Hopfian property for semigroupsturnbull.mcs.st-and.ac.uk/~nik/talks/york13.pdfHopfian property for...
Transcript of Hopfian property for semigroupsturnbull.mcs.st-and.ac.uk/~nik/talks/york13.pdfHopfian property for...
Hopfian property for semigroups
School of Mathematics and Statistics, University of St Andrews
York, 23 January 2013
Infinity bias
When an infinite set is not defined negatively as:
◮ X is infinite if it is not finite.
it is invariably defined so:
◮ X is infinite if there is a proper injection X → X .
and not as:
◮ X is infinite if there is a proper surjection X → X .
University of St Andrews Nik Ruskuc: Hopfian semigroups
Combinatorial algebra
◮ Foundational concepts:◮ generators;◮ defining relations;◮ decidability problems (word problem).
◮ Properties (finiteness conditions):◮ periodicity;◮ local finiteness;◮ residual finiteness;◮ hopfian property.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Typical questions
◮ Examples◮ classes of positive examples;◮ specific examples of negative examples;
◮ behaviour under constructions (e.g. direct products);
◮ substructures (subgroups of finite index);
◮ relationships to other properties.
University of St Andrews Nik Ruskuc: Hopfian semigroups
A case study: residual finiteness (1)
DefinitionA is residually finite if for any two distinct a, b ∈ A there exists afinite B and a homomorphism f : A → B such that f (a) 6= f (b).
Example
All of the following are residually finite:
◮ finite groups (finiteness condition);
◮ free groups;
◮ finitely generated abelian groups.
Example (Baumslag)
The group 〈a, b|a−1b2a = b3〉 is not residually finite. (b and a−1bacommute in every finite quotient.)
RemarkAlso, infinite simple groups are not residually finite.
University of St Andrews Nik Ruskuc: Hopfian semigroups
A case study: residual finiteness (2)
Proposition
G × H residually finite iff G & H residually finite.
[Aside: Is this a result about: (a) groups; (b) semigroups; or (c)general algebraic systems?]
Proposition
G – a group; H ≤ G; [G : H] <∞.G residually finite iff H residually finite.
Theorem (folklore)
A finitely presented residually finite group has a decidable wordproblem.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopfian property
Definition (Hopf ’31)
A is hopfian if every onto endomorphism A → A is actually anisomorphism.
DefinitionA is hopfian if A is not isomorphic to any of its proper quotients.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopfian (semi)groups
All of the following (semi)groups are hopfian:
◮ finite (finiteness condition);
◮ f.g. free semigroups;
◮ f.g. free groups (Nielsen);
◮ infinite simple groups;
◮ f.g. commutative (semi)groups.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Non-hopfian examples
Example
Infinite direct product P = A× A× . . . is not hopfian because of
P → P, (x1, x2, x3, . . . ) 7→ (x2, x3, . . . ).
QuestionIs every finitely generated group hopfian?
Example (Baumslag, Solitar ’62)
The group 〈a, b|a−1b2a = b3〉 is not hopfian. (a 7→ a, b 7→ b2.)
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopfian property and residual finiteness
Theorem (Malcev ’40)
A finitely generated residually finite group G is hopfian.
Proof
◮ Suppose θ : G ։ G .
◮ For n ∈ N let H1, . . . ,Hk be all subgroups of index n.
◮ Check: [G : Hiθ−1] = n.
◮ θ−1 permutes H1, . . . ,Hk .
◮ ker θ ≤ Hi for all i . And all n.
◮ r.f. ⇒ ker θ = 1.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subgroups
Example (Baumslag, Solitar ’62)
The group 〈a, b|a−1b12a = b18〉 is hopfian, but its subgroup 〈a, b6〉of index 6 is isomorphic to 〈a, b1|a
−1b21a = b31〉 and is non-hopfian.
Theorem (Hirshon ’69)
G – a f.g. group, H ≤ G, [G : H] <∞.If H is hopfian then G is hopfian too.
RemarkNot known whether f.g. assumption can be removed.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups (Rees index)
TheoremS – semigroup; T ≤ S, |S \ T | <∞.Then S satisfies property P iff T satisfies P, where P is any of thefollowing:
◮ finite generation [Jura ’78];
◮ finite presentability [NR ’98];
◮ decidable word problem, periodicity, local finiteness [NR ’98];
◮ residual finiteness [NR, Thomas ’98];
◮ automaticity [Hoffmann, NR, Thomas ’02];
◮ finite complete rewriting system [Wang ’98;Wong, Wong ’11].
RemarkThe proof is always different from the group analogues, and oftenharder.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.
xy = yx = y
(x higher than y)
T
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.
T hopfian (identity is the onlyonto endomorphism).
xy = yx = y
(x higher than y)
T
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.
T hopfian (identity is the onlyonto endomorphism).
xy = yx = y
(x higher than y)
T
1
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.
T hopfian (identity is the onlyonto endomorphism).
T 1 non-hopfian (b1 7→ 1, bi+1 7→ bi ).
xy = yx = y
(x higher than y)
T
1
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.
T hopfian (identity is the onlyonto endomorphism).
T 1 non-hopfian (b1 7→ 1, bi+1 7→ bi ).
xy = yx = y
(x higher than y)
T
1
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.
T hopfian (identity is the onlyonto endomorphism).
T 1 non-hopfian (b1 7→ 1, bi+1 7→ bi ).
T 1 ∪ S hopfian (again, only identity).
xy = yx = y
(x higher than y)
T
1
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.
T hopfian (identity is the onlyonto endomorphism).
T 1 non-hopfian (b1 7→ 1, bi+1 7→ bi ).
T 1 ∪ S hopfian (again, only identity).
RemarkNone finitely generated.
xy = yx = y
(x higher than y)
T
1
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups & endomorphisms
TheoremS – finitely generated semigroup;T < S; |S \ T | <∞.For every endomorphism θ : S → S we haveTθ 6= S.
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
TheoremS – f.g. semigroup; T ≤ S; |S \ T | <∞.If T is hopfian then S is hopfian as well.
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
◮ Suppose θ : S ։ S . <∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
◮ Suppose θ : S ։ S .
◮ Consider: {Tθi : i = 0, 1, 2, . . . };note |S \ Tθi | ≤ |S \ T |.
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
◮ Suppose θ : S ։ S .
◮ Consider: {Tθi : i = 0, 1, 2, . . . };note |S \ Tθi | ≤ |S \ T |.
◮ F.g. ⇒ finitely many subsemigroups offixed finite complement ⇒ Tθi = Tθj forsome i 6= j .
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
◮ Suppose θ : S ։ S .
◮ Consider: {Tθi : i = 0, 1, 2, . . . };note |S \ Tθi | ≤ |S \ T |.
◮ F.g. ⇒ finitely many subsemigroups offixed finite complement ⇒ Tθi = Tθj forsome i 6= j .
◮ Tψ2 = Tψ for some ψ = φi .
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
◮ Suppose θ : S ։ S .
◮ Consider: {Tθi : i = 0, 1, 2, . . . };note |S \ Tθi | ≤ |S \ T |.
◮ F.g. ⇒ finitely many subsemigroups offixed finite complement ⇒ Tθi = Tθj forsome i 6= j .
◮ Tψ2 = Tψ for some ψ = φi .
◮ Check: T ⊆ Tψ ( calculation of sizes).
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
◮ Suppose θ : S ։ S .
◮ Consider: {Tθi : i = 0, 1, 2, . . . };note |S \ Tθi | ≤ |S \ T |.
◮ F.g. ⇒ finitely many subsemigroups offixed finite complement ⇒ Tθi = Tθj forsome i 6= j .
◮ Tψ2 = Tψ for some ψ = φi .
◮ Check: T ⊆ Tψ ( calculation of sizes).
◮ Previous Theorem: T = Tψ.
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
◮ Suppose θ : S ։ S .
◮ Consider: {Tθi : i = 0, 1, 2, . . . };note |S \ Tθi | ≤ |S \ T |.
◮ F.g. ⇒ finitely many subsemigroups offixed finite complement ⇒ Tθi = Tθj forsome i 6= j .
◮ Tψ2 = Tψ for some ψ = φi .
◮ Check: T ⊆ Tψ ( calculation of sizes).
◮ Previous Theorem: T = Tψ.
◮ T hopfian: ψ|T bijective.
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Cofinite subsemigroups and Hopf
Proof
◮ Suppose θ : S ։ S .
◮ Consider: {Tθi : i = 0, 1, 2, . . . };note |S \ Tθi | ≤ |S \ T |.
◮ F.g. ⇒ finitely many subsemigroups offixed finite complement ⇒ Tθi = Tθj forsome i 6= j .
◮ Tψ2 = Tψ for some ψ = φi .
◮ Check: T ⊆ Tψ ( calculation of sizes).
◮ Previous Theorem: T = Tψ.
◮ T hopfian: ψ|T bijective.
◮ |S \ T | <∞: ψ bijective, hence φbijective.
<∞
T
S
University of St Andrews Nik Ruskuc: Hopfian semigroups
Only one more . . .
Maitre D: And finally, monsieur, a wafer-thin mint.
Mr Creosote: No.
Maitre D: Oh sir! It’s only a tiny little thin one.
Monty Python’s Meaning of Life
University of St Andrews Nik Ruskuc: Hopfian semigroups
Only one less . . .
TheoremThere exists a finitely generated hopfian semigroup S whichcontains a non-hopfian semigroup T with |S \ T | < 1.
University of St Andrews Nik Ruskuc: Hopfian semigroups
Semigroup actions
X – set; S – semigroup;
X × S → X , (x , s) 7→ x · s;
(x · s) · t = x · (st) (x ∈ X , s, t ∈ S).
RemarkAction = homomorphism into the full transformation semigroup =representation by transformations.
RemarkAlgebraic structures in their own right; hence: generators foractions; homomorphisms of actions; hopfian actions;. . . .
University of St Andrews Nik Ruskuc: Hopfian semigroups
A non-hopfian action of F3
Proposition
The rank 3 free semigroup F3 admits a cyclic non-hopfian action.(xi 7→ xi−1, yi 7→ yi−1, z1 7→ y0, zi+1 7→ zi )
x−2 x
−1 x0 x1 x2
y−2 y
−1 y0 y1 y2
z1 z2
0
University of St Andrews Nik Ruskuc: Hopfian semigroups
Extending an act
Proposition
F – free semigroup; X – cyclic F -act.There exists a hopfian F -act extension Y of X with |Y \ X | = 1.
x−2 x
−1 x0 x1 x2
y−2 y
−1 y0 y1 y2
z1 z2
0X
University of St Andrews Nik Ruskuc: Hopfian semigroups
Extending an act
Proposition
F – free semigroup; X – cyclic F -act.There exists a hopfian F -act extension Y of X with |Y \ X | = 1.
y0
x−2 x
−1 x0 x1 x2
y−2 y
−1 y0 y1 y2
z1 z2
0X
University of St Andrews Nik Ruskuc: Hopfian semigroups
Extending an act
Proposition
F – free semigroup; X – cyclic F -act.There exists a hopfian F -act extension Y of X with |Y \ X | = 1.
y0
x−2 x
−1 x0 x1 x2
y−2 y
−1 y0 y1 y2
z1 z2
0X
University of St Andrews Nik Ruskuc: Hopfian semigroups
A construction
S – semigroup; X – an S-act.S [X ] := S ·∪X with multiplication:
◮ s ∗ t = st (s, t ∈ S);
◮ s ∗ x = x (s ∈ S , x ∈ X );
◮ x ∗ s = x · s (s ∈ S , x ∈ X );
◮ y ∗ x = x (x , y ∈ X ).
S
s ∗ t = st
Xs ∗ x = y ∗ x = x
x ∗ s = x · s
University of St Andrews Nik Ruskuc: Hopfian semigroups
A construction
S – semigroup; X – an S-act.S [X ] := S ·∪X with multiplication:
◮ s ∗ t = st (s, t ∈ S);
◮ s ∗ x = x (s ∈ S , x ∈ X );
◮ x ∗ s = x · s (s ∈ S , x ∈ X );
◮ y ∗ x = x (x , y ∈ X ).
LemmaF – free semigroup; X – an F -act.The semigroup F [X ] is hopfian iff X is ahopfian F -act.
S
s ∗ t = st
Xs ∗ x = y ∗ x = x
x ∗ s = x · s
University of St Andrews Nik Ruskuc: Hopfian semigroups
Putting it together
University of St Andrews Nik Ruskuc: Hopfian semigroups
Putting it together
◮ F – f.g. free semigroup; F3
University of St Andrews Nik Ruskuc: Hopfian semigroups
Putting it together
◮ F – f.g. free semigroup;
◮ X – a cyclic non-hopfian F -act;
◮ F [X ] is non-hopfian;
F3
x−2 x
−1 x0 x1 x2
y−2 y
−1 y0 y1 y2
z1 z2
0X
University of St Andrews Nik Ruskuc: Hopfian semigroups
Putting it together
◮ F – f.g. free semigroup;
◮ X – a cyclic non-hopfian F -act;
◮ F [X ] is non-hopfian;
◮ Y := X ∪ {y0} – a hopfianextension;
◮ F [Y ] is hopfian;
F3
y0
x−2 x
−1 x0 x1 x2
y−2 y
−1 y0 y1 y2
z1 z2
0X
Y
University of St Andrews Nik Ruskuc: Hopfian semigroups
Putting it together
◮ F – f.g. free semigroup;
◮ X – a cyclic non-hopfian F -act;
◮ F [X ] is non-hopfian;
◮ Y := X ∪ {y0} – a hopfianextension;
◮ F [Y ] is hopfian;
◮ |F [Y ] \ F [X ]| = |Y \ X | = 1.
F3
y0
x−2 x
−1 x0 x1 x2
y−2 y
−1 y0 y1 y2
z1 z2
0X
Y
University of St Andrews Nik Ruskuc: Hopfian semigroups
Some questions
QuestionDoes there exist a finitely presented hopfian semigroup S whichcontains a cofinite non-hopfian subsemigroup? (A sharperexample.)
Green index – a common generalisation of group index and finitecomplement [Gray, NR ’08].
QuestionIs it true that if a finitely generated semigroup S has a hopfiansubsemigroup T of finite Green index then S itself must behopfian? (Combination of [Hirshon 69] and [VM&NR].)
QuestionIf S is a hopfian semigroup and T a finite commutative semigroup,is S × T necessarily hopfian? (Yes for groups [Hirshon ’69].)
University of St Andrews Nik Ruskuc: Hopfian semigroups
Thank you
- H. Hopf, Beitrage zur Klassifizierung der Flachenabbildungen, J.Reine Angew. Math. 165 (1931), 225–236.
- A.I. Malcev, On isomorphic matrix representations of infinitegroups, Mat. Sb. 8 (1940), 405–422.
- G. Baumslag, D. Solitar, Some two-generator one-relatornon-hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199–201.
- R. Hirshon, Some theorems on Hopficity, Trans. Amer. Math.Soc. 141 (1969), 229–244.
- W. Magnus, Residually finite groups, Bull. Amer. Math. Soc. 75(1969), 305–316.
- R. Gray, N. Ruskuc, Green index and finiteness conditions forsemigroups, J. Algebra 320 (2008), 3145–3164.
- V. Malcev, N. Ruskuc, On hopfian cofinite subsemigroups,submitted.
University of St Andrews Nik Ruskuc: Hopfian semigroups