Hopfian property for semigroupsturnbull.mcs.st-and.ac.uk/~nik/talks/york13.pdfHopﬁan property for...

Hopfian property for semigroups

Nik [email protected]

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.


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.


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.


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.


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.


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.


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.)


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.


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.


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.


Hopf & subsemigroups: an exampleThe rest of the talk: joint work with Victor Malcev.



xy = yx = y

(x higher than y)

T



T hopfian (identity is the onlyonto endomorphism).

xy = yx = y

(x higher than y)

T




xy = yx = y

(x higher than y)

T

1




T 1 non-hopfian (b1 7→ 1, bi+1 7→ bi ).

xy = yx = y

(x higher than y)

T

1





xy = yx = y

(x higher than y)

T

1

S





T 1 ∪ S hopfian (again, only identity).

xy = yx = y

(x higher than y)

T

1

S





T 1 ∪ S hopfian (again, only identity).

RemarkNone finitely generated.

xy = yx = y

(x higher than y)

T

1

S


Cofinite subsemigroups & endomorphisms

TheoremS – finitely generated semigroup;T < S; |S \ T | <∞.For every endomorphism θ : S → S we haveTθ 6= S.

<∞

T

S


Cofinite subsemigroups and Hopf

TheoremS – f.g. semigroup; T ≤ S; |S \ T | <∞.If T is hopfian then S is hopfian as well.

<∞

T

S



Proof

<∞

T

S



Proof

◮ Suppose θ : S ։ S . <∞

T

S



Proof

◮ Suppose θ : S ։ S .

◮ Consider: {Tθi : i = 0, 1, 2, . . . };note |S \ Tθi | ≤ |S \ T |.

<∞

T

S



Proof



◮ F.g. ⇒ finitely many subsemigroups offixed finite complement ⇒ Tθi = Tθj forsome i 6= j .

<∞

T

S



Proof




◮ Tψ2 = Tψ for some ψ = φi .

<∞

T

S



Proof





◮ Check: T ⊆ Tψ ( calculation of sizes).

<∞

T

S



Proof






◮ Previous Theorem: T = Tψ.

<∞

T

S



Proof







◮ T hopfian: ψ|T bijective.

<∞

T

S



Proof







◮ T hopfian: ψ|T bijective.

◮ |S \ T | <∞: ψ bijective, hence φbijective.

<∞

T

S


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


Only one less . . .

TheoremThere exists a finitely generated hopfian semigroup S whichcontains a non-hopfian semigroup T with |S \ T | < 1.


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;. . . .


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


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


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


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


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


Putting it together


Putting it together

◮ F – f.g. free semigroup; F3


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


Putting it together




◮ 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


Putting it together




◮ 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


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].)


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.


Hopfian property for semigroupsturnbull.mcs.st-and.ac.uk/~nik/talks/york13.pdfHopﬁan property for...

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