LIAFA, CNRS and University Paris Diderot
The kernel of a monoid morphism
Jean-´Eric Pin1
1LIAFA, CNRS and University Paris Diderot
October 2009, Valencia
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Outline
(1) Kernels and extensions (2) The synthesis theoem (3) The finite case
(4) Group radical and effective characterization (5) The topological approach
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Basic definitions
An elementeof a semigroup isidempotentif e2=e. The set of idempotents of a semigroupSis denoted byE(S).
A semigroup isidempotentif each of its elements is idempotent (that is, ifE(S) =S). Asemilatticeis acommutative and idempotent monoid.
Avariety of finite monoidsis a class of finite monoids closed under takingsubmonoids,quotient monoids andfinite direct products.
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Part I
Kernels and extensions
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The kernel of a group morphism
Letπ:H→Gbe asurjectivegroup morphism.
Thekernelofπis the group
T = Ker(π) =π−1(1) andHis anextensionofGbyT.
Thesynthesis problemin finite group theory consists in constructingHgivenGandT.
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The kernel of a group morphism
Letπ:H→Gbe asurjectivegroup morphism.
Thekernelofπis the group
T = Ker(π) =π−1(1) andH is anextensionofGbyT.
Thesynthesis problemin finite group theory consists in constructingHgivenGandT.
◮Is there a similar theory forsemigroups?
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A specific example
A monoidM is anextension of a group by a semilatticeif there a surjective morphismπfromM onto a groupGsuch thatπ−1(1)is asemilattice.
• How tocharacterizethe extensions of a group by a semilattice?
• Is there asynthesis theoremin this case?
• In the finite case, what is thevarietygenerated by the extensions of a group by a semilattice?
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The difference between semigroups and groups
Letπ:H→Gbe a surjectivegroupmorphism and letK=π−1(1). Thenπ(h1) =π(h2) iff
h1h−12 ∈K.
Ifπ:M→Gbe a surjectivemonoidmorphism and K=π−1(1), there is in general no way to decide whetherπ(m1) =π(m2), givenK.
For this reason, the notion of akernel of a monoid morphismhas to be stronger. . .
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The kernel category of a morphism
LetGbe a group and letπ:M→Gbe a surjective morphism. Thekernel categoryKer(π) ofπhasGas its object set and for allg, h∈G Mor(u, v)={(u, m, v)∈G×M×G|uπ(m) =v}
Note thatMor(u, u)is a monoid equal toπ−1(1) and thatGacts naturally (on the left) onKer(π):
u v
gu gv
m
gm
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A first necessary condition Proposition
Letπbe a surjective morphism from a monoidM onto a groupGsuch thatπ−1(1)is asemilattice.
Thenπ−1(1) =E(M)and theidempotentsofM commute.
Proof. Asπ−1(1)is a semilattice,π−1(1)⊆E(M).
Ifeis idempotent, thenπ(e)is idempotent and therefore is equal to1. ThusE(M)⊆π−1(1).
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A second necessary condition
LetMbe a monoid withcommuting idempotents.
• It isE-unitaryif for alle, f∈E(M)and x∈M, one of the conditionsex=f or xe=f implies thatxisidempotent.
• It isE-denseif, for each x∈M, there are elementsx1andx2inM such thatx1xand xx2are idempotent.
Note that any finite monoid isE-dense, since every element has anidempotent power. But(N,+)is notE-densesince its unique idempotent is0.
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A second necessary condition (2) Proposition
Letπbe a surjective morphism from a monoidM onto a groupGsuch thatπ−1(1) =E(M). Then M isE-unitary dense.
Proof. Ifex=f thenπ(e)π(x) =π(f), that is π(x) = 1. Thusx∈E(M)andM isE-unitary.
Letx∈M and letg=π(x). Letx¯be such that π(¯x) =g−1. Thenπ(¯xx) = 1 =π(x¯x). Therefore
¯
xxandx¯xareidempotent. ThusMisE–dense.
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The fundamental group π
1(M )
LetF(M)be thefree groupwith basisM. Then there is a natural injectionm→(m)fromMinto F(M). Thefundamental groupπ1(M)ofMis the group with presentation
hM|(m)(n)=(mn)for allm, n∈Mi Fact. IfM is anE-densemonoid withcommuting idempotents, thenπ1(M)is the quotient ofMby the congruence∼defined byu∼viff there exists an idempotentesuch thateu=ev.
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Characterization of extensions of groups Theorem (Margolis-Pin, J. Algebra 1987)
LetM be a monoid whoseidempotents form a subsemigroup. TFCAE:
(1) there is a surjective morphismπ:M→G onto agroupGsuch thatπ−1(1) =E(M), (2) the surjective morphismπ:M→Π1(M)
satisfiesπ−1(1) =E(M), (3) M isE-unitary dense.
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Part II
The synthesis theorem
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Categories
Notation: uandvareobjects,x,y,p,q,p+x, p+x+yaremorphisms,p,q,x+y,y+xare loops.
u v
x y
p q
For each objectu, there is a loop0ubased onu such that, for every morphismxfromutov, 0u+x=xandx+ 0v=x.
Thelocal monoidatuis the monoid formed by the loopsbased onu.
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Groups acting on a category (1)
Anactionof a groupGon a categoryC is given by a group morphism fromGinto the automorphism group ofC. We writegxfor the result of the action ofg∈Gon anobjectormorphismx. Note that for allg∈Gandp, q∈C:
• g(p+q) =gp+gq,
• g0u= 0gu.
The groupGactsfreelyonCifgx=ximplies g= 1. It actstransitivelyif the orbit of any object ofC underGisObj(C).
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The monoid C
uLetGbe a group actingfreelyandtransitivelyon a categoryC. Letube an object ofC and let
Cu={(p, g)|g∈G, p∈Mor(u, gu)}
ThenCuis amonoidunder the multiplication defined by(p, g)(q, h) = (p+gq, gh).
u gu
hu ghu
p
q p+gq gq
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A property of he monoid C
uProposition
LetGbe a group actingfreelyandtransitivelyon a category. Then for each objectu, the monoidCuis isomorphic toC/G.
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The synthesis theorem
Theorem (Margolis-Pin, J. Algebra 1987)
LetM be a monoid. The following conditions are equivalent:
(1) M is anextension of a group by a semilattice, (2) M isE-unitary densewithcommuting
idempotents,
(3) M is isomorphic toC/G, whereGis a group actingfreelyandtransitivelyon a connected, idempotent and commutativecategory.
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The covering theorem
LetMandN be monoids withcommuting idempotents. A cover is a surjective morphism γ:M→N which induces anisomorphismfrom E(M)toE(N).
Theorem (Fountain, 1990)
EveryE-densemonoid with commuting idempotents has anE-unitary dense cover with commuting idempotents.
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Part III The finite case
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Closure properties Proposition
The class ofextensions of groups by semilatticesis closed under takingsubmonoidsanddirect product.
Proof. Letπbe a surjective morphism from a monoidMonto a groupGsuch thatπ−1(1)is a semilattice. IfN be a submonoid ofM, thenπ(N) is asubmonoidofGand hence isgroupH. ThusN is an extension ofH andπ−1(1)∩N is a
semilattice.
Direct products: easy.
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Variety generated by finite extensions
LetVbe the variety generated byextensionsof groupsbysemilattices.
A monoid belongs to Viff it is aquotientof an extensionof a group by a semilattice.
M
N G
π−1(1) =E(M) γ π
The monoidN belongs toV.
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Variety generated by finite extensions
LetVbe the variety generated byextensionsof groupsbysemilattices.
A monoid belongs toViff it is aquotientof an extensionof a group by a semilattice.
M
N G
π−1(1) =E(M) γ π
The monoidN belongs toV.
◮This diagram is typical of arelational morphism.
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Relational morphisms
LetM andN be monoids. Arelational morphism fromM toN is a mapτ :M→ P(N)such that:
(1) τ(s)isnonemptyfor alls∈M, (2) τ(s)τ(t)⊆τ(st)for alls, t∈M, (3) 1∈τ(1).
Examples of relational morphisms include:
•Morphisms
•Inverses of surjective morphisms
•The composition of two relational morphisms
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Graph of a relational morphism
The graphRofτ is a submonoid ofM×N. Let α:R→Mandβ:R→N be theprojections.
Thenαissurjectiveandτ =β◦α−1.
M N
R⊆M×N
α β
τ
α(m, n)=m τ(m)=β(α−1(m)) β(m, n)=n τ−1(n)=α(β−1(n))
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An example of relational morphism
LetQbe a finite set. LetS(Q)thesymmetric group onQand letI(Q)be the monoid of allinjective partial functionsfromQtoQunder composition.
Letτ :I(Q)→S(Q)be the relational morphism defined byτ(f)={Bijections extendingf}
1 2 3 4
f 3 - 2 -
h1 3 1 2 4 h2 3 4 2 1
τ(f) ={h1, h2}
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Relational morphisms Proposition
Letτ:M→N be a relational morphism. IfT is a subsemigroupofN, then
τ−1(T)={x∈M|τ(x)∩T 6=∅}
is a subsemigroup ofM.
In our example,τ−1(1)is asemilatticesince τ−1(1)={f∈I(Q)|theidentityextendsf}
={subidentitiesonQ} ≡(P(Q),∩)
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Finite extensions and relational morphisms
A monoid belongs to Viff it is aquotientof an extension of a group by a semilattice.
M
N G
π−1(1)semilattice γ(π−1(1))semilattice γ π
τ
Proposition
A monoidN belongs toViff there is arelational morphismτ fromN onto agroupGsuch that τ−1(1)is asemilattice.
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Finite extensions and relational morphisms (2)
Consider the canonical factorization ofτ:
N G
R⊆N×G
α β
τ
Thenαinduces aisomorphismfromβ−1(1)onto τ−1(1)since
β−1(1)={(n,1)∈R|1∈τ(n)}
τ−1(1)={n∈N |1∈τ(n)}
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A non effective characterization
Theorem (Margolis-Pin, J. Algebra 1987)
LetN be a finite monoid. TFCAE (1) N belongs toV,
(2) N is aquotientof anextension of a group by a semilattice,
(3) N iscoveredby anextension of a group by a semilattice,
(4) there is arelational morphismτ fromN onto agroupGsuch thatτ−1(1)is asemilattice.
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The finite covering theorem Theorem (Ash, 1987)
Everyfinitemonoid with commuting idempotents has afiniteE-unitary coverwith commuting idempotents.
Corollary
The varietyVis the variety of finite monoids with commuting idempotents.
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Part IV Group radical
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Group radical of a monoid
LetMbe a finite monoid. Thegroup radicalofM is the set
K(M)= \
τ:M→G
τ−1(1) where the intersection runs over the set of all relational morphisms fromM into afinite group.
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Universal relational morphisms Proposition
For each finite monoidM, there exists a finite groupGand a relational morphismτ:M→G such that K(M) =τ−1(1).
Proof. There are only finitely many subsets ofM.
ThereforeK(M)=τ1−1(1)∩ · · · ∩τn−1(1)where τ1:M→G1, . . . , τn:M→Gn. Let
τ :M→G1× · · · ×Gnbe the relational
morphism defined byτ(m) =τ1(m)× · · · ×τn(m).
Thenτ−1(1) =K(M).
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Another characterization of V Theorem
LetMbe a finite monoid. TFCAE:
(1) M belongs toV, (2) K(M) is asemilattice,
(3) TheidempotentsofM commuteand K(M) =E(M).
◮Is there an algorithm to computeK(M)?
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Ash’s small theorem Theorem (Ash 1987)
IfM is a finite monoid withcommuting idempotents, thenK(M) =E(M).
Corollary
The varietyVis the variety of finite monoids with commuting idempotents.
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Ash’s big theorem
Denote byD(M)the least submonoidT ofM closed underweak conjugation: ift∈T and a¯aa=a, thenat¯a∈T and¯ata∈T.
Theorem (Ash 1991)
For each finite monoidM, one hasK(M) =D(M).
Corollary
One can effectively computeK(M).
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Part V
The topological approach
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The pro-group topology
Thepro-group topologyonA∗[onF G(A)] is the least topology such that every morphism fromA∗ on afinite(discrete) group is continuous.
Proposition
LetLbe a subset ofA∗andu∈A∗. Thenu∈L iff, for every morphismβfromA∗onto a finite groupG,β(u)∈β(L).
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A topological characterization of K(M ) Theorem (Pin, J. Algebra 1991)
Letα:A∗→Mbe surjective morphism. Then m∈K(M)iff1∈α−1(m).
1∈α−1(m) ⇐⇒ for allβ:A∗→G,1∈β(α−1(m))
⇐⇒ for allτ :M→G,1∈τ(m)
⇐⇒ for allτ :M→G,m∈τ−1(1)
⇐⇒ m∈K(M)
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Finitely generated subgroups of the free group Theorem (M. Hall 1950)
Every finitely generated subgroup of the free group isclosed.
Theorem (Ribes-Zalesskii 1993)
LetH1, . . . , Hnbe finitely generated subgroups of the free group. ThenH1H2· · ·Hnisclosed.
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Computation of the closure of a set
Theorem (Pin-Reutenauer, 1991 (mod R.Z.))
There is a simple algorithm to compute theclosure of a givenrational subsetof the free group.
Theorem (Pin, J. Algebra 1991 (mod P.R.))
There is a simple algorithm to compute theclosure of a givenrational languageof the free monoid.
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Another proof of Ash’s big theorem
Theorem (Pin, Bull. Austr. Math. 1988)
Given the simple algorithm to compute theclosure of arational language, one hasK(M) =D(M).
Therefore, Ribes-Zalesskii’s theorem givesanother proofof Ash’s big theorem.
Theorem
Given adecidablevarietyV, the variety generated byV-extensions of groups isdecidable.