The Homology groups of right pointed sets over a partially commutative monoid

THE HOMOLOGY GR OUPS OF RIGHT POINTED SETS O VER A P AR TIALL Y COMMUT A TIVE MONOID Lopatkin V. Abstract W e study the homology of p oin ted sets ov er a partially commutativ e monoid. 2000 Mathematics Sub ject Classification 18G10, 55U99 Keyw ords: free partially comm utative monoid, homology of small categories, async hronous transition systems. In tro duction In this pap er w e consider the homology groups of righ t pointed sets o v er a partially comm utative monoid M ( E , I ). Definition. Supp ose E is finite set, I ⊆ E × E is an irreflexive and symmetric relation. A monoid given by a set of generators E and relations ab = ba for all ( a, b ) ∈ I is called fr e e p artial ly c ommutative and denoted b y M ( E , I ) [1]. If ( a, b ) ∈ I then members a, b ∈ E are said to b e c ommutating gener ators [1]. In this paper we denote b y X • a pointed set, in other words X • = X ∪ {∗} , here {∗} is a se lected elemen t. The p oin ted set X • o ver the partially comm utative monoid M ( E , I ) is also called (see [2]) M ( E , I )-set X • . Motiv ation and bases result The motiv ation for the research of homology groups of the M ( E , I )-sets came from a desire to find top ology’s in v ariants for asynchronous transition systems. M. Bednarczyk [3] has introduced asynchr onous tr ansition systems to the mod- eling the concurrent pro cesses. In [4] it was prov ed that the category of asyn- c hronous transition systems admits an inclusion in to the category of p ointed sets ov er free partially commutativ e monoids. Th us asynchronous transition systems ma y b e considered as M ( E , I )-sets. It w as shown in [5] that if the homology of pointed the M ( E , I )-set are isomorphic to the homology of a p oin t, then the partially commutativ e monoid M ( E , I ) consist of one elemen t, and the p oin ted set consist of one p oint. Then, w e get a follo wing question; supp ose that, we hav e tw o p ointed M ( E , I )-sets X • and Y • , if their homology are isomorphic, do es it follo w that this M ( E , I )-sets are isomorphic? In this pap er w e giv e the negative answer to this question. Denoting Let A be a small category . W e denote b y Ob A and Mor A the classes of ob jects and morphisms in A , respectively . By A op w e denote the opp osite category . 1 Denote by Set ∗ the category of p oin ted sets: each its ob jects is a set X with a selected element, written ∗ and called the ”base point“; its morphisms are maps X → Y whic h send the base point of X to that of Y . Suc h maps are called b ase d . W e denote ob jects of the category Set ∗ b y X • . The set X • \ {∗} denote b y X . W e’ll b e consider any monoid as the small category with the one ob ject. This exert influence on our terminology . In particular a right M -set X will b e considered and denoted as a functor X : M op → Set (the v alue of X at the unique ob ject will b e denoted b y X ( M ) or shortly X .) Morphisms of righ t M - sets are natural transformations. In this paper w e consider a case, when p oin ted set X • is finite. Let us denote by X • n a finite p ointed set that consist of n + 2 elemen ts { x 0 , x 1 , . . . , x n , ∗} . Let us assume that X • − 1 = {∗} . Supp ose that, we hav e the right p ointed M ( E , I )-set X • n . Let us construct a category Cat ∗ [ M ( E , I ) , X • n ], or shortly Cat ∗ [ X • n ], which ob jects are elements of p ointed set X • n and morphisms are triples ( x, µ, x 0 ) which x, x 0 ∈ X • n and µ ∈ M ( E , I ) satisfying to x · µ = x 0 . W e assume that the morphism  x, 1 M ( E ,I ) , x  is iden tical morphism of category Cat ∗ [ M ( E , I ) , X • n ], here 1 M ( E ,I ) is iden tit y elemen t of monoid M ( E , I ). The comp osition of morphisms ( x, µ, x 0 ), ( x 0 , µ 0 , x 00 ) is a morphism of form ( x, µ · µ 0 , x 00 ). Sometimes we denote this triples b y x µ − → x 0 . By Ab w e denote the category of ab elian groups and homomorphisms. Let A b e a small category . Denote b y ∆ A Z : A → Ab, or shortly ∆ Z , the functor whic h has constan t v alues ∆ Z ( a ) = Z at a ∈ Ob A and ∆ Z ( α ) = 1 Z at α ∈ Mor A . F or an y functor F : A → Ab, here A is a small category , denote b y lim − → A k F v alues of the left satellites of the colimit lim − → A : Ab A → Ab. It is w ell [7, Prop. 3.3, Aplication 2] that there exists an isomorphism of left satellites of the colimit lim − → A : Ab A → Ab and the functors H n ( C ∗ ( A , − )) : Ab A → Ab. Since the category Ab A has enough pro jectiv es, these satellites are natural isomorphic to the left derived functor of lim − → A : Ab A → Ab. Denote the v alues H n ( C ∗ ( A , − )) of satellites at F ∈ Ab A b y lim − → A n F . 1 Basic notions and definitions W e’ve assume that, the set of generators of the monoid M ( E , I ) is finite (see In- tro duction), hence this monoid M ( E , I ) satisfying to locally b ounded condition [2]. Then for this monoids there are a following [2, Theorem 3.1] Theorem 1.1 L et M ( E , I ) b e a fr e e p artial ly c ommutative monoid, X : M ( E , I ) op → Set is a right M ( E , I ) -set, F : Cat ∗ [ M ( E , I ) , X ] → Ab is a functor. Then the homolo gy gr oups lim − → Cat ∗ [ M ( E ,I ) , X ] n F ar e isomorphic to the homolo gy gr oups of the chain c omplex 0 ← − M x ∈X F ( x ) d 1 ← − M ( x,e 1 ) F ( x ) d 2 ← − M ( x,e 1 e 2 ) , e 1 2 , her e H ∗ ( E , M ) is homolo gy gr oups of the simplicial schema ( E , M ) . In p artial ly, let us assume that, the ge ometric r e alization of nonr elation gr aph of the monoid M ( E , I ) home omorphic to the lens sp ac e S 2 / Z p , then we get fol lowing isomorphism lim − → Cat ∗ [ X • 0 ] 2 Z [ x 0 ] ∼ = Z p . The isomorphism of form lim − → Cat ∗ [ X • 0 ] n Z [ x 0 ] ∼ = H n − 1 ( E , M ), where n > 2, can b e generalized b y follo wing Theorem 1.2 Supp ose that, we have the p ointe d set X 0 = { x 0 , ∗} over the fr e e p artial ly c ommutative monoid M ( E , I ) , then ther e is a fol lowing isomorphism for n > 1 lim − → Cat ∗ [ X • 0 ] n Z [ x 0 ] ∼ = e H n − 1 ( E , M ) , her e e H n − 1 ( E , M ) is r e duc e d homolo gy of the simplicial schema ( E , M ) . 3 Pro of. F or n > 2, this isomorphism follows from Example 1.1. Supp ose that, n > 1. Let us consider a follo wing diagram . . . d 3 / / L e 1 0, the c hain complex of the p ointed M ( E , I )-set X 0 , with co efficien ts in the functor Z [ x 0 ], is isomorphic to the augmen tation complex ( C n − 1 ( E , M ) , d n − 1 ) of the simplicial s c hema. It means that, there is a following isomorphism, for n > 1 lim − → Cat ∗ [ X • 0 ] n Z [ x 0 ] ∼ = e H n − 1 ( E , M ) . Q.E.D. F rom this theorem we get follo wing Corollary 1.3 The one-dimension homolo gy of the p ointe d M ( E , I ) -set X • 0 with c o efficients in the functor Z [ x 0 ] hasn ’t the torsion sub gr oups. Pro of is trivial. Let us consider the monoid M ( E , I ) as the small category with the one ob ject. In this case we can to find the homology of the partially commutativ e monoid M ( E , I ). Indeed, follow Theorem 1.1 we get (see [2, Example 4.1]) Example 1.2 L et M ( E , I ) b e a fr e e p artial ly c ommutative monoid which acts on the one-p oint set X • − 1 = { X } . In this c ase by The or em 1.1 the homolo gy gr oups lim − → Cat ∗ [ X • − 1 ] s ∆ Z ar e isomorphic to the homolo gy of the chain c omplex 0 ← Z d 1 ← − M e Z d 2 ← − M e 1 1 lim − → Cat ∗ [ X • n ] s ∆ Cat ∗ [ X • n ] Z ∼ = lim − → Cat ∗ [ X • n ] s Z [ x 0 , . . . , x n ] ⊕ lim − → Cat ∗ [ X • − 1 ] s ∆ Cat ∗ [ X • − 1 ] Z . Pro of. F rom Theorem 1.1 it isn’t hard to see that, lim − → Cat ∗ [ X • − 1 ] s ∆ Cat ∗ [ X • − 1 ] Z ∼ = lim − → Cat ∗ [ X • n ] s Z [ ∗ ] ∼ = lim − → Cat ∗ [ X • − 1 ] s Z [ ∗ ] . Let us consider a functor Ret : Cat ∗ [ X • n ] → Cat ∗ [ X • − 1 ]. This functor takes all ob jects of the category Cat ∗ [ X • n ] to one ob ject of the category Cat ∗ [ X • − 1 ], further, on morphisms of form ( x p , e, x q ), here x p 6 = ∗ , this functor is zero, and on other morphisms it is identical. It can easily b e chec ked that the functor Ret is left in verse to inclusion functor Inj : Cat ∗ [ X • − 1 ] ⊆ Cat ∗ [ X • n ]. And since the functor RetInj : Cat ∗ [ X • − 1 ] → Cat ∗ [ X • − 1 ] is indicate functor, then the homomorphism Ret s Inj s is indicated automorphism of the group lim − → Cat ∗ [ X • − 1 ] s Z [ ∗ ], for s > 1, and moreo ver there is a following isomorphism lim − → Cat ∗ [ X • n ] s ∆ Z ∼ = Im Inj s ⊕ Ker Ret s . This completes the pro of. Let us sho w that there is a following Prop osition 2.2 Supp ose that, we have the p ointe d M ( E , I ) -set, wher e the monoid M ( E , I ) acts ful ly over the X • n , and assume that the gr aph G [ M ( E , I ) , X • n ] is the r o oting tr e e with the r o ot in ∗ , then ther e is a fol lowing isomorphism, for s > 1 lim − → Cat ∗ [ X • n ] s Z [ x 0 , x 1 , . . . , x n ] ∼ =  lim − → Cat ∗ [ X • 0 ] s Z [ x 0 ]  n +1 . Pro of. Let us consider the category Cat ∗ [ X • n ]. F rom conditions i) – ii) it follo ws that, there exist x, x 0 ∈ Ob Cat ∗ [ X • n ], and for all e ∈ E , there are a following equations x · e = x 0 , moreov er there not exist x 00 ∈ Ob Cat ∗ [ X • n ], that x 00 · e = x . Without loss of generalit y it can b e assumed that x = x n , and x n · e = x n − 1 for all e ∈ E . Then, it is easily shown that the category Cat ∗ [ { x 0 , . . . , x n − 1 , ∗} ] = Cat ∗ [ X • n − 1 ] is full sub category of the category Cat ∗ [ X • n ]. Let us denote by 6 Q [ x n ] the inclusion functor Q [ x n ] : Cat ∗ [ X • n − 1 ] ⊂ Cat ∗ [ X • n ]. Let us intro- duce a follo wing functor P [ x n ] : Cat ∗ [ X • n ] → Cat ∗ [ X • n − 1 ]. This functor w e define in the following wa y; P [ x n ]( x n ) = x n − 1 , and on other ob jects this func- tor is indicated, further, on morphisms, w e assume that P [ x n ]( x n , e, x n − 1 ) =  x n − 1 , 1 M ( E ,I ) , x n − 1  , on other morphisms this functor is indicated. W e see that the functor P [ x n ] is left inv erse to the inclusion functor Q [ x n ]. And since the functor P [ x n ] Q [ x n ] : Cat ∗ [ X • n − 1 ] → Cat ∗ [ X • n − 1 ] is indicated functor, thus, for s > 1, the homomorphism P s [ x n ] Q s [ x n ] is indicated automorphism of the group lim − → Cat ∗ [ X • n − 1 ] s Z [ x 0 , . . . , x n ], and moreo ver there is a following isomorphism, for s > 1 lim − → Cat ∗ [ X • n ] s Z [ x 0 , . . . , x n ] ∼ = Im Q s [ x n ] ⊕ Ker P s [ x n ] . It is obvious that Im Q s [ x n ] ∼ = lim − → Cat ∗ [ X • n − 1 ] s Z [ x 0 , . . . , x n ]. Also, it is not hard to pro ve that Ker P s [ x n ] ∼ = lim − → Cat ∗ [ { x n ,x n − 1 } ] s Z [ x 0 , . . . , x n ] ∼ = lim − → Cat ∗ [ X • 0 ] s Z [ x 0 , . . . , x n ]. F urther, from theorem 1.1 it follow that lim − → Cat ∗ [ X • n − 1 ] s Z [ x 0 , . . . , x n ] ∼ = lim − → Cat ∗ [ X • n − 1 ] s Z [ x 0 , . . . , x n − 1 ] lim − → Cat ∗ [ X • 0 ] s Z [ x 0 , . . . , x n ] ∼ = lim − → Cat ∗ [ X • 0 ] s Z [ x 0 ]. Thu s, w e hav e a following isomor- phism lim − → Cat ∗ [ X • n ] s Z [ x 0 , . . . , x n ] ∼ = lim − → Cat ∗ [ X • n − 1 ] s Z [ x 0 , . . . , x n − 1 ] ⊕ lim − → Cat ∗ [ X • 0 ] s Z [ x 0 ] . In other hand, in the same wa y , w e’ll hav e corresp onding isomorphism for cat- egories Cat ∗ [ X • n − 1 ], Cat ∗ [ X • n − 2 ] e.t.c. And finally , w e’ll hav e isomorphism, for s > 1 lim − → Cat ∗ [ X • n ] s Z [ x 0 , x 1 , . . . , x n ] ∼ =  lim − → Cat ∗ [ X • 0 ] s Z [ x 0 ]  n +1 Q.E.D. The follo wing theorem is our main result of this pap er. Theorem 2.3 Supp ose, we have the p ointe d M ( E , I ) -set X • n = { x 0 , x 1 , . . . , x n , ∗} , which satisfaing c onditions i) – ii), then ther e is a fol lowing isomorphism, for s > 1 H s ( X • n ; Z ) ∼ =  e H s − 1 (( E , M ); Z )  n +1 ⊕ Z p s , her e p s is the numb er of subsets { e 1 , . . . , e s } ⊆ E c onsisting of mutual ly c ommu- tating gener ators, e H s − 1 (( E , M )) is r e duc e d homolo gy of the simplicial schema ( E , M ) . Pro of. Since, conditions i) – ii) are fulfilled, then from prop osition 2.2 it follo ws that, w e hav e a following isomorphism lim − → Cat ∗ [ X • n ] s ∆ Z ∼ =  lim − → Cat ∗ [ X • 0 ] s Z [ x 0 ]  n +1 ⊕ lim − → Cat ∗ [ X • − 1 ] s ∆ Z . In outher hand, from theorem 1.2 it follows that, there is a isomorphism, for s > 1 lim − → Cat ∗ [ X • 0 ] s Z [ x 0 ] ∼ = e H s − 1 ( E , M ) . 7 But, we already kno w (see Example 1.2) that lim − → Cat ∗ [ X • − 1 ] s ∆ Z ∼ = Z p s . This completes the pro of. Example 2.1 Assume that we have the p ointe d M ( E , I ) -set X • 3 = { x 0 , x 1 , x 2 , x 3 , ∗} with ful ling action of the monoid M ( E , I ) . This M ( E , I ) -set is shown in figur e 2. x 2 { e }   x 0 { e } / / x 1 { e } / / ∗ { e } d d x 3 { e } F F Figure 2: Here w e sketc hy sho w full action of the monoid M ( E , I ) ov er the p oin ted set X • 3 . W e assume that e runs ov er all set E . L et us c alculate the homolo gy of this M ( E , I ) -set. Supp ose that the ge ometric r e alization of the simplicial schema ( E , M ) is home omorphic to a top olo gy sp ac e V . Thus, fr om the or em 2.3 we get the fol lowing isomorphism, for s > 1 H s ( X • 3 ) ∼ = ( H s − 1 ( V )) 4 ⊕ Z p s . F rom Theorem 2.3 we get a follo wing Corollary 2.4 Supp ose that, we have the p ointe d M ( E , I ) -set X • n which satis- fying c onditions i) – ii), then the inte ger one-dimension homolo gy gr oup of this M ( E , I ) -set hasn ’t torsion sub gr oup. Pro of. The proof follows from Corollary 1.3 and Theorem 2.3. F rom Theorem 2.3 follo ws that, differ ent M ( E , I )-sets can ha ve isomorphic homology . Indeed, let us show this b y follo wing Example 2.2 In figur e 3 is shown differ ent M ( E , I ) -sets. We assume that, the set E c onsist of s elements. But this differ ent M ( E , I ) -sets have isomorphic homolo gy. Inde e d, let us denote the M ( E , I ) -set fr om left figur e by L , and the M ( E , I ) -set fr om right figur e, we denote by R . Then, fr om The or em 2.3, we get the fol lowing isomorphisms, for n > 1 H n ( L ; Z ) ∼ =  e H n − 1 ( E , M )  2 ⊕ Z p n ; H n ( R ; Z ) ∼ =  e H n − 1 ( E , M )  2 ⊕ Z p n ; The nul l-dimension homolo gy also isomorphic as M ( E , I ) -sets have c ommon c onne cte d c omp onents. 8 x 0 e 1 ) ) e s 9 9 . . . x 1 e 1 ( ( e s : : . . . ∗ . . . e 1 { { e s n n x 0 e 1   e s , , ... x 1 e 1 v v e s   ... ∗ ... e 1 c c e s ; ; Figure 3: Here is shown different actions of the monoid M ( E , I ) o ver the pointed set X • 1 = { x 0 , x 1 , ∗} . Concluding remarks W e’ve see that, different M ( E , I )-sets can ha ve isomorphic homology groups. This means that, homology groups is not enough for research of top ology’s in v ariants of M ( E , I )-sets (asynchronous transition systems). Thus there arise a motiv ation for researc h of others ob jects of algebraic top ology for pointed M ( E , I )-sets. References [1] Diekert V., M´ etivier Y. Partial Commutation and T r ac es. // Handb ook of formal languages. V. 3. Springer-V erlag, 1997. P . 457–533. [2] Husainov A. On the Cubical Homology Groups of F ree Partially Com- m utative Monoids // New Y ork: Cornell Univ, Preprint, 2006. 47 pp. h [3] Bednarczyk M. A. Categoris of Async hronous Systems Ph. D. Thesis, Unicersit y of Syssex, rep ort 1/88, 1988, 222p. h ttp://www.ipipan.gda.pl/ ∼ matek [4] Husainov A. A., Tk achenk o V. V. Homology groups of asynchronous tran- sition systems. Mathematical modeling and the near questions of mathe- matics. Collection of the scientifics works. Khabarovsk: KhHPU, 2003. P . 23-33. (Russian) h ttp://www.knastu.ru/husaino v site/index.html [5] A.A. Khusaino v, V.E. Lopatkin, I.A. T reshev, “Algebraic top ology ap- proac h to mathematical mo del analysis of concurrent computational pro- cesses”, Sib. Zh. Ind. Mat., 11:1 (2008), 141-152. (Russian) [6] Winskel G., Nielsen M. Mo dels for Concurrency .//Handb o ok of Logic in Computer Science, V ol. IV, ed. Abramsky , Gabbay and Maibaum. Oxford Univ ersity Press, 1995. P .1 – 148 [7] P . Gabriel, M. Zisman, Calculus of fr actions and homotopy the ory , Springer, Berlin, 1967. 9 Lopatkin Viktor wickktor@gmail.com Komsomolsk on Am ure State T echnical Univ ersit y Russia 10