The homology groups of a partial monoid action category презентация

sets and partial maps
Let M be a monoid and let S be a set.
Partial right action of M on S is a homomorphism of monoids Mop → PSet(S,S)
Example 1.
M=N2 ={apbq : p, q=0, 1, 2, …},
S= {s0, s1, s2, s3, s4}
s0a=s1, s0b=s2
s1a=s4, s1b=s3
s2a=s3

actions has objects s∈S. Morphisms s1→s2 are triples (s1, μ, s2). Composition is defined by
(s2, μ2, s3) (s1, μ1, s2)= (s1, μ1 μ2, s3).
In Example 1,
K(S) is a partially ordered
set with
s0s1s2

be an irrefexive symmetric relation (of independence)

Free partially commutative monoid (or trace monoid) M(E,I) is a quotient monoid E*/(≡) where E* is a monoid of all words in alphabet E and (≡) is smallest equivalence relation for which
uabv ≡ ubav, for all (a,b)∈I, u∈ E*, v∈ E*

Trace monoid M(E,I) is called to be locally finite dimensional if E does not contain infinite subsets of pairwise independent elements

a free commutative monoid

E ={a,b}, I = ∅,
M(E,I) ≅ {a,b}* is a free monoid

a finite sequence of actions
a1 a2 … an
Transposing neighboring pairs of independent elements, it can be reduced to Foata normal form
[b1b2 …bp] [bp+1… bq]… [br+1… bn]
(actions bounded square brackets are performed concurrently)

can be constructed by gluing cells. We assign to each object c∈ObC
the point. Assign
to each morphism c0→с1
the segment, to each
pair c0→с1 →с2
the triangle and etc.

K(S) for a monoid partial action on a set S.
Application of the Baues-Wirsching homology for building the algorithms for computing the homology groups of the space B(K (S)) and directed homology groups for a partial trace monoid action.

→Ab be the colimit fuctor. For any functor F: C →Ab considered as an object in AbC , there exists a projective resolution
0 ←F ← P0 ← P1 ← … ← Pn ← Pn+1 ← …
and a complex С* of Abelian groups
0 ← colimC P0 ← colimC P1 ← …
…← colimC Pn ← colimC Pn+1 ← …
The groups Hn(C,F)≅Hn(С*), n≥0, are called
homology groups of the category C
with coefficients in F.

factorization category Fact(C) are all morphisms of C. Morphisms
in Fact(C) are given by commutative diagrams




Example 2. Let M be a monoid considered as the category. Then Ob(Fact (M) )= M. Its morphisms from α to β are pairs (f,g) of f, g ∈ M for which gαf=β.
Proposition 1. Fact(C)≅Fact(Cop)

Fact(C)op→Ab,
Hn(Fact(C)op,F) are called Baues-Wirsching homology groups of C with coefficients in contravariant natural systems F on C.

If C=M is a monoid, then Hn(Fact(M)op,F) are Leech homology groups of a diagram F

a monoid M.
Let K(S) be the action category with morphisms
Consider a functor F: Fact(K(S))op →Ab and the functor
U: K(S) →Mop defined as U(s1,μ,s2)= μ.
Theorem. Hn(Fact(K(S))op,F)≅ Hn(Fact(M)op,L(F))
Here L(F) is a left Kan extension of the functor F along the functor
Fact(K(S))op→Fact(M)op,
determined by the functor U.

trace monoid.
For n= 0, 1, 2, … , introduce sets
Tn(E,I)={(a1,…, an)| ai for all 1 ≤ i < j ≤ n}
In particular, T0(E,I)={1}.
For an arbitrary set S, denote by LS a free Abelian group generated by S.

over M(E,I) is a set with a partial action of a trace monoid.
Corollary 1. Let S be a polygon over a locally finite dimensional trace monoid M(E,I). Groups Hn(B(K(S))) isomorphic to homology groups of complex

Here sμ ∈S denotes that sμ is defined.

complex of vector spaces. Then
Hn=Ker dn / Im dn+1 = kdimCn-r(dn)-r(dn+1) where r(A) is denoted the rank of matrix A
Proposition 2. Let A be a matrix with integer entries. Then there are matrixes S, T, D, for which
A= T°D°S 2) |det S|=|det T|=1
3) D is a diagonal matrix
with δ1 | δ2 | … | δr

D is Smith normal form of A


If k=Z is a ring of integers, then the homology groups can be computing by the Smith normal form:
Hn=Ker dn / Im dn+1 = ZdimCn-r(dn)-r(dn+1)×Z/δ1Z×…× Z/ δr Z , r=r(dn+1)

H1=Z3-1-1 = Z

values Z (group of integers) on objects s and defined at morphisms as

Groups Hn0(S,M(E,I)) =def Hn(K(S), Δ0Z)
and Hn1(S,M(E,I)) =опр Hn(K(S)op, Δ1Z) are called
directed homology groups.

groups Hn0(S,M(E,I)) are isomorphic to homology groups of the complex consisting of the same Abelian groups but has the following differentials

Hn1(S,M(E,I)) can be computed by a similar complex with differentials

H10
C0=L{s0,s1,s2 }, C1=L{(s0,a), (s0,b)},






We obtain H00=Z3-1=Z2, H10=Z2-1 = Z .

C1=L{(s0,a), (s0,b)},





Obtain H00=Z3-2=Z, H10=Z2-2 = 0.

if the action category does not has morphisms with codomain s. It is final if there are not morphisms with domain s.
Groups H00(S,M(E,I)) and H01 (S,M(E,I)) are free. Initial elements generated
H01 (S,M(E,I)), final elements generated H00 (S,M(E,I))
In Figure: rk H01 (S,M(E,I))=1
rk H00 (S,M(E,I))=2

and
where is a functor defined as


Similarly defined the functor .
We obtain decomposition the directed homology groups. Local homology groups of the category for the partial trace monoid action M(E,I) on S are defined as

S be a polygon over M(E,I). For s∈S, the groups
can be computed by the following complex

be a polygon over M(E,I). For s∈S, we consider a simplicial scheme ∇(s) with the set of vertexes ∇0(S,E,I.s)= {a0∈E | sa0∈S}.
Its sets of n-simplexes equal

Recall that

computing the groups Hn0

Let S be a polygon over M(E,I). For s∈S, the groups can be computed by the following complex

Example 6.

Let S be a polygon over M(E,I). For s∈S, we can remove the condition e1<…

There are isomorphisms

is open if and only if the corresponding morphism of asynchronous systems is PomΛ -open.

Let S be a polygon over M(E,I). Label function λ: E→Λ is a map such that
(a1,a2)∈I ⇒ λ (a1)≠ λ (a2 ).
Let A=(S,M(E,I),s0 , Λ, λ) be a polygon over M(E,I) with a label function. An element
s∈S is reachable if (∃μ∈M(E,I)) s= s0 μ . Let S(s0 ) the set of reachable s.
Definition. A morphism of polygons with label functions A→A’ is open, if it is given by
a pair of (total) maps σ: S→S’, η: E→E’ , satisfying the following conditions
σ(s0)= s’0 ,
η save labels, i.e. λ’ °η = λ.
(e1,e2)∈I ⇒ (η(e1),η (e2) ∈I’
if s is reachable, then η: {s | se ∈S} → {e’ | σ(s)e’ ∈S’} surjective
if s is reachable, then η: {(e1,e2)∈I | se1e2 ∈S} → {(e’1,e’2)∈I’ | σ (s)e’1e’2 ∈S’} is surjective

label functions λ: E→Λ and λ’: E’→Λ are bisimulation equivalent, then ∀s∈S(s0) ∃ s’∈S’(s0’) and ∀s’∈S(s0’) ∃ s∈S(s0) for which

Pointed polygons A1 and A2 with labels in Λ are bisimulation equivalent if there is a pointed polygon A and open morphisms A1 ← A → A2.
Consider the following complex LΛ:

Label function gives the chain morphism

Λ ={a, b, c}
E={a1, a2 ,b, c}, E’={a ,b, c}
λ(a1)= λ(a2)= a λ’(a)= a
λ(b)=b λ’(b)=b
λ(c)=c λ’(c)=c

Trace ab and ac.
Complexes

actions
Applications of the homology groups for the studying the topology of state spaces of concurrent systems