Reduced Incidence algebras description of cobweb posets and KoDAGs

Reduced Incidence algebras description of c ob w eb p osets and KoD A Gs Ew a Krot-Sieniaw sk a Institute of Computer Scie nce, Bia lystok Univ ersit y PL-15-887 Bia lystok, ul.Sosnow a 64, POLAND e-mail: ew akrot@wp.pl, ew akrot@ii.u wb.e du.pl Abstract The notion of reduced incidence alg ebra of an arbitra r y cobw eb p oset is delivered. KEY WORDS: cobw eb po set, incidence algebra of lo cally finite p oset, order com- patible equiv alence relatio n, reduced incidenc e a lgebra . AMS C la ssification num bers : 06A06, 06A07, 0 6A11, 11C08, 11B37 Presented a t Gian-Carlo Rota Polish Semina r : http://ii.u wb.edu.pl/akk/sem/s e m rota.htm 1 Cob w eb p osets The family of th e so called cob w eb po sets Π has be en inv ented by A.K.K w a´ sniewski few years ago (for refer ences s e e : [5, 6]). These structures are such a gener alization of the Fib onacci tree growth that allo ws joint combinatorial interpretation for all of them under the admissibility condition (se e [7, 8]). Let { F n } n ≥ 0 be a natural n um ber s v alued sequence with F 0 = 1 (with F 0 = 0 being exceptional as in case of Fibonacci num bers ). An y sequence satisfying this prop ert y uniquely designates cobw eb p oset defined as follows. F or s ∈ N 0 = N ∪ { 0 } let us to define levels of Π: Φ s = {h j, s i , 1 ≤ j ≤ F s } , (in case o f F 0 = 0 level Φ 0 corres p onds to the empt y ro ot {∅} ). ) Then Definition 1. Corr esp onding c obweb p oset is an infinit e p artial ly or der e d set Π = ( V , ≤ ) , wher e V = [ 0 ≤ s Φ s 1 ar e the elements ( vertic es) of Π and the p artial or der r elatio n ≤ on V for x = h s, t i , y = h u, v i b eing elements of c obweb p oset Π is define d by formula ( x ≤ P y ) ⇐ ⇒ [( t < v ) ∨ ( t = v ∧ s = u )] . Obviously a n y cobw eb p oset can b e repres en ted, via its Hasse diag ram, as infinite directed graf Π = ( V , E ), where set V of its vertices is defined as a bov e and E = { ( h j, p i , h q, ( p + 1) i ) } ∪ { ( h 1 , 0 i , h 1 , 1 i ) } , where 1 ≤ j ≤ F p and 1 ≤ q ≤ F ( p +1) stays for set of (dire cted) edges. The K w a ´ sniewski cobw eb p osets under co nsideration represented by graphs are ex- amples of o derable directed acyclic graphs (oDA G) whic h we start to ca ll from now in brief: KoD A Gs. These are s tructures of universal impor tance fo r the whole of mathematics - in particular for discrete ”‘mathemagics”’ [h ttp://ii.uwb.edu.pl/akk/ ] and co mput er sciences in general (quotation from [7 , 8] ): F or any g iv en natural n umber s v alued sequence the graded (lay ered) cobw e b p osets‘ D A Gs ar e equiv alently representations of a chain of bi- nary rela tions. Every relation o f the cobw eb p oset chain is biunivocally represented b y the uniquely designated complete bipar tite digr aph-a digraph whic h is a di-bicliq ue designated by the very given sequence. The cobw eb p oset is then to be identified with a c hain of di- bicliques i.e. by definition - a chain of complete bipartite one direction digraphs. An y chain of relations is therefor e o btainable fr om the co b w eb p oset chain of co mplete r elations via deleting arcs (ar ro ws) in di-bicliques. According to the definition abov e arbitrary cobweb p oset Π = ( V , ≤ ) is a gra ded po set (ranked p oset) a nd for s ∈ N 0 : x ∈ Φ s − → r ( x ) = s, where r : Π → N 0 is a r ank function on Π. Let us then define Kwa ´ sniews ki finite cobw eb s ub- posets as follows Definition 2. L et P n = ( V n , ≤ ) , ( n ≥ 0) , for V n = [ 0 ≤ s ≤ n Φ s and ≤ b eing t he induc e d p artial or der r elation on Π . Its easy to see that P n is ranked po set with rank function r as above. P n has a unique minimal elemen t 0 = h 1 , 0 i ( w ith r (0) = 0). Mor eo v er Π a nd all P n s are lo cally finite, i.e. for a n y pair x, y ∈ Π, the se gmen t [ x, y ] = { z ∈ Π : x ≤ z ≤ y } is finite. Let us rec a ll that one defines the incidence algebra o f a lo cally finite partially ordered set P as follows (see [9, 10, 11]): I ( P ) = I ( P, R ) = { f : P × P − → R ; f ( x, y ) = 0 unl ess x ≤ y } . 2 The sum of tw o such functions f and g and multiplication by scalar ar e defined as usual. The pro duct h = f ∗ g is defined as follows: h ( x, y ) = ( f ∗ g )( x, y ) = X z ∈ P : x ≤ z ≤ y f ( x, z ) · g ( z , y ) . It is immediately verified that this is an asso ciative algebr a (with an identit y element δ ( x, y ), the Kr o nec k er delta), ov er any asso ciative ring R. In [4] the incidence alg ebra of an arbitra r y cobw e b p oset Π ( or its subp osets P n ) uniquely designated by the natural num bers v alued sequence { F n } n ≥ 0 , was considered by the prese n t author . The explicit for m ula s for s o me typical elemen ts of incidence algebra I (Π) o f Π wher e delivered there. So for x, y being so me a rbitrary elemen ts of Π such that x = h s, t i , y = h u, v i , ( s, u ∈ N , t, v ∈ N 0 ), 1 ≤ s ≤ F t and 1 ≤ u ≤ F v one has: (1) ζ function of Π b eing a character istic function of par tial or der in Π ζ ( x, y ) = ζ ( h s, t i , h u, v i ) = δ ( s, u ) δ ( t, v ) + ∞ X k =1 δ ( t + k , v ) , (1) one c a n also verify , that ζ k enum erates all m ultic hains of leng th k , (2) M¨ obius function of Π being a inv e rse of ζ µ ( x, y ) = δ ( t, v ) δ ( s, u ) − δ ( t + 1 , v ) + ∞ X k =2 δ ( t + k, v )( − 1) k v − 1 Y i = t +1 ( F i − 1) , (2) (3) function ζ 2 = ζ ∗ ζ coun ting the num b er of elemen ts in the segment [ x, y ] ζ 2 ( x, y ) = car d [ x, y ] = v − 1 X i = t +1 F i ! + 2 , (3) (4) function η η ( x, y ) = ∞ X k =1 δ ( t + k, v ) =  1 t < v 0 w p.p. , (4) (5) function η k ( x, y ) , ( k ∈ N ) counting the n um ber of chains of length k , (with ( k + 1 ) elements) from x to y η k ( x, y ) = X x n. Definition 3. L et [ x, y ] ∈ α k, n . F or l ∈ N 0 one c an define the incidenc e c o efficients in R (Π) as fol lows  k , n l  = |{ z ∈ [ x, y ] : [ x, z ] ∈ ( k , l ) ∧ [ z , y ] ∈ ( l , n ) }| . (12) The the following formula holds. Prop osition 1.  k , n l  =  F l k ≤ l ≤ n 0 otherv ise (13) Now one can define the pr oduct ∗ in R (Π) as follows. Prop osition 2. L et [ x, y ] ∈ ( k , n ) , ( k ≤ n ) and f , g ∈ R (Π) . The n ( f ∗ g )( k , n ) = X l ≥ 0 F l f ( k , l ) g ( l , n ) (14) with the assumption that ( k , n ) = ∅ , for l < k or l > n . Pr o of. ( f ∗ g )( k, n ) = ( f ∗ g )( x, y ) = X x ≤ z ≤ y f ( x, z ) g ( z , y ) = X k ≤ l ≤ n X { z :[ x,z ] ∈ ( k, l ) , [ z ,y ] ∈ ( l, n ) } f ( x, z ) g ( z , y ) = X k ≤ l ≤ n F l f ( k , l ) g ( l , n ) = X l ≥ 0 F l f ( k , l ) g ( l , n ) , 5 So we hav e prov ed Theorem 1. L et F = { F n } n ≥ 0 z F 0 = 1 , b e an a rbitr ary natur al numb ers value d se quenc e with F 0 = 1 (with F 0 = 0 b eing exc eptional). The the numb ers F n ( n ≥ 0 ) ar e the incidenc e c o efficients in the st anda r d r e duc e d algebr a R (Π) of c obweb p oset Π u n iquely designate d by the se qu en c e F = { F n } n ≥ 0 . One can also show the following Theorem 2. L et f ∈ R (Π) . Then fo r x, y ∈ Π such that [ x, y ] ∈ ( k , n ) t he value f ( x, y ) dep ends on r ( x ) = k and r ( y ) = n ) only, i.e. f ( x, y ) = f ( k , n ) = f ( r ( x ) , r ( y )) . (15) F r om the definition of p artial or der on Π one c an also infer that for x, y ∈ Π satisfying r ( x ) = r ( y ) and for f ∈ I (Π) , one has f ( x, y ) = δ ( x, y ) . It is known that all elemen ts of I (Π) mentioned abov e, i.e. functions: ζ , µ , ζ 2 , ζ k , η , η k , C , C − 1 , χ , χ k , M , M − 1 are the elements of a n ar bitrary reduced incidence algebra R (Π , ∼ ), (i.e. mo dulo an arbitra ry order co mpatible equiv alence relatione ∼ on S (Π)). Then the next results follows immediately from this fact and ab o v e theore ms . Corollary 1. L et ( k , n ) ∈ T . Then: ζ ( k , n ) =  1 k ≤ n 0 k > n ; (16) ζ 2 ( k , n ) =      n − 1 X i = k F i ! + 2 k ≤ n 0 k > n ; (17) η ( k, n ) = δ k n ; (19) η s ( k , n ) =    X k n ; (20) C ( k , n ) =    1 k = n − 1 k < n 0 k > n ; (21) 6 χ ( k , n ) = δ ( k + 1 , n ); (22) χ s ( k , n ) = δ ( k + s, n ) · F k +1 F k +2 ...F n − 1 ; (23) µ ( k , n ) =      ( − 1) n − k n − 1 Y i = k +1 ( F i − 1) k ≤ n 0 k > n . (24) Corollary 2. The standar d r e duc e d incidenc e algebr a R (Π) is the maximal ly r e- duc e d incidenc e algebr a R (Π) , i.e. the smal lest r e duc e d incidenc e algebr a on Π . Equivalently t he e quivalenc e r elation define d by (10) is the maximal element in the lattic e of al l or der c omp atible e quivalenc e r elations on S (Π) . Ac knowledgement s Discussions with Participan ts of Gian-Carlo Ro ta Polish Seminar, ht tp://ii.uwb.edu.pl/akk/sem/sem rota.htm are hig hly apprec iated. 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