The Torsion of Homology Groups of M(E,I)-sets

In this paper, following [1,2], we consider the homology groups of right pointed sets over a partially commutative monoid M (E, I).

Definition. Let E be a set and I ⊆ E × E an irreflexive and summetric relation. A monoid guven by a set of generators E and relations ab = ba for all (a, b) ∈ I is called free partially commutative [3] and denoted by M (E, I). If (a, b) ∈ I then the members a, b ∈ E are said to be commuting generators.

The motivation for the research of homology groups of the M (E, I)-sets came from a desire to find topology’s invariants for asynchronous transition systems. M. Bednarczyk [4] has introduced asynchronous transition systems to the modeling the concurrent processes. In [5] it was proved that the category of asynchronous transition systems admits an inclusion into the category of pointed sets over free partially commutative monoids. Thus asynchronous transition systems may be considered as M (E, I)-sets. C n F . We’ll be consider any monoid as the small category with the one object. This exert influence on our terminology. In particular a right M -set X will be considered and denoted as a functor X : M op → Set (the value of X at the unique object will be denoted by X(M ) or shortly X.) Morphisms of right M -sets are natural transformations.

Let us show that there is following Theorem 1.1 Let X • m and X • m+1 are M (E, I)-sets, here m -1, then there exist an isomorphism

k ∆Z is exact with respect to first argument, we have homomorphisms;

From this long exact sequence we can make up the following shot exact sequence

Indeed, this follows from long exact sequence and homomorphism’s theorem;

using [2, Theorem 3.1, Example 3.2], we’ll complete the proof of theorem.

From this thoerem, we get the following Corollary 1.2 Let X • m and X • m+1 are M (E, I)-sets, then there exist a isomorphism

Tor

), here m -1 and k 1.

For one-dimensional homology groups, we get the following Theorem 1.3 One-dimensional homology groups of M (E, I)-sets form X • m are free, here m -1.

Proof. Indeed, using the theorem 1.1, where m = 0, we get the isomorphism

) is free. Further, using induction on m and theorem 1.1, we’ll complete the proof of this theorem.

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