combinatorial methods in study of structure of inverse...

Combinatorial Methods in Study ofStructure of Inverse Semigroups

Tatiana Jajcayova

Comenius UniversityBratislava, Slovakia

Graphs, Semigroups, and Semigroup Acts 2017, BerlinOctober 12, 2017

Tanya Jajcay Comenius University Combinatorial Methods in Inverse Semigroups

Presentations

A presentation for an algebraic structure is ”contracted”information about the structure allowing for a complete recovery ofits original multiplication table.

Formally, a presentation is a pairA = 〈X |R〉

with the relations R being equations between expressions formed ofthe generators from X .

For example,

V4 = 〈a, b|a2 = b2 = e, ab = ba〉


Presentations

A presentation for an algebraic structure is ”contracted”information about the structure allowing for a complete recovery ofits original multiplication table.

Formally, a presentation is a pairA = 〈X |R〉

with the relations R being equations between expressions formed ofthe generators from X .

For example,

V4 = Gp〈a, b|a2 = b2 = e, ab = ba〉

A = Gp〈X |R〉A = Inv〈X |R〉A = InvM〈X |R〉


Starting from a presentation

A = 〈X |R〉

Starting from a presentation, there are many interesting

I decision problems to study (”word problem”, ”isomorphismproblem”, . . .),

I and for decidable such problems it is interesting to investigatetheir complexity.

Tools:graphs, automata related to the presentation,

actions on these graphs


Starting from a presentation

A = 〈X |R〉

Starting from a presentation, there are many interesting

I decision problems to study (”word problem”, ”isomorphismproblem”, . . .),

I and for decidable such problems it is interesting to investigatetheir complexity.

Tools:graphs, automata related to the presentation,actions on these graphs


Inverse semigroups

Inverse Semigroup

I associative binary operation

I existence of generalized inverses:

a · a−1 · a = a

a−1 · a · a−1 = a−1


Inverse semigroups

Group -

I associative binary operation permutations, symmetries,

I identity bijections,...

I inverses: a · a−1 = a−1 · a = e

Semigroup - associative operation concatenation of strings

Inverse Semigroup

I associative binary operation strings, paths in graphs,

I existence of inverses: transition semigroups,

a · a−1 · a = a partial transformations,

a−1 · a · a−1 = a−1 do/undo proceses

Inverse Monoid is an inverse semigroup with the identity.


Inverse semigroups

S = Inv〈X |R〉 ifS = (X ∪ X−1)+/τ

where τ is the smallest congruence containing the relation R andVagner’s relations:

{(uu−1u, u)|u ∈ S} ∪ {(uu−1vv−1, vv−1uu−1)|u, v ∈ S}.


Why inverse semigroups?

While groups can be represented as symmetries:

Theorem (Cayley)

Every group can be embedded in the set of one to onetransformations on a set.

Inverse semigroups can be represented as partial symmetries:

Theorem (Vagner-Preston)

Every inverse semigroup can be embedded in the set of partial oneto one transformations on a set.


Schutzenberger (Cayley) graph

Let S = Inv〈X |R〉

Definition (Schutzenberger graph)

Let w be a word in (X ∪ X−1)+. The Schutzenberger graph of wrelative to the presentation Inv〈X |R〉 is the graph SΓ(X ,R,wτ)whose vertices are the elements of the R-class Rwτ of wτ in S ,and whose edges are of the form

{(v1, x , v2) | v1, v2 ∈ Rwτ and v1(x τ) = v2}.

sv1

sv2

-xτ

if v2 = v1 · xτ .

Think strongly connected components of Cayley graphs.


Schutzenberger graphs

Schutzenberger graphs

I generalization of Munn trees

I tool to approach algorithmic and structural problems ininverse semigroups

I connected components of the Cayley graph of H containingwτ

I deterministic inverse word graphs


Structure of inv. semigroups: Maximal subgroups

One of the most basic structural question concerning inversesemigroups is the classification of the maximal subgroups of agiven semigroup S .


Maximal Subgroups:

An element a is called an idempotent if it satisfies the propertya2 = a.

For example,if S is a group, it has only one idempotent - the identity.

In general, inverse semigroups can have many idempotents.


Schutzenberger graphs - Stephen’s theorem

Schutzenberger graph SΓ(X ,R, e) has many nice properties...[Stephen, 1994]

I one especially useful for the study of structure:

Ge∼= Aut(SΓ(X ,R, e))


Schutzenberger graphs of inverse semigroups

Applying Stephen’s theorem assumes that we already know theSchutzenberger graphs for the given words and inverse semigroup.

BUTin general, we do not know any effective procedure for constructingthe Schutzenberger graphs.


Stephen’s iterative procedure.

Elementary expansion:- sewing on a relation r = s

s sv1 v2-r

s

Elementary determination:-edge folding

sx

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HHHjx

HHHHHH

s

s−→fold

6s

x

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s sv1 v2-r

s


sx

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HHHjx

HHHHHH

s

s

−→fold6

sx

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s sv1 v2-r

s


sx

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��

HHHjx

HHHHHH

s

s−→fold

6

sx

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s sv1 v2-r

s


sx

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HHHjx

HHHHHH

s

s−→fold

6s

x

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Schutzenberger graphs - iterative procedure

In this way we get a directed system of inverse graphs

Γ1 → Γ2 → . . .→ Γi → . . .

whose directed limit is the Schutzenberger graph SΓ(X ,R,w).

In general, this is:

· infinite

· complicated

· not transparent what is the best way to get to the limit

· never ending

·...

Goal: Introduce some order for some classes of inverse semigroups


HNN-extensions

HigmanNeumannNeumann - extensions

t−1at = aφ for ∀a ∈ A


HNN-extensions - groups

For example, the fundamental group of a surface with a handle isan HNN-extension of the fundamental group of the surface withoutthe handle attached.


Definition of HNN-extensions for inverse semigroups

Definition (A.Yamamura)

Let S = Inv〈X | R〉 be an inverse semigroup.Let A,B be inverse subsemigroups of S ,ϕ : A −→ B be an isomorphism

Then

S∗ = Inv〈X , t | R, t−1at = aϕ,

t−1t = f , tt−1 = e, ∀a ∈ A〉

is called the HNN-extension of S associated with ϕ.

e ∈ A ⊆ eSe and f ∈ B ⊆ fSf (or e /∈ A ⊆ eSe and f /∈ B ⊆ fSffor some e, f ∈ E (S)).

:-)


Definition of HNN-extensions for inverse semigroups



Then

S∗ = Inv〈X , t | R, t−1at = aϕ, t−1t = f , tt−1 = e, ∀a ∈ A〉



:-)


Definition of HNN for inverse semigroups



ThenS∗ = Inv〈X , t | R ∪ RHNN〉



S ↪→ S∗


Amalgams

If S1 = Inv〈X1|R1〉, S2 = Inv〈X2|R2〉 with X1 ∩ X2 = ∅

S1 ∗U S2 = Inv〈X |R1,R2,Rw 〉 = Inv〈X |R〉

where X = X1 ∪ X2, Rw = {(ω1(u), ω2(u)) : u ∈ U}


Structure: HNN-extensions - Tools

S = Inv〈X , t|R ∪ RHNN〉

a part of the word graph over X ∪{t} may look something like this:


In the special case when S = Inv〈X , t|R ∪ RHNN〉, a part of theword graph over X ∪ {t} may look something like this:


⇓


The tree structure of lobe graphs

Theorem (T.Jajcayova)

The lobe graph T (Γ) of a Schutzenberger graph Γ relative to thepresentation Inv〈X , t |R ∪ RHNN〉 is an oriented tree.


The tree structure of lobe graphs


Bass-Serre: Structure Theorem

Recall:Ge∼= Aut(SΓ(X ,R, e))

Theorem (Structure theorem of Bass-Serre theory)

Let G be a group acting without inversions on a connected graphX . Then X is a tree if and only if Φ : π(G,G \ X ) −→ G is anisomorphism.

Thus, the Structure theorem of Bass-Serre theory allows one tofind a presentation for a group G provided that an action of G ona tree is known.


Bass-Serre: Group acting on a graph

X = (Vert(X ),Edge(X ), α, ω)

The action of G on X preserves the incidence structure of X :

α(g · y) = g · α(y)ω(g · y) = g · ω(y)g · y = g · y

If the action satisfies the additional condition

y 6= g · y ,

for all y ∈ Edge(X ) and all g ∈ G , we say it is without inversions.


Bass-Serre: Quotient graphs

The quotient graph of the action of G on X is the graph

G \ X = (Vert(G \ X ),Edge(G \ X ))

Vert(G \ X ) - the set of orbits of G of the vertices of X ,Edge(G \ X ) - the set of orbits of G of the edges of X ,

with the incidence relation:α(G · y) = G · α(y)ω(G · y) = G · ω(y)G · y = G · y

Theorem (Serre, 1980)

Let G be a group acting on a connected tree X without inversions.Then every subtree T of G \ X can be a lifted to a subtree of X .


Bass-Serre: Graph of groups

LetG : v 7→ Gv

G : y 7→ Gy such that Gy = Gy .

together with group monomorphismσy : Gy −→ Gα(y) andτy : Gy −→ Gω(y)

satisfying the relation σy = τy .

Then the graph X together with the group assignment G is calleda graph of groups (G(−),X ).


Bass-Serre: Fundamental group of a graph of groups

Let

(G(−),X ) be a graph of groups,T be any maximal subtree of the underlying graph X

The fundamental group π(G(−),X ,T )

is generated:

by the disjoint union of vertex groups Gv

and by the edges of X ,

subject to the relations:

{y = y−1, y−1σy (a)y = τy (a) for all y ∈ Edge(X ) and a ∈ Gy}

∪{y = 1 for all y ∈ T}.


Bass-Serre: Fundamental group: Examples

Example 1.Let X be any graph, and let Gv = Gy =<1> for all vertices andedges of X (the monomorphisms σy and τy being obviously thetrivial mappings). The fundamental group of (G(−),X ) is the freegroup generated by the positive edges of X not in a maximal tree.

Example 2.Take X to be a tree and let Gy = 1 for all y ∈ Edge(X ). Then themaximal tree T is X itself. The fundamental group π(G(−),X ) isthe free product of the groups Gv (v ∈ Vert(X )).

Example 3.Let X be a segment, that is, X consists of one edge y and twovertices v1 and v2. Then the fundamental group π(G(−),X ) is thefree product of Gv1 and Gv2 amalgamating Gy .


Bass-Serre: Graphs of groups from action

Let G be a group acting on a connected non-empty graph X ,and let Y = G \ X be the quotient graph of X under the action ofG .

We construct a graph of groups (G,Y ).

The general idea:

assign to the vertex G · v the stabilizer group Gv of the vertex v ofX under the action of G .Similarly, to the edge G · y assign the stabilizer group Gy .

Problem:we need to pick a specific stabilizer allowing us to defineembedding monomorphisms.


Bass-Serre: Structure Theorem

Theorem (Structure theorem of Bass-Serre theory)

Let G be a group acting without inversions on a connected graphX . Then X is a tree if and only if Φ : π(G,G \ X ) −→ G is anisomorphism.

Thus, the Structure theorem of Bass-Serre theory allows one tofind a presentation for a group G provided that an action of G ona tree is known.


Characterization of the Schutzenberger graphs


Let S∗ be a lower bounded HNN-extension. The Schutzenbergergraphs of S∗ relative to the presentation Inv〈X ∪ {t}|R ∪ RHNN〉are precisely the complete T -graphs that possess a host.

· Schutzenberger graphs contain a special subgraph with onlyfinitely many lobes that contains the information for thewhole graph.

· Schutzenberger graphs of HNN-extensions have tree like lobestructure


Characterization of maximal subgroups of l.b. HNN’s


Let S∗ be a lower bounded HNN-extension, and let e be anidempotent of S . Then the maximal subgroup of S∗ containing eis isomorphic to the fundamental group of the graph of groups(H(−),Ze).


Let S∗ be a lower bounded HNN-extension. Let e be anidempotent of S∗ that is not D-related to any element of S . Thenthe maximal subgroup of S∗ containing e is isomorphic to asubgroup H of S whose quotient H/ ∼A and H/ ∼B is finite.


Maximal subgroups of amalgams of finite inv. semigroups

Theorem (A.Cherubini, T.Jajcayova, E. Rodaro)

Let e ∈ E (S1 ∗U S2), S1, S2 finite inverse semigroups, and supposethat e is not DS1∗US2-related to any idempotent of S1 or S2. ThenHS1∗US2e is a homomorphic image of a subgroup of the maximal

subgroup HSkg , for some g ∈ E (Sk) and some k ∈ {1, 2}.


Let e ∈ E (S1 ∗U S2), S1,S2 finite inverse semigroups, then there isan algorithm to compute a presentation for HS1∗US2

e .


Maximal subgroups of amalgams of finite inv. semigroups


Let e ∈ E (S1 ∗U S2), S1, S2 finite inverse semigroups, and supposethat e is not DS1∗US2-related to any idempotent of S1 or S2. ThenHS1∗US2e is a homomorphic image of a subgroup of the maximal

subgroup HSkg , for some g ∈ E (Sk) and some k ∈ {1, 2}.


Let e ∈ E (S1 ∗U S2), S1, S2 finite inverse semigroups, then there isan algorithm to compute a presentation for HS1∗US2

e .


Maximal subgroups

A. Cherubini, T. B. Jajcayova and E. Rodaro, Maximal subgroups of amalgams offinite inverse semigroups, Semigroup Forum 90, No. 2 (2015), 401–424.

T. B. Jajcayova, Schutzenberger Automata for HNN-extensions of Inverse Monoids,

submitted to Information and Computation.


Future work

I apply existing tools to study maximal subgroups of otherclasses of inverse semigroups

I apply existing tools to study different (structural/algorithmic)questions

I adjust tools for other varieties of algebras


Thank you!


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