A Proof of the Factorization Forest Theorem

A Pro of of the F actorization F orest Theorem Manfred Kufleitner 1 , 2 1 LaBRI, Unive rsit´ e de Bordeaux and CNRS 351, Cours de la Lib ´ eration F-33405 T alence cedex, F rance 2 FMI, Univ ersit¨ at Stut tgart Unive rsit¨ atsstr. 38 D-70569 Stuttgart, German y Abstract W e sho w that for every homomorphism Γ + → S where S is a finite semigroup there exists a f actorization forest of height ≤ 3 | S | . The pro o f is based on Green’s relations. 1 In tro duction F actorization for e sts where introduced by Simo n [5, 7 ]. An impor tant prop erty of finite semigro ups is that they admit factorization for ests of finite height. This fact is ca lle d the F actor ization F ore st Theorem. It can b e consider ed a s an Ramsey-type prop erty of finite semigroups. There exists differen t p ro ofs of this fact of differen t difficult y and with different b ounds on the height. The first pro of of t he F actoriza tion F orest Theor em is due to Simon [7]. He showed that for every finite s emigroup S there exis ts a factoriza tion for est of height ≤ 9 | S | . The pro of relie s on s everal different techniques. It uses gr aph c o lorings , Green’s relations, and a de c o mp osition tec hnique inspir ed b y the Rees-Suschk ewitsch Theorem on completely 0-simple semigroups. In [8] Simon gav e a simplified pro of relying on the Krohn-Rho des dec o mp osition. The b ound shown is 2 | S | +1 − 2. A concis e pr o of has b een given b y Chalopin and Leung [1]. The pro of relies o n Green’s relations and yields the b ound 7 | S | on the height. Independently o f this work, Colcombet has also shown a b ound of 3 | S | for the heig ht of factorization forests [2]. He uses a generaliza tion of the F actorization F ores t The o rem in terms of R amseyan splits . The pro of also relies on Green’s relatio ns . A v ariant of our pro of for the sp ecial case of a p erio dic monoids has b een shown in [3] with a bo und of 3 | S | . The main be nefit of that pro of is that is uses very little machinery . The pro o f in this pa p e r ca n b e seen as an extension o f that pr o of. The main to ol a re again Green’s relations. W e o nly r equire basic results from the theor y of finite s e migroups which can b e found in standar d textbo o ks such as [4]. A lower b ound of | S | was shown f or rectangular bands in [6] and also in [1]. The s a me b o und has als o be e n shown for g r oups [1]. Ther efore, the upp er bo und o f 3 | S | reduces the ga p b etw een the lower and the upp er b ound. 2 The F actorization F orest Theorem Let S be a finite semigro up. A factori zation for est o f a homomorphism ϕ : Γ + → S is a function d which maps ev er y word w with length | w | ≥ 2 to a 1 factorization d ( w ) = ( w 1 , . . . , w n ) o f w = w 1 · · · w n with n ≥ 2 and w i ∈ Γ + and such that n ≥ 3 implies ϕ ( w 1 ) = · · · = ϕ ( w n ) is idempo tent in S . The height h of a word w is defined a s h ( w ) = ( 0 if | w | ≤ 1 1 + max { h ( w 1 ) , . . . , h ( w n ) } if d ( w ) = ( w 1 , . . . , w n ) W e call the tree defined by the “bra nching” d fo r the word w the factorization tr e e of w . T he height h ( w ) is the height of this tree. The height o f a fa ctorization forest is the supr emum over the heights of all words. F ac toriztion F orest Theorem (Simon [7]). L et S b e a finite monoid. Eve ry homomorph ism ϕ : Γ + → S has a factorization for est of height ≤ 3 | S | . Pr o of: Let [ w ] = ϕ ( w ). W e show that for every w ∈ Γ + there exists a fac- torization tree of height h ( w ) ≤ 3 |{ x ∈ S | [ w ] ≤ J x }| . First, w e p erform a n induction on the cardina lity o f the set { x ∈ S | [ w ] ≤ J x } ; then within o ne J - class we re fine this pa rameter. Let w ∈ Γ + with | w | ≥ 2. Then w has a unique factorization w = w 0 a 1 w 1 · · · a m w m with a i ∈ Γ and w i ∈ Γ ∗ satisfying the following tw o co nditions: ∀ 1 ≤ i ≤ m : [ a i w i ] J [ w ] and ∀ 0 ≤ i ≤ m : w i = ε ∨ [ w ] < J [ w i ] The idea is tha t we s uccessively choose a i w i ∈ Γ + from r ight to left to be the shortest non-empty word such that [ a i w i ] J [ w ]. Let w ′ i = a i w i for 1 ≤ i ≤ m . F or ea ch 1 ≤ i < m define a pair ( L i , R i ) where L i is the L -class of [ w ′ i ] and R i is the R -class of [ w ′ i +1 ]. Every suc h pair repr esents an H -clas s within the J -class of [ w ]. All H - classes within this J -cla ss contain the s ame num b er n o f elements. Let h ′ ( w ) = h ( w ) − 3 · |{ x ∈ S | [ w ] < J x }| W e can think o f h ′ as the heigh t of a tree where w e additionally allow words v ∈ Γ + with [ w ] < J [ v ] as leafs. Within the J -class o f w we perfo rm an induction o n the ca rdinality of the s e t { ( L i , R i ) | 1 ≤ i < m } in order to show h ′ ( w ) ≤ 3 n · |{ ( L i , R i ) | 1 ≤ i < m }| Since n · |{ ( L i , R i ) | 1 ≤ i < m } | ≤ |{ x ∈ S | [ w ] J x }| this yields the desired bo und for the height h ( w ). If every pair ( L, R ) o ccurs at most twice then w e hav e m − 1 ≤ 2 · |{ ( L i , R i ) | 1 ≤ i < m } | . W e define a factoriza tion tree for w by d ( w ) = ( w 0 w ′ 1 , w ′ 2 · · · w ′ m ) d ( w 0 w ′ 1 ) = ( w 0 , w ′ 1 ) d ( w ′ i · · · w ′ m ) = ( w ′ i , w ′ i +1 · · · w ′ m ) for 2 ≤ i < m d ( w ′ i ) = ( a i , w i ) for 1 ≤ i ≤ m Since [ w ] < J [ w i ], b y induction ev e ry w i has a factorization tr ee of height h ( w i ) < 3 |{ x | [ w i ] ≤ J x }| ≤ 3 |{ x | [ w ] < J x }| . This yields: h ′ ( w ) ≤ m ≤ 3 n · |{ ( L i , R i ) | 1 ≤ i < m }| Note that the height do es not increase if some of the w i are empty . No w supp ose there exists a pair ( L, R ) ∈ { ( L i , R i ) | 1 ≤ i < m } o c c ur ring (at least) three times. Let i 0 < · · · < i k be the sequence of all po s itions with ( L, R ) = ( L i j , R i j ). By c o nstruction we have k ≥ 2. Let c w j = w ′ i j − 1 +1 · · · w ′ i j for 1 ≤ j ≤ k . F or a ll 1 ≤ j ≤ ℓ ≤ k we hav e 2 • [ c w j · · · c w ℓ ] ≤ L [ w ′ i ℓ ] L [ w ′ i 0 ]. • [ c w j · · · c w ℓ ] ≤ R [ w ′ i j − 1 +1 ] R [ w ′ i 0 +1 ]. • [ w ′ i ℓ ] ≤ J [ c w j · · · c w ℓ ] ≤ J [ w ] J [ w ′ i ℓ ] J [ w ′ i 0 ] J [ w ′ i 0 +1 ] b y a s sumption on the factor ization. Thu s for all 1 ≤ j ≤ ℓ ≤ k and 1 ≤ j ′ ≤ ℓ ′ ≤ k we get • [ c w j · · · c w ℓ ] L [ w ′ i 1 ] L [ c w j ′ · · · c w ℓ ′ ] and • [ c w j · · · c w ℓ ] R [ w ′ i 1 +1 ] R [ c w j ′ · · · c w ℓ ′ ] and ther efore • [ c w j · · · c w ℓ ] H [ c w j ′ · · · c w ℓ ′ ] Therefore, all [ c w j ] denote elements in the sa me H -cla ss H and since k ≥ 2 the class H is a group. W e co nsider the following set of elements in H induced by prop er pr efixes P ( c w 1 · · · c w k ) = { [ c w 1 · · · c w j ] | 1 ≤ j < k } F or the pair ( L , R ) we show by induction on | P ( c w 1 · · · c w k ) | that h ′ ( w ) ≤ 3 | P ( c w 1 · · · c w k ) | + 3 n |{ ( L i , R i ) | 1 ≤ i < m }| Suppo se every element x ∈ P ( c w 1 · · · c w k ) ⊆ H o cc urs a t most twice. Then k − 1 ≤ 2 | P ( c w 1 · · · c w k ) | . W e co ns truct the following factor ization tre e for w : d ( w ) = ( w 0 w ′ 1 · · · w ′ i 1 , w ′ i 1 +1 · · · w ′ m ) d ( w 0 w ′ 1 · · · w ′ i 1 ) = ( w 0 w ′ 1 · · · w ′ i 0 , c w 1 ) d ( w ′ i 0 +1 · · · w ′ m ) = ( c w 2 · · · c w k , w ′ i k +1 · · · w ′ m ) d ( c w i · · · c w k ) = ( c w i , [ w i +1 · · · c w k ) for 2 ≤ i < k By inductio n on the num b er of pairs ( L i , R i ) there exist factor ization trees for the words w 0 w ′ 1 · · · w ′ i 0 , w ′ i k +1 · · · w ′ m , and all c w i of heig ht ≤ 3 n |{ ( L i , R i ) | 1 ≤ i < m } \ { ( L, R ) }| + 3 |{ x | [ w ] < J x }| This yie lds h ′ ( w ) − 3 n |{ ( L i , R i ) | 1 ≤ i < m }| ≤ k ≤ 3 | P ( c w 1 · · · c w k ) | Now suppose there exists an element x ∈ P ( c w 1 · · · c w k ) ⊆ H tha t o ccurs at least three times. Let j 0 < · · · < j t be the seq ue nc e of all p os itions with x = [ c w 1 · · · c w j i ]. By co nstruction we hav e t ≥ 2. It follows that [ \ w j i +1 · · · [ w j i +1 ] = e = e 2 where e is the neutral element of the group H . Let v i = \ w j i − 1 +1 · · · c w j i for 1 ≤ i ≤ t . W e constr uc t the following factorizatio n tree for w : d ( w ) = ( w 0 · · · c w i 0 , \ w i 0 +1 · · · w ′ m ) d ( \ w i 0 +1 · · · w ′ m ) = ( v 1 · · · v t , \ w i t +1 · · · w ′ m ) d ( v 1 · · · v t ) = ( v 1 , . . . , v t ) W e hav e x ∈ P ( c w 1 · · · c w k ) \ P ( c w 1 · · · c w i 0 ) and xP ( \ w j i − 1 +1 · · · c w j i ) ⊆ P ( c w 1 · · · c w k ) but x 6∈ xP ( \ w j i − 1 +1 · · · c w j i ). Hence, by induction on the ca rdinality of the prefix sets, ther e exis t factor ization forests for w 0 · · · c w i 0 , \ w i t +1 · · · w ′ m and the v i of height ≤ 3 | P ( c w 1 · · · c w k ) | − 3 + 3 n |{ ( L i , R i ) | 1 ≤ i < m } \ { ( L, R ) }| + 3 |{ x | [ w ] < J x }| This yie lds a fac torization tree for w with the de s ired height b ound. ✷ 3 Ac kno wledgement. I would like to thank V olker Diekert for numerous dis- cussions o n this topic. References [1] J´ er´ emie Chalo pin and Hing Leung. O n factoriza tion forests of finite height. The or etic al Computer Scienc e , 3 10(1-3 ):489– 4 99, 2 0 04. [2] Thoma s Colcombet. F actorisa tion forests fo r infinite w or ds . In Er zs´ ebet Csuha j-V arj ´ u and Zo lt´ an ´ Esik, editor s, F u ndamentals of Computation The- ory, 16th International Symp osium, FCT 2007, Budap est , Hu n gary, August 27-30, 2007 , Pr o c e e dings , v olume 463 9 of L e ctur e Notes in Computer Sci- enc e , pa g es 226– 237. Springer-V er la g, 2 007. [3] V olker Diekert a nd Manfred Kufleitner . O n first-or der fra gments for words and Mazurkiewicz traces: A survey . 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