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Hopfian property for semigroups
Nik Ruskucnik@mcs.st-and.ac.uk
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
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