Combinatorial Methods in Study of Structure of Inverse Semigroups

Transcription

1 Combinatorial Methods in Study of Structure of Inverse Semigroups Tatiana Jajcayová Comenius University Bratislava, Slovakia Graphs, Semigroups, and Semigroup Acts 2017, Berlin October 12, 2017

2 Presentations A presentation for an algebraic structure is contracted information about the structure allowing for a complete recovery of its original multiplication table.

3 Presentations A presentation for an algebraic structure is contracted information about the structure allowing for a complete recovery of its original multiplication table. Formally, a presentation is a pair A = X R with the relations R being equations between expressions formed of the generators from X.

4 Presentations A presentation for an algebraic structure is contracted information about the structure allowing for a complete recovery of its original multiplication table. Formally, a presentation is a pair A = X R with the relations R being equations between expressions formed of the generators from X. For example, V 4 = a, b a 2 = b 2 = e, ab = ba

5 Presentations A presentation for an algebraic structure is contracted information about the structure allowing for a complete recovery of its original multiplication table. Formally, a presentation is a pair A = X R with the relations R being equations between expressions formed of the generators from X. For example, V 4 = Gp a, b a 2 = b 2 = e, ab = ba A = Gp X R A = Inv X R A = InvM X R

6 Starting from a presentation A = X R Starting from a presentation, there are many interesting decision problems to study ( word problem, isomorphism problem,...), and for decidable such problems it is interesting to investigate their complexity. Tools: graphs, automata related to the presentation,

7 Starting from a presentation A = X R Starting from a presentation, there are many interesting decision problems to study ( word problem, isomorphism problem,...), and for decidable such problems it is interesting to investigate their complexity. Tools: graphs, automata related to the presentation, actions on these graphs

8 Inverse semigroups Inverse Semigroup associative binary operation existence of generalized inverses: a a 1 a = a a 1 a a 1 = a 1

9 Inverse semigroups Group - associative binary operation identity inverses: a a 1 = a 1 a = e Semigroup - associative operation Inverse Semigroup associative binary operation existence of inverses: a a 1 a = a a 1 a a 1 = a 1 permutations, symmetries, bijections,... concatenation of strings strings, paths in graphs, transition semigroups, partial transformations, do/undo proceses

10 Inverse semigroups Group - associative binary operation identity inverses: a a 1 = a 1 a = e Semigroup - associative operation Inverse Semigroup associative binary operation existence of inverses: a a 1 a = a a 1 a a 1 = a 1 permutations, symmetries, bijections,... concatenation of strings strings, paths in graphs, transition semigroups, partial transformations, do/undo proceses Inverse Monoid is an inverse semigroup with the identity.

11 Inverse semigroups S = Inv X R if S = (X X 1 ) + /τ where τ is the smallest congruence containing the relation R and Vagner s relations: {(uu 1 u, u) u S} {(uu 1 vv 1, vv 1 uu 1 ) u, v S}.

12 Why inverse semigroups? While groups can be represented as symmetries: Theorem (Cayley) Every group can be embedded in the set of one to one transformations on a set.

13 Why inverse semigroups? While groups can be represented as symmetries: Theorem (Cayley) Every group can be embedded in the set of one to one transformations 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 one to one transformations on a set.

14 Schützenberger (Cayley) graph Let S = Inv X R Definition (Schützenberger graph) Let w be a word in (X X 1 ) +. The Schützenberger graph of w relative to the presentation Inv X R is the graph SΓ(X, R, wτ) whose vertices are the elements of the R-class R wτ of wτ in S, and whose edges are of the form {(v 1, x, v 2 ) v 1, v 2 R wτ and v 1 (x τ) = v 2 }. v 1 xτ v2 if v 2 = v 1 xτ. Think strongly connected components of Cayley graphs.

15 Schützenberger graphs Schützenberger graphs generalization of Munn trees tool to approach algorithmic and structural problems in inverse semigroups connected components of the Cayley graph of H containing wτ deterministic inverse word graphs

16 Structure of inv. semigroups: Maximal subgroups One of the most basic structural question concerning inverse semigroups is the classification of the maximal subgroups of a given semigroup S.

17 Structure of inv. semigroups: Maximal subgroups One of the most basic structural question concerning inverse semigroups is the classification of the maximal subgroups of a given semigroup S.

18 Maximal Subgroups: An element a is called an idempotent if it satisfies the property a 2 = a.

19 Maximal Subgroups: An element a is called an idempotent if it satisfies the property a 2 = a. For example, if S is a group, it has only one idempotent - the identity.

20 Maximal Subgroups: An element a is called an idempotent if it satisfies the property a 2 = a. For example, if S is a group, it has only one idempotent - the identity. In general, inverse semigroups can have many idempotents.

21 Schützenberger graphs - Stephen s theorem Schützenberger graph SΓ(X, R, e) has many nice properties... [Stephen, 1994] one especially useful for the study of structure: G e = Aut(SΓ(X, R, e))

22 Schützenberger graphs of inverse semigroups Applying Stephen s theorem assumes that we already know the Schützenberger graphs for the given words and inverse semigroup.

23 Schützenberger graphs of inverse semigroups Applying Stephen s theorem assumes that we already know the Schützenberger graphs for the given words and inverse semigroup. BUT in general, we do not know any effective procedure for constructing the Schützenberger graphs.

24 Stephen s iterative procedure. Elementary expansion: - sewing on a relation r = s

25 Stephen s iterative procedure. Elementary expansion: - sewing on a relation r = s v r 1 v 2

26 Stephen s iterative procedure. Elementary expansion: - sewing on a relation r = s s v r 1 v 2

27 Stephen s iterative procedure. Elementary expansion: - sewing on a relation r = s s v r 1 v 2 Elementary determination: -edge folding x x

28 Stephen s iterative procedure. Elementary expansion: - sewing on a relation r = s s v r 1 v 2 Elementary determination: -edge folding x x fold

29 Stephen s iterative procedure. Elementary expansion: - sewing on a relation r = s s v r 1 v 2 Elementary determination: -edge folding x x fold x

30 Schützenberger graphs - iterative procedure In this way we get a directed system of inverse graphs Γ 1 Γ 2... Γ i... whose directed limit is the Schützenberger 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

31 HNN-extensions HigmanNeumannNeumann - extensions t 1 at = aφ for a A

32 HNN-extensions - groups For example, the fundamental group of a surface with a handle is an HNN-extension of the fundamental group of the surface without the handle attached.

33 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 1 at = aϕ,

34 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 1 at = aϕ, t 1 t = f, tt 1 = e, a A is called the HNN-extension of S associated with ϕ.

35 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 e A ese and f B fsf (or e / A ese and f / B fsf for some e, f E(S)). Then S = Inv X, t R, t 1 at = aϕ, t 1 t = f, tt 1 = e, a A is called the HNN-extension of S associated with ϕ.

36 Definition of HNN 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 e A ese and f B fsf (or e / A ese and f / B fsf for some e, f E(S)). Then S = Inv X, t R R HNN is called the HNN-extension of S associated with ϕ.

37 Definition of HNN 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 e A ese and f B fsf (or e / A ese and f / B fsf for some e, f E(S)). Then S = Inv X, t R R HNN is called the HNN-extension of S associated with ϕ. S S

38 Amalgams If S 1 = Inv X 1 R 1, S 2 = Inv X 2 R 2 with X 1 X 2 = S 1 U S 2 = Inv X R 1, R 2, R w = Inv X R where X = X 1 X 2, R w = {(ω 1 (u), ω 2 (u)) : u U}

39 Structure: HNN-extensions - Tools S = Inv X, t R R HNN a part of the word graph over X {t} may look something like this:

40 In the special case when S = Inv X, t R R HNN, a part of the word graph over X {t} may look something like this:

41 In the special case when S = Inv X, t R R HNN, a part of the word graph over X {t} may look something like this:

42

43 The tree structure of lobe graphs Theorem (T.Jajcayová) The lobe graph T (Γ) of a Schützenberger graph Γ relative to the presentation Inv X, t R R HNN is an oriented tree.

44 The tree structure of lobe graphs

45 Bass-Serre: Structure Theorem Recall: G e = Aut(SΓ(X, R, e))

46 Bass-Serre: Structure Theorem Recall: G e = Aut(SΓ(X, R, e)) Theorem (Structure theorem of Bass-Serre theory) Let G be a group acting without inversions on a connected graph X. Then X is a tree if and only if Φ : π(g, G \ X ) G is an isomorphism.

47 Bass-Serre: Structure Theorem Recall: G e = Aut(SΓ(X, R, e)) Theorem (Structure theorem of Bass-Serre theory) Let G be a group acting without inversions on a connected graph X. Then X is a tree if and only if Φ : π(g, G \ X ) G is an isomorphism. Thus, the Structure theorem of Bass-Serre theory allows one to find a presentation for a group G provided that an action of G on a tree is known.

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

49 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

50 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.

51 Bass-Serre: Graph of groups Let G : v G v G : y G y such that G y = G y. together with group monomorphism σ y : G y G α(y) and τ y : G y G ω(y) satisfying the relation σ y = τ y. Then the graph X together with the group assignment G is called a graph of groups (G( ), X ).

52 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 G v and by the edges of X, subject to the relations: {y = y 1, y 1 σ y (a)y = τ y (a) for all y Edge(X ) and a G y } {y = 1 for all y T }.

53 Bass-Serre: Fundamental group: Examples Example 1. Let X be any graph, and let G v = G y =<1> for all vertices and edges of X (the monomorphisms σ y and τ y being obviously the trivial mappings). The fundamental group of (G( ), X ) is the free group generated by the positive edges of X not in a maximal tree.

54 Bass-Serre: Fundamental group: Examples Example 1. Let X be any graph, and let G v = G y =<1> for all vertices and edges of X (the monomorphisms σ y and τ y being obviously the trivial mappings). The fundamental group of (G( ), X ) is the free group generated by the positive edges of X not in a maximal tree. Example 2. Take X to be a tree and let G y = 1 for all y Edge(X ). Then the maximal tree T is X itself. The fundamental group π(g( ), X ) is the free product of the groups G v (v Vert(X )). Example 3. Let X be a segment, that is, X consists of one edge y and two vertices v 1 and v 2. Then the fundamental group π(g( ), X ) is the free product of G v1 and G v2 amalgamating G y.

55 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 of G. We construct a graph of groups (G, Y ). The general idea: assign to the vertex G v the stabilizer group G v of the vertex v of X under the action of G. Similarly, to the edge G y assign the stabilizer group G y.

56 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 of G. We construct a graph of groups (G, Y ). The general idea: assign to the vertex G v the stabilizer group G v of the vertex v of X under the action of G. Similarly, to the edge G y assign the stabilizer group G y. Problem: we need to pick a specific stabilizer allowing us to define embedding monomorphisms.

57 Bass-Serre: Structure Theorem Theorem (Structure theorem of Bass-Serre theory) Let G be a group acting without inversions on a connected graph X. Then X is a tree if and only if Φ : π(g, G \ X ) G is an isomorphism. Thus, the Structure theorem of Bass-Serre theory allows one to find a presentation for a group G provided that an action of G on a tree is known.

58 Characterization of the Schützenberger graphs Theorem (T.Jajcayová) Let S be a lower bounded HNN-extension. The Schützenberger graphs of S relative to the presentation Inv X {t} R R HNN are precisely the complete T -graphs that possess a host.

59 Characterization of the Schützenberger graphs Theorem (T.Jajcayová) Let S be a lower bounded HNN-extension. The Schützenberger graphs of S relative to the presentation Inv X {t} R R HNN are precisely the complete T -graphs that possess a host. Schützenberger graphs contain a special subgraph with only finitely many lobes that contains the information for the whole graph. Schützenberger graphs of HNN-extensions have tree like lobe structure

60 Characterization of maximal subgroups of l.b. HNN s Theorem (T.Jajcayová) Let S be a lower bounded HNN-extension, and let e be an idempotent of S. Then the maximal subgroup of S containing e is isomorphic to the fundamental group of the graph of groups (H( ), Z e ). Theorem (T.Jajcayová) Let S be a lower bounded HNN-extension. Let e be an idempotent of S that is not D-related to any element of S. Then the maximal subgroup of S containing e is isomorphic to a subgroup H of S whose quotient H/ A and H/ B is finite.

61 Maximal subgroups of amalgams of finite inv. semigroups Theorem (A.Cherubini, T.Jajcayová, E. Rodaro) Let e E(S 1 U S 2 ), S 1, S 2 finite inverse semigroups, and suppose that e is not D S 1 U S 2 -related to any idempotent of S 1 or S 2. Then H S 1 U S 2 e is a homomorphic image of a subgroup of the maximal subgroup H S k g, for some g E(S k ) and some k {1, 2}.

62 Maximal subgroups of amalgams of finite inv. semigroups Theorem (A.Cherubini, T.Jajcayová, E. Rodaro) Let e E(S 1 U S 2 ), S 1, S 2 finite inverse semigroups, and suppose that e is not D S 1 U S 2 -related to any idempotent of S 1 or S 2. Then H S 1 U S 2 e is a homomorphic image of a subgroup of the maximal subgroup H S k g, for some g E(S k ) and some k {1, 2}. Theorem (A.Cherubini, T.Jajcayová, E. Rodaro) Let e E(S 1 U S 2 ), S 1, S 2 finite inverse semigroups, then there is an algorithm to compute a presentation for H S 1 U S 2 e.

63 Maximal subgroups A. Cherubini, T. B. Jajcayová and E. Rodaro, Maximal subgroups of amalgams of finite inverse semigroups, Semigroup Forum 90, No. 2 (2015), T. B. Jajcayová, Schützenberger Automata for HNN-extensions of Inverse Monoids, submitted to Information and Computation.

64 Future work apply existing tools to study maximal subgroups of other classes of inverse semigroups apply existing tools to study different (structural/algorithmic) questions adjust tools for other varieties of algebras

65 Thank you!