B (p A 2). If A i rivial, hen each edge of G B under hi operaion of muliplicaion by A collape ino one verex and equenially applying hi operaion o all edge reul in he graph AB which coni of a ingle verex wih no edge, ha i, reul in a rivial graph. If A i infinieimal and non-rivial, hen each edge e = (u, v) i replaced by a copy of A wih Ae = Ae idenified wih u collaped wih v. Thu he number of verice produced by an edge e = (u, v) beide he collaped verex u = v i (p A 1). Hence he oal number of verice in AB i 1 + q B (p A 1). Lemma Flow graph muliplicaion i aociaive. Proof. Given flow graph A = (G A, A, A ), B = (G B, B, B ), C = (G C, C, C ), we wan o how: (AB)C = A(BC). We define a bijecion Λ beween he verice (AB)C and he verice of A(BC), and hen how ha Λ repec he edge relaion. Le Λ : ((v 1, (v 2, w 2 )), e 3 ) (v 1, ((v 2, e 3 ), (w 2, e 3 ))), where v 1 i any verex in A, v 2 and w 2 are wo verice in B, and e 3 i any edge in C. An edge in (AB)C i of he form ( ((v 1, (v 2, w 2 )), e 3 ), ((v 1, (v 2, w 2 )), e 3 ) ), where (v 1, v 1) E[G A ]. The image of hi edge under Λ i ( v 1, ((v 2, e 3 ), (w 2, e 3 )) ) which i an edge in A(BC). Hence (AB)C i a ubgraph of A(BC). Proceeding in he ame way uing Λ 1, one can how ha A(BC) i a ubgraph of (AB)C. The Lemma i proved.

12 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 12 One can check ha R 1 F 2 F 2 R 1. Thu we obain Lemma Flow graph muliplicaion i no commuaive. The nex lemma follow immediaely from Definiion 2.12 and 3.9. Lemma Le m, n be naural number. Then i(n m) = i(n)i(m). Definiion 3.14 (Scalar exponeniaion of flow graph). Given a flow graph A, and a poiive naural number k in N, we define righ-exponeniaion inducively a follow: A 1 = A A k = A k 1 A. Lef-exponeniaion i defined analogouly. A we have een, i aociaive, and o he wo noion coincide. We hall ubequenly conider only righ-exponeniaion by ineger calar Zero Divior and Uni The nex lemma how ha F ha no member which behave like zero divior. Lemma Given flow graph G and H: GH = 0 H = 0 or G = 0. (3) Proof. If G = 0 hen GH = HG = 0. For he revere implicaion, we appeal o Lemma 3.10, noing ha GH = 0 implie q G q H = 0, o eiher q G = 0 or q H = 0. I follow ha eiher H = 0 or G = 0. The nex Lemma how ha he only uni are 1 and 1. Lemma Given flow graph G and H: GH = 1 G = 1 = H or G = 1 = H. (4) Proof. By Lemma 3.10, we know ha q G q H = 1, hence q G = 1 and q H = 1. I follow ha G, H {F 1, F 1, S 1, S 1, R 1 }. Since 1 i no infinieimal, i follow ha G, H {F 1, F 1} Then ince he reul follow. F 1 F 1 = F 1 F 1 = F 1 1 F 1 F 1 = F 1 F 1 = 1

13 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A Srucural Uniary Operaor The unary operaion of (Definiion 2.4) and (Definiion 2.7) inerac nicely wih he binary operaion of addiion and muliplicaion. The following ideniie are eay o verify: 1. Nilpoency of and operaion: A = A = (A ) 2. Diribuiviy of and over addiion and muliplicaion: (A+B) = A +B (A+B) = B +A (AB) = A B (AB) = AB 3. Muliplicaive definiion of and : A = (1 )A A = A(1 ) 4. Commuaiviy of and operaion: (A ) = (A ) = 1 A1. Noe ha hi ideniy i he juificaion for he erm elf-conjugae in Definiion 2.9, ince if A = A, hen A = 1 A1, and hu a elf-conjugae graph A i iomorphic o ielf conjugaed by he only non-ideniy uni Ideniy Lemma The flow graph 0 def = F 0 i he unique one-ided ideniy on each ide wih repec o +. Tha i, for all flow graph A, G F, A+G = A G = 0 G+A = A. Proof. If G = 0 hen A+G = G+A = A. For he revere implicaion, we appeal o Remark 3.8, noing ha A+G = A implie p A + p G 1 = p A and q A + q G = q A. Hence p G = 1 and q G = 0, o G = 0. An analogou argumen how ha G+A = A implie G = 0. We noe F n = F n for all n. Conidering addiion, F n +F m = (F n+f m) = (F m +F n ) = F m+n.

14 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 14 Conidering muliplicaion, F n F m = F n F m = F n F m = F mn F n F m = F n F m = F mn F n F m = F nf m = F mn Thee obervaion are mirrored in he naural number, where for any n, m, we have ha n + m = ( n + m) nm = ( n)( m) n( m) = ( n)m = (nm). Thu, we ugge viewing he and operaion a wo differen kind of negaion on flow graph, conidering F n = Fn o be differen inerpreaion of he number5 n. Following hi meaphor, he reverible flow graph J and reflecive flow graph H have he propery of being iomorphic o heir own negaion. We hall ee ha he behavior of muliplicaion will aify cerain ideniie a long a he parameer lie ouide of hee wo pahological e, in much he ame way ha cerain muliplicaive ideniie hold for he naural number a long a cerain parameer are aumed o be nonzero. We now conider he righ muliplicaive ideniy. Noe ha for any flow graph H, H = H1 and H = H1. So if H i a reflecive flow graph hen H = H1, hence 1 and 1 are boh righ ideniie on H. The nex lemma how ha on F \H, here i a unique righ ideniy, 1. Lemma Le G, H be non-rivial flow graph wih G I and H H. Then HG = H G = 1. (5) Proof. If G = 1 hen HG = GH = H. For he revere implicaion, we appeal o Lemma 3.10, noing ha HG = H implie p G + q G (p H 2) = p H and q H q G = q H. Hence q G = 1, p G = 2 and o G {1, 1 S 1, S1 }. Bu G canno be S 1 or S1 ince G i no infinieimal. Likewie, G canno be 1 ince H1 = H and H H ince H H. I follow ha G = 1. We now urn o he exience of lef ideniy. Noe ha if H J, hen 1H = H = H = 1 H o boh 1 and 1 are lef ideniie on J. If H {S n n N} hen S 1 H = H, o boh 1 and S 1 are lef ideniie on {S n n N}. If H {Sn n N} hen S1H = H, o boh 1 and S1 are lef ideniie on {S n n N}. The nex lemma how ha on F \ (J {S n n N} {Sn n N}) here i a unique lef ideniy, 1. 5 The meaphor hold only up o a poin, however, ince for n > m > 0, we have F n +F m F n m. Cancellaion i no wineed beween poiive and negaive flow graph.

15 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 15 Lemma Le H S1 H. Then G, H be non-rivial flow graph wih H J and S 1 H H and GH = H G = 1. (6) Proof. If G = 1 hen GH = H. For he revere implicaion, we appeal o Lemma a. Suppoe G I. Then GH = H implie 1+q H (p G 1) = p H and q H q G = q H. Hence q G = 1, o G {1, 1, R 1, S 1, S1}. Since G I, i canno be 1 or 1. Likewie, G canno be R 1 ince GH = H implie R 1 H = R qh and o H = R qh, conradicing ha H J. Suppoe G = S 1 (or S1 ) hen GH = H implie ha H = S q H (or H = Sq H ) which conradic he hypohei S 1 H H (or H S1 H). Hence G canno be in I. b. So now we conider G I. Then GH = H implie p H + q H (p G 2) = p H and q H q G = q H. Hence q G = 1 which implie G {1, 1, R 1, S 1, S1 }. Since G I, i mu be 1 or 1. Bu G canno be 1 ince 1 H = H and H H ince H J. I follow ha G = Infinieimal The following obervaion moivae our choice of he erm infinieimal for flow graph whoe ource and arge verice coincide. Propoiion Le B and C be non-rivial flow graph. Then B+C i infinieimal, if and only if boh B and C are infinieimal. Proof. If B (rep. C) i no infinieimal hen B B (rep. C C ), hence B+C B+C. So B+C i no infinieimal. If B and C are infinieimal hen B = B and C = C hence B+C = B+C. So B+C i infinieimal. The nex Propoiion how ha wih repec o muliplicaion, he e of infinieimal behave, in ome ene, like a prime ideal inide F. Propoiion Le G and H be non-rivial flow graph, hen GH i infinieimal if and only if a lea one of he wo facor i infinieimal.

16 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 16 Proof. If H or G i infinieimal, hen GH = GH in GH and HG = HG in HG. Hence GH and HG are boh infinieimal. On he oher hand, uppoe G and H are non-rivial flow graph ha are boh noninfinieimal. Then GH GH in GH and HG HG in HG. Hence GH and HG are boh non-infinieimal. The reader may wih o compare he above Propoiion wih aerion (3) of Lemma 3.15 which howed ha {0} alo behave, in ome ene, like a prime ideal inide F. The nex wo propoiion how ha H and J behave like one-ided ideal in F. Propoiion Le G be any flow graph and H be a reflecive flow graph. Then GH i a reflecive flow graph. Proof. (GH) = GH = GH, ince H = H. Propoiion Le G be a reverible flow graph and H be any flow graph. Then GH i a reverible flow graph. Proof. (GH) = G H = GH, ince G = G Infinieimalizing Unary Operaor I i alo poible o define naural infinieimalizing unary operaion on flow graph. We inroduce he following: Definiion Given a flow graph A, define A + a he graph A wih he arge moved down o coincide wih he ource. A a he graph A wih he ource moved up o coincide wih he arge. A a he graph A wih he ource and arge node idenified. The following ideniie are eay o verify: 5. Idempoency: Given wo operaion x, y {+,, }: (A x ) y = A x. More generally: applying +, or o an infinieimal ha no effec, and A = A = A + = A if and only if A i infinieimal.

17 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 17 The infinieimalizing operaor inerac nicely wih he unary operaion of (Definiion 2.4) and (Definiion 2.7). The following ideniie are eay o verify: 6. Commuaiviy of wih any operaion x {+,, }: (A x ) = (A ) x 7. Ineracion of wih infinieimalizing operaion: (A ) + = A = (A ) (A ) = A + = (A + ) (A ) = A = (A ) 8. Muliplicaive definiion of +, and : A + = A1 + = AS 1 A = A1 = AS 1 A = A1 = AR 1 9. Non-diribuiviy of +, and over addiion and muliplicaion. Taking A = B = F 2, one ee: (A+B) + A + +B + (A+B) A +B (A+B) A +B (AB) + A + B + (AB) A B (AB) A B 10. Lef-ideniie oher han 1 on ar, ani-ar, and roe: S k A = k1 + A = S k E[GA ] Sk A = k1 A = Sk E[G A ] R k A = k1 A = R k E[GA ]. Taking k = 1 and A o be a ar, ani-ar, or roe (repecively), he above ideniie how ha S 1, S 1 and R 1 ac a lef ideniie on he e of ar, ani-ar, and roe (repecively).

18 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A Order Given our repreenaion of he naural number n by he flow graph F n in Definiion 2.12, comparing he order of wo number n 1 and n 2 (a flow graph) require imply comparing he lengh of he correponding chain graph F n1 and F n2. To generalize hi o all of F, however, we canno refer o lengh. In wha follow, we preen wo poible inerpreaion of in F. To avoid confuion, we denoe hee diinc inerpreaion by he ymbol, and hee are referred o a he rong and induced order repecively Srong Order We now define an ordering on F. Given wo flow graph A and B, informally, we ay ha A B iff wo copie of G A appear in G B ; one a a neighborhood of B and one a a neighborhood of B. The nex definiion make hi aemen precie. Definiion 4.1 (Srong order). Given wo flow graph A = (G A, A, A ) and B = (G B, B, B ), we ay A B iff here are graph embedding 6 where φ : G A G B and φ : G A G B which aify φ ( A ) = B and φ ( A ) = B. Conider he comparion of F 3 and F 5 depiced in Figure 9; clearly F 3 F 5. F 3 F F 3 5 F 5 Figure 9: Sandard rong ordering of naural number (repreened a flow graph). The proof of he following lemma i immediae. Lemma 4.2. Le m, n be naural number. Then n m i(n) i(m). Figure 10 illurae a more general example in which rong order i ued o compare wo elemen of F which are no graphical naural number. 6 Thee embedding are merely direced graph embedding whoe image need no be an induced ubgraph of G B ).

19 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 19 A A B A A B B B Figure 10: General rong ordering of flow graph. The nex Propoiion follow immediaely from Lemma 3.3, 3.17, 3.13, 3.18, and 4.2. Propoiion 4.3. Under he embedding i : n F n, he andard model N = N, 0, 1,, +, i a ubmodel of F = F, 0, 1,, +,, where 0 = F 0, 1 = F 1, and he relaion +, and reinerpre +, and inide F Induced Order We now give an alernae ordering on F. Given wo flow graph A and B, informally, we ay ha A B iff B can be ranformed ino A by a erie of edge conracion 7. The nex wo definiion make hi aemen precie. Definiion 4.4 (Edge conracion). Given a flow graph A = (G A, A, A ) and an edge e = (u, v) E[G A ], he flow graph A/e i obained from A by deleing e in G A and idenifying verex u wih v. If u or v wa he ource (rep. arge) of A, hen he idenified verex u v will be aken a he ource (rep. arge) of A/e. The nex wo obervaion conider he effec of conracing an edge e = (u, v) in a flow graph A = (G A, A, A ). Obervaion 4.5. E[G A/e ] = E[G A ] 1. If e i a non-loop edge hen V [G A/e ] = V [G A ] 1; if e i a loop edge hen V [G A/e ] = V [G A ]. Definiion 4.6. Le A = (G A, A, A ) be a flow graph, where G A = (V, E). Fix X E and define an equivalence relaion R X on he verice of A by aking (v 1, v 2 ) R X iff v 1 and v 2 are in he ame conneced componen of (V, X). We define G/R X o be he graph obained by conidering he quoien of he edge relaion E by he equivalence relaion R X. Noe ha he verex e of G/R X i {[v] v V }. 7 The induced order wa he oucome of dicuion held when hee reul were preened a he Ciy Univeriy of New York Logic Workhop, Sepember 2004.

20 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 20 Definiion 4.7 (Induced order). Given wo flow graph A = (G A, A, A ) and B = (G B, B, B ), we ay A B iff here i e of edge X E[G B ] uch ha B/R X i iomorphic o A. The proof of he following lemma i immediae. Lemma 4.8. Le m, n be naural number. Then n m i(n) i(m). Obervaion 4.9. Given wo verice u and v in V [G A ], he diance beween u and v in G\e doe no exceed he diance beween u and v in G. There i no obviou relaionhip beween induced order and he afforemenioned rong order. Figure 13 how flow graph A and B for which he rong order relaionhip A B hold. However, ince d B ( B, B ) = 1 and d A ( A, A ) = 2, by Obervaion 4.9 no equence of edge conracion can ranform B ino A, and hence A B. In he revere direcion, le A = F 1 +R 1 +F 1 and B = F 2 +R 1 +F 2. Then A B ince each F 2 ummand in B can be edge conraced o become F 1. Noe ha A conain a unique verex wih a loop edge aached, and hi verex i a diance 1 from A and A. In conra, in B here i a unique verex wih a loop edge aached, and hi verex i a diance 2 from B and B. I follow ha A B. The nex Propoiion follow from Lemma 3.3, 3.17, 3.13, 3.18, and 4.8. Propoiion Under he embedding i : n F n, he andard model N = N, 0, 1,, +, i a ubmodel of F = F, 0, 1,, +,, where 0 = F 0, 1 = F 1, and he relaion +, and reinerpre +, and inide F. The unary operaion of (Definiion 2.4) and (Definiion 2.7) inerac nicely wih he wo order and. The following aerion are eaily verified. A B A B A B A B A B A B A B A B A B A + B + A B A + B + A B A B A B A B A B A B A B A B The la hree implicaion are no reverible, ince: If A = F 1, B = F + 1, hen A + B + and A + B + bu A B and A B. If A = F 1, B = F 1, hen A B and A B bu A B and A B. If A = F 1, B = F 1, hen A B and A B bu A B and A B.

21 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A Oher embedding of N ino F Propoiion 4.3 and 4.10 how ha he e of all graphical naural number {F n n N} induce a ubmodel of F ha i iomorphic o N. There are oher embedding of N ino F. For example, conider he e of roe R n (Definiion 2.10). A a ubrucure of he flowgraph, hee are iomorphic o N, ince R n+m = R n +R m = R m +R n, and R mn = R m R n = R n R m for all n, m, N. Noe ha R 1 i no a muliplicaive ideniy on all of F, bu i i on he ube of roe. Alernaively, we can embed N ino he infinieimal uing eiher he ar S n or he ani-ar Sn (Definiion 2.11). Le u define: i F : n F n i R : n R n i S : n S n i S : n Sn. By carrying ou a imilar analyi for hee funcion, one can how ha i F, i R, i S, i S : N F are embedding of rucure, and hu he ubmodel induced by heir image in F (i.e. e of all ani-pah (F n), roe, ar, and ani-ar) are each iomorphic o he naural number. A we hall ee, however, here are aeheic advanage o he mapping which repreen he naural number n by he flow graph F n (e.g. Propoiion 5.12, pp. 26). 5. Properie of Flow Graph In hi ecion we how ha lef-diribue over + bu doe no righ-diribue. We define lef and righ diviibiliy of flow graph, and how ha righ diviibiliy diribue over +, bu lef diviibiliy doe no. We inroduce he noion of a prime flow graph, and how ha he concep of lef-prime and righ-prime coincide. Finally, we explore he properie and relaionhip of he differen order, and decribe he ineracion beween he order inroduced in Secion 4 and he operaion of + and Muliplicaive Properie Lemma 5.1 (Lef-diribuiviy of over +). For any flow graph A, B, C, C(A+B) = CA+CB

22 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 22 Proof. Fix e E[G C(A+B) ]. Then define β 0 (e) = Λ C,A+B (e). Noe ha β 0 (e) = (f, e ), where f i an edge in E[G C ] and e i an edge in E[G A+B ]. Define β 1 : E[G A+B ] E[G A ] E[G B ] o ha β 1 (e) = { σ + 1 A B (e) if e Im(σ + 1 τ + 1 A B (e) if e Im(τ + 1 A B ) A B ). Then β 1 β 0 map E[G C(A+B) ] injecively ino (E[G C ] E[G A ]) (E[G C ] E[G B ]). Define β 2 by aking β 2 (e) = { Λ 1 C,A (e) if e E[G C] E[G A ] Λ 1 C,B (e) if e E[G C] E[G B ]. Then β 2 map (E[G C ] E[G A ]) (E[G C ] E[G B ]) ino E[G CA ] E[G CB ] injecively. Finally, define β 3 by aking β 3 (e) = { σ + CA (e) CB if e E[G CA] τ + CA CB (e) if e E[G CB ]. Then β 3 map E[G CA ] E[G CB ] injecively ino E[G (CA)+(CB) ]. The compoie map β 3 β 2 β 1 β 0 map he edge of C(A+B) injecively ino he edge of CA+CB, and i he deired flow graph iomorphim demonraing he claimed equaliy. Le A be he flow graph coniing of a direced cycle of lengh 3 aking ource and arge verice o be any wo diinc verice on hi cycle. Oberve ha (F 1 +F 1 )A = F 2 A, while (F 1 A)+(F 1 A) = A+A = 2A = AF 2. Referring o Figure 11, we ee ha AF 2 F 2 A. A A F 2 F 2 F 2 A Figure 11: (F 1 +F 1 )A AF 1 +AF 1. Lemma 5.2 (Non Righ-diribuiviy of muliplicaion over addiion). There exi flow graph A, B, and C, (B+C)A BA+CA

23 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 23 Remark 5.3 (Violaion of lef and righ cancellaion). Le B be non-reflecive and A be reverible. Since A i reverible, A = A, o hen BA = BA = B1 A = B A. Since B i no reflecive, B B violaing righ cancellaion. For example, aking B = 1, we ge ha 1A = A = A = 1 A, bu 1 1. If A be reflecive, and B be non-reverible. Since A i reflecive, A = A, o hen AB = A B = A1 B = AB. Since B i no reverible, B B violaing lef cancellaion. For example, aking B = 1, we ge ha A1 = A = A = A1, bu 1 1. Definiion 5.4. Given flow graph A, B a lea one of which i non-rivial, we define A/B a he e of flow graph C for which A = BC. Analogouly, we define A\B a he e of flow graph C for which A = CB. If A/B = 0 (rep. A\B = 0) hen we ay ha A i no righ-diviible (rep. no lef-diviible) by B. Noe ha he e A/B and A\B may have ize bigger han one. For example, if A = A, hen A = 1 A = 1A, o A\A conain boh 1 and 1. If A = A, hen A = A1 = A1, o A/A conain boh 1 and 1. By convenion, we ay ha 0/0 and 0\0 are undefined. Clearly if m and n are andard ineger hen F m i righ-diviible by F n iff F m i lef-diviible by F n iff k N for which F m = F n F k = F k F n iff m i diviible by n. We exend muliplicaion of flow graph o muliplicaion of e of flow graph in he obviou way: Definiion 5.5. Given wo nonempy e of flow graph A and B, we define A B = {AB A A, B B}, A +B = {A+B A A, B B}. Noe ha for all flow graph A, C, if A/C hen here exi a flow graph B uch ha A/C B/C A/B a e; imply ake B = C. In conra, he nex lemma concern cancellaion in produc: Lemma 5.6. For all flow graph A, B, C, if B/C and A/B hen B/C A/B A/C. Proof. Suppoe K 2 B/C and K 1 A/B. Then by Definiion 5.4, B = CK 2 and A = BK 1, which implie A = (CK 2 )K 1 which by by Lemma 3.11 implie A = C(K 2 K 1 ), o K 2 K 1 A/C. Thu, B/C A/C A/C. Noe ha Lemma 5.6 i no an equaliy, ha i B/C A/B need no equal A/C even if B/C and A/B. To ee hi, fix n 3 odd, and ake i > 2, j = (n 1)/2 1, and k = n. Pu A = R ijk, B = R ij and C = R i. Then A/C i he e of all flow graph having jk edge, while B/C (rep. A/B) i he e of all flow graph having j (rep. k)

24 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 24 edge. Thu, o how ha he incluion in Lemma 5.6 i omeime a proper incluion, i uffice o how ha no every flow graph having jk edge can be expreed a he produc of wo flow graph which have j and k edge, repecively. Take, for example, he flow graph G obained by conidering any ournamen 8 on n verice, wih diinc ource and arge. Since he number of edge in G i n(n 1)/2 = jk, we know ha G i in A/C. We claim ha G canno be expreed a a produc HK where H ha j edge and K ha k edge, i.e. ha G i no in B/C A/B. Suppoe oward conradicion ha a facorizaion G = HK wa poible. Since G wa conruced o be non-infinieimal, by Propoiion 3.21, boh H and K mu be non-infinieimal. Then by Lemma 3.10, we have ha he number of verice p and he number of edge q in H, K and G are relaed by he expreion p K + q K (p H 2) = p G, which in he pecific eing become p K + n(p H 2) = n. (7) Examining equaion (7) we ee ha if p H 2 hen p K > n 1 = q K 1, violaing ha K i conneced, and if p H 3 hen p K < 0, violaing ha K i a flow graph. I follow ha no uch facorizaion of G exi, and hu he incluion in Lemma 5.6 i omeime proper. There i an analogou reul o Lemma 5.6 concerning lef-diviibiliy, namely for all flow graph A, B, C, if B\C and A\B hen A\B B\C A\C (8) I i unclear wheher analogou example can be conruced o demonrae ha he incluion in he lef diviibiliy analogue (8) i proper. The auhor conjecure ha expreion (8) i acually an equaliy. Lemma 5.7 (Rericed diribuiviy of righ-diviibiliy over +). For all flow graph A, B, C, A/B +C/B (A+C)/B Proof. Suppoe K 1 A/B and K 2 C/B. Then by Definiion 5.4, A = BK 1 and C = BK 2. Thu A+C = (BK 1 )+(BK 2 ) which by Lemma 5.1, i B(K 1 +K 2 ). Thi implie ha K 1 +K 2 belong o (A+C)/B conain. Thu A/B +C/B i conained in (A+C)/B. To ee ha Lemma 5.7 i no necearily an equaliy, ha i A/B +C/B i no equal o (A+C)/B, conider he following example. Le A = C = F 1 and B = F 2. Since F 1 +F 1 = F 2 F 1 i mean F 1 (A + C)/B. Now F 1 canno be in A/B +C/B ince only poibiliy i ha F 1 = F 0 +F 1 or F 1 = F 1 +F 0 and F 0 i no in A/B or A/C. 8 By ournamen we mean a complee graph in which every wo verice u and v are conneced by eiher he edge (u, v) or he edge (v, u).

25 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 25 Obervaion 5.8 (Non-diribuiviy of lef-diviibiliy over +). Noe ha Lemma 5.2 can be ued o conruc example ha demonrae non-diribuiviy of lef-diviibiliy over +. For example, le B be a direced cycle of lengh 3 wih any wo diinc verice a B and B. Take A = F 2 B. Then F 2 A\B. Now ake C = B. Then F 1 C\B and o F 2 +F 1 = F 3 A\B +C\B. Since A+C F 3 B hi mean ha F 2 +F 1 = F 3 (A+C)\B. Thu, he example how ha for ome A, B, and C, he e (A\B) +(C\B) i no conained in he e (A+C)\B. In Lemma 3.16, we deermined ha 1 and 1 are he only uni in he e of flow graph. Thi moivae he following definiion of a prime flow graph: Definiion 5.9. A flow graph A i called prime if A i neiher rivial nor a uni, and A = BC implie ha eiher B or C i a uni. If we conider Definiion 5.9 in he cae when A i aumed o be non-infinieimal, we ee ha B and C mu boh be non-infinieimal and hence, E[G B ] = 1 (rep. E[G C ] = 1) implie ha B = 1 or 1 (rep. C = 1 or 1 ). Accordingly, le u ay a flow graph A i righ-prime if for all flow graph B, A/B implie one of he following hold: RP 1) RP 2) RP 3) RP 4) B = 1 and A A/B. B = A and 1 A/B. B = A and 1 A/B. B = 1 and A A/B. Likewie, le u ay ha a flow graph A i lef-prime if for all flow graph C, A\C implie one of he following hold: LP 1) LP 2) LP 3) LP 4) C = A and 1 A\C. C = 1 and A A\C. C = A and 1 A\C. C = 1 and A A\C. Noe ha a naural number n i prime iff he non-infinieimal flow graph F n i prime iff F n i righ-prime iff F n i lef-prime. More generally: Lemma Le A be a non-infinieimal flow graph. Then A i righ-prime iff A i lef-prime.

26 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 26 Proof. Suppoe A i righ-prime. To how ha A i lef-prime we mu how ha for all flow graph C, A\C implie ha a lea one of LP1-LP4 hold. Suppoe B A\C; hen A = BC, and hence C A/B. Since A i aumed o be righ prime, we know ha a lea one of RP1-RP4 hold for B. Suppoe ha RP1 hold. Then B = 1 and A = C A/B, which implie LP1 hold. Similarly one can how ha if RPi hold, hen LPi hold (for i = 2, 3, 4). Thu A i lef-prime. A imilar argumen how ha lef-prime implie righ-prime. The previou lemma how ha he noion of prime, lef-prime and righ-prime coincide on non-infinieimal. However, an infinieimal flow graph can be prime while being neiher lef-prime nor righ-prime. To ee hi, le A be any infinieimal flow graph for which V [G A ] > 1 and E[G A ] i prime. Fix a verex V [G A ], for which A, A, and ake A o be he flow graph (G A, A, ). Then A S 1 = A, o A i neiher lef-prime, nor righ prime. Bu A i prime, ince by Lemma 3.10, any facorizaion of A ino a produc BC mu aify E[G A ] = E[G A ] E[G A ]. Indeed, any flow graph wih a prime number of edge i necearily a prime flow graph. Definiion Given a e of flow graph S F, we ay ha A i cenral in S if A S and for all flow graph B S, we have ha AB = BA. The e of all flow graph ha are cenral in S i denoed a Z(S). Propoiion Z(F ) = {0, 1}. Proof. Suppoe A i cenral. Then AS 1 = S 1 A. Since AS 1 = A + and S 1 A = S E[GA ], i follow ha A + i a ar, and hu A (viewed a a direced graph) i alo a ar. Thi implie ha A (viewed a a direced graph) i an aniar. Since A i cenral, A1 = 1 A; bu A1 = A1 = A and 1 A = A. Thu A = A. So A (viewed a a direced graph) i alo an aniar. Thi mean A a direced graph i a ar wih in-degree() in A equal o ou-degree() in A and in-degree() in A equal o ou-degree() in A. I follow ha A ha a mo one edge. Bu hi i poible if and only if A i 0 or Order Properie In hi ecion we explore he relaionhip beween rong order (denoed ), and induced order (denoed ). While hee order coincide on he graphical naural number, only induced order i ani-ymmeric on all of F, and only he rong order and induced order are raniive. We conider everal andard law ha govern he relaionhip beween, + and in N, and how ha hee law coninue o hold for induced order bu everal are violaed under he rong and induced order.

27 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A Srong Order We begin by decribing he properie of F under he rong order. Lemma 5.13 (Srong Order Preervaion). For flow graph A, B, C, if A B hen CA CB. Proof. Le A = (G A, A, A ), B = (G B, B, B ). Since A B here are graph embedding φ : G A G B and φ : G A G B which aify φ ( A ) = B and φ ( A ) = B. Define γ : E[G C ] E[G A ] E[G C ] E[G B ] by Then he compoie map (f, e) (f, φ (e)). Φ C : E[G CA] Λ C,A E[G C] E[G A ] γ E[G C] E[G B ] Λ 1 C,B E[G BC] define an embedding of G CA G CB which ake CA o CB. An analogou conrucion can be carried ou o produce a map Φ C which embed G CA G CB and end CA o CB. Lemma 5.14 (Srong Order Violaion). There exi flow graph A, B and C for which A B bu: Proof. See Figure 12. (i) A+C B+C (ii) C+A C+B (iii) AC BC. We conider poible ani-ymmery of. Suppoe A B and B A. There i a graph embedding φ : G A G B which aifie φ ( A ) = B. Hence V [G A ] = V [φ (G A )] V [G B ] and E[G A ] = E[φ (G A )] E[G B ]. Since B A, here i a graph embedding ψ : G B G A which aifie ψ ( B ) = A. So V [G B ] = V [ψ (G B )] V [G A ] and E[G B ] = E[ψ (G B )] E[G A ]. I follow ha φ i acually an iomorphim from G A o G B aifying φ ( A ) = B. A imilar argumen how ha here i an iomorphim φ from G A o G B aifying φ ( A ) = B. To conclude ha A = B require a ingle flow graph iomorphim π from A o B, aifying boh π( A ) = B and π( A ) = B. Indeed in ome cae, no uch iomorphim may exi. Example Le G A be a direced cycle of lengh 4, and ake A, A o be any wo verice in V [G A ] ha are diance 2 apar. Pu G B iomorphic o G A, aking B, B o be wo verice in V [G B ] ha are diance 1 apar. Then i i eay o verify ha (G A, A, A ) = A B = (G B, B, B ) and B A. Clearly, however, A B a flow graph (ee Figure 13).

28 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 28 A B A + C B + C C C + A C + B A C B C Figure 12: Srong order violaion: (i). A+C B+C, (ii). C+A C+B, and (iii). AC BC. A B Figure 13: An example which demonrae ha he rong order i no aniymmeric. The previou example prove he nex lemma. Lemma 5.16 (Non-aniymmery of rong order ). There exi flow graph A and B for which A B and B A bu A B. Lemma 5.17 (Traniiviy of rong order ). For all flow graph A, B, C A B and B C implie A C. Proof. A B: i.e. here are graph embedding φ : G A G B and φ : G A G B which aify φ ( A ) = B and φ ( A ) = B. B C: i.e. here are graph embedding θ : G B G C and θ : G B G C which aify θ ( B ) = C and θ ( B ) = C. We wan o how A C: i.e. here are graph embedding α : G A G C and α : G A G C which aify α ( A ) = C and α ( A ) = C. Pu α = θ φ and α = θ φ Induced Order We now inveigae he properie of F under he induced order.

29 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 29 Fir, noe ha ince flow graph are conneced direced graph, hen any flow graph A aifie 0 A. Lemma 5.18 (Induced Order Preervaion). For flow graph A, B, C, if A B hen (i) A+C B+C (ii) C+A C+B (iii) CA CB (iv) AC BC. Proof. (i, ii) Since A B, edge of B can be conraced o yield A. When hi equence of conracion i applied o B+C, i yield A+C. When hi equence of conracion i applied o C+B, i yield C+A. (iii, iv) The flow graph CB i obained by replacing each edge e in B wih a graph C e ha i iomorphic o C. Since A B, here i a equence of edge e 1, e 2,..., e k for which he equence B 0 = B, B i = B i 1 /e i (for i = 1, 2,..., k), end wih B k = A. We hall conrac CB in phae, where a phae i, we collape C ei o a poin. Thi i poible ince 0 C. A he end of hi proce, CB ha been ranformed ino CA. The argumen which how (iv) i enirely analogou. Lemma 5.19 (Aniymmery of induced order ). For all flow graph A and B A B and B A A = B. Proof. Since A B, E[G A ] E[G B ] and ince B A, E[G B ] E[G A ]. I follow ha E[G B ] = E[G A ]. I follow ha no edge conracion are required o ranform B ino A, hence A and B are iomorphic a flow graph. Lemma 5.20 (Traniiviy of induced order ). For all flow graph A, B, C A B and B C implie A C. Proof. If ome equence of edge conracion ranform C ino B, and ome equence of edge conracion ranform B ino A, hen he concaenaion of hee wo equence demonrae ha A C Summary of Order Properie Table 1 ummarize properie of he rong and induced order (when ubiued for ). Noe ha he induced order aifie all lied properie, hough he ignificance of hi fac hould perhap be miigaed by he fac ha boh he = and empy relaion alo aify all he properie on he li.

30 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 30 Properie Srong Order Induced Order = = A B = (AC) (BC) Fale True A B = (CA) (CB) True True A B = (A+C) (B+C) Fale True A B = (C+A) (C+B) Fale True A B and B A = A = B Fale True A B and B C = A C True True Table 1: Properie of F under rong and induced order. 6. Concluion and Fuure Work Our fuure reearch will conider he rucural properie of flow graph and decribe T h(f), including for rericed ube of F ha can be defined in erm of rucural conrain, e.g. he e of all ree, direced acyclic graph, ec. Some queion we are preenly conidering are lied below. i. Characerize +-commuing pair, i.e. under wha condiion on flow graph A and B doe A+B = B+A? ii. Graph +-Irreducible Decompoiion Conjecure. Every flow graph i uniquely expreible (up o well-defined reordering) a he um of +-irreducible flow graph. iii. Characerize pair which commue wih repec o muliplicaion, i.e. under wha condiion on flow graph A and B doe AB = BA? iv. Graph Prime Facorizaion Conjecure. Every flow graph i uniquely expreible (up o ome well-defined reordering and applicaion of unary rucural operaor) a he produc of prime flow graph. Acknowledgemen The auhor would like o hank he referee for many inighful obervaion and recommendaion ha led o coniderable refinemen in he heory. We are graeful o Marin Kaabov for hi inciive commen on an early verion of hi paper. Apec of hee reul were preened a he Ciy Univeriy of New York Logic Workhop in Sepember We hank he logician preen a he eminar, epecially Joel Hamkin and Roman Koak for heir conrucive commen and uggeion which improved he preenaion of he paper. We are epecially graeful o Joel Hamkin for hi many idea and conribuion which imulaed he developmen of hi heory, and paricularly for uggeing he induced order.

31 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A12 31 Reference [1] Kiran R. Bhuani and Bilal Khan. A meric on he cla of conneced imple graph of given order. Aequaione Mahemaicae, 66(3): , December [2] Kiran R. Bhuani and Bilal Khan. Diance beween graph uing graph labeling. Ar Combinaoria, 77:45 52, [3] George Canor. Uber unendliche, lineare Punkmannigfaligkeien, Arbeien zur Mengenlehre au dem Jahren Teubner-Archiv zur Mahemaik, Leipzig, Germany, [4] K. Cieielki. Se Theory for he Working Mahemaician. Cambridge Univeriy Pre, [5] W. Imrich and S. Klavzar. Produc Graph, Srucure and Recogniion. John Wiley & Son Inc., [6] Bilal Khan and Kiran R. Bhuani. A meric repreening dilaion beween conneced imple graph of a given order. preprin. [7] J. E. Rubin. Theory for he Mahemaician. Holden-Day, New York, [8] P. Suppe. Axiomaic Se Theory. Dover, New York, [9] Dougla B. We. Inroducion o Graph Theory. Prenice Hall, 2nd ediion, 2001.