Monoids and Their Cayley Graphs

Transcription

1 Monoids and Their Cayley Graphs Nik Ruskuc School of Mathematics and Statistics, University of St Andrews NBGGT, Leeds, 30 April, 2008

2 Instead of an Apology Then, as for the geometrical analysis of the ancients and the algebra of the moderns, besides the fact that they extend to very abstract matters which seem to be of no practical use, the former is always so tied to the inspection of figures that it cannot exercise the understanding without greatly tiring the imagination, while, in the latter, one is so subjected to certain rules and numbers that it has become a confused and obscure art which oppresses the mind instead of being a science which cultivates it.

3 Instead of an Apology Then, as for the geometrical analysis of the ancients and the algebra of the moderns, besides the fact that they extend to very abstract matters which seem to be of no practical use, the former is always so tied to the inspection of figures that it cannot exercise the understanding without greatly tiring the imagination, while, in the latter, one is so subjected to certain rules and numbers that it has become a confused and obscure art which oppresses the mind instead of being a science which cultivates it. (Descartes, 1637)

4 Contents Cayley Graphs, Green s Equivalences Schützenberger Groups Generators and Presentations Rees Index Boundaries Conclusion

5 Cayley Graphs

6 Cayley Graphs M = A a monoid, with a generating set.

7 Cayley Graphs M = A a monoid, with a generating set. The Cayley graph of M w.r.t. A:

8 Cayley Graphs M = A a monoid, with a generating set. The Cayley graph of M w.r.t. A: vertices: M;

9 Cayley Graphs M = A a monoid, with a generating set. The Cayley graph of M w.r.t. A: vertices: M; edges: x a xa, x M, a A.

10 Cayley Graphs M = A a monoid, with a generating set. The Cayley graph of M w.r.t. A: vertices: M; edges: x a xa, x M, a A. Notation: Γ r (M, A) = Γ r (M) = Γ(M).

11 Cayley Graphs M = A a monoid, with a generating set. The Cayley graph of M w.r.t. A: vertices: M; edges: x a xa, x M, a A. Notation: Γ r (M, A) = Γ r (M) = Γ(M). r stands for right.

12 Example: The Bicyclic Monoid B

13 Example: The Bicyclic Monoid B B = b, c bc = 1 = {c i b j : i, j 0}.

14 Example: The Bicyclic Monoid B B = b, c bc = 1 = {c i b j : i, j 0}. i j

15 Example: The Bicyclic Monoid B B = b, c bc = 1 = {c i b j : i, j 0}. i j b

16 Example: The Bicyclic Monoid B B = b, c bc = 1 = {c i b j : i, j 0}. i j b c

17 Example: The Bicyclic Monoid B B = b, c bc = 1 = {c i b j : i, j 0}. i j b c

18 Green s R-equivalence

19 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M)

20 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M) iff xm ym.

21 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M) iff xm ym. R is a pre-order.

22 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M) iff xm ym. R is a pre-order. xry x R y & y R x

23 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M) iff xm ym. R is a pre-order. xry x R y & y R x xm = ym.

24 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M) iff xm ym. R is a pre-order. xry x R y & y R x xm = ym. R x - the R-class of x

25 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M) iff xm ym. R is a pre-order. xry x R y & y R x xm = ym. R x - the R-class of x (= the strongly connected component of x).

26 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M) iff xm ym. R is a pre-order. xry x R y & y R x xm = ym. R x - the R-class of x (= the strongly connected component of x). R x R y x R y - partial order.

27 Green s R-equivalence x R y iff there is a directed path from x to y in Γ(M) iff xm ym. R is a pre-order. xry x R y & y R x xm = ym. R x - the R-class of x (= the strongly connected component of x). R x R y x R y - partial order. A monoid is a partially ordered set of R-classes.

28 Left/Right Duality

29 Left/Right Duality Left Cayley graph Γ l (M, A): x a ax, x M, a A. Strongly connected components: L-classes.

30 Left/Right Duality Left Cayley graph Γ l (M, A): x a ax, x M, a A. Strongly connected components: L-classes. xly Mx = My.

31 Left/Right Duality Left Cayley graph Γ l (M, A): x a ax, x M, a A. Strongly connected components: L-classes. xly Mx = My. A monoid is a partially ordered set of L-classes.

32 Left/Right Duality Left Cayley graph Γ l (M, A): x a ax, x M, a A. Strongly connected components: L-classes. xly Mx = My. A monoid is a partially ordered set of L-classes. One more relation: H = R L.

33 R-Classes of B

34 R-Classes of B b c

35 R-Classes of B b c

36 L-Classes of B

37 L-Classes of B b c

38 L-Classes of B b c

39 Schützenberger Group

40 Schützenberger Group M a monoid; H an H-class.

41 Schützenberger Group M a monoid; H an H-class. Stabiliser: Stab r (H) = {a M : Ha = H}.

42 Schützenberger Group M a monoid; H an H-class. Stabiliser: Stab r (H) = {a M : Ha = H}. Stab r (H) acts on H.

43 Schützenberger Group M a monoid; H an H-class. Stabiliser: Stab r (H) = {a M : Ha = H}. Stab r (H) acts on H. Σ r (H) = Stab r (H) factored by the kernel of this action.

44 Schützenberger Group M a monoid; H an H-class. Stabiliser: Stab r (H) = {a M : Ha = H}. Stab r (H) acts on H. Σ r (H) = Stab r (H) factored by the kernel of this action. Proposition Σ r (H) is a group acting regularly on H.

45 Schützenberger Group M a monoid; H an H-class. Stabiliser: Stab r (H) = {a M : Ha = H}. Stab r (H) acts on H. Σ r (H) = Stab r (H) factored by the kernel of this action. Proposition Σ r (H) is a group acting regularly on H. Σ r (H) is called the right Schützenberger group of H.

46 Schützenberger Group M a monoid; H an H-class. Stabiliser: Stab r (H) = {a M : Ha = H}. Stab r (H) acts on H. Σ r (H) = Stab r (H) factored by the kernel of this action. Proposition Σ r (H) is a group acting regularly on H. Σ r (H) is called the right Schützenberger group of H. There is an analogous construction for the left Schützenberger group Σ l (H) of H.

47 Properties of Schützenberger Groups

48 Properties of Schützenberger Groups Σ r (H) = Σ l (H)

49 Properties of Schützenberger Groups Σ r (H) = Σ l (H)(= Σ(H)).

50 Properties of Schützenberger Groups Σ r (H) = Σ l (H)(= Σ(H)). Σ(H) = H.

51 Properties of Schützenberger Groups Σ r (H) = Σ l (H)(= Σ(H)). Σ(H) = H. If xry then Σ(H x ) = Σ(H y ).

52 Properties of Schützenberger Groups Σ r (H) = Σ l (H)(= Σ(H)). Σ(H) = H. If xry then Σ(H x ) = Σ(H y ). If xly then Σ(H x ) = Σ(H y ).

53 Properties of Schützenberger Groups Σ r (H) = Σ l (H)(= Σ(H)). Σ(H) = H. If xry then Σ(H x ) = Σ(H y ). If xly then Σ(H x ) = Σ(H y ). If H contains an idempotent then H is a (maximal sub)group of M

54 Properties of Schützenberger Groups Σ r (H) = Σ l (H)(= Σ(H)). Σ(H) = H. If xry then Σ(H x ) = Σ(H y ). If xly then Σ(H x ) = Σ(H y ). If H contains an idempotent then H is a (maximal sub)group of M and H = Σ(H).

55 Properties of Schützenberger Groups Σ r (H) = Σ l (H)(= Σ(H)). Σ(H) = H. If xry then Σ(H x ) = Σ(H y ). If xly then Σ(H x ) = Σ(H y ). If H contains an idempotent then H is a (maximal sub)group of M and H = Σ(H). A monoid is a union of groups, many of which are isomorphic, and some of which are actually in the monoid.

56 Example M = a, a 1, b aa 1 = a 1 a = 1, b 3 = b 2, a 2 b = ba 2, bab = 0

57 Example M = a, a 1, b aa 1 = a 1 a = 1, b 3 = b 2, a 2 b = ba 2, bab = 0 = {a i, a i b j a k : i Z, j = 1, 2, k = 0, 1} {0}.

58 Example M = a, a 1, b aa 1 = a 1 a = 1, b 3 = b 2, a 2 b = ba 2, bab = 0 = {a i, a i b j a k : i Z, j = 1, 2, C a i k = 0, 1} {0}. a 2i b a 2i b a a 2i+1 b a 2i+1 b a a 2i b 2 a 2i b 2 a a 2i+1 b 2 a 2i+1 b 2 a 0

59 Example M = a, a 1, b aa 1 = a 1 a = 1, b 3 = b 2, a 2 b = ba 2, bab = 0 = {a i, a i b j a k : i Z, j = 1, 2, C a i k = 0, 1} {0}. a 2i b a 2i b a a 2i+1 b a 2i+1 b a a 2i b 2 a 2i b 2 a a 2i+1 b 2 a 2i+1 b 2 a 0

60 Example M = a, a 1, b aa 1 = a 1 a = 1, b 3 = b 2, a 2 b = ba 2, bab = 0 = {a i, a i b j a k : i Z, j = 1, 2, k = 0, 1} {0}. C a i a 2i b a 2i b a Σ(H b ) = C a 2i+1 b a 2i+1 b a a 2i b 2 a 2i b 2 a a 2i+1 b 2 a 2i+1 b 2 a 0

61 Example M = a, a 1, b aa 1 = a 1 a = 1, b 3 = b 2, a 2 b = ba 2, bab = 0 = {a i, a i b j a k : i Z, j = 1, 2, k = 0, 1} {0}. C a i a 2i b a 2i b a Σ(H b ) = C Σ(H b 2) = C = Hb 2. a 2i+1 b a 2i+1 b a a 2i b 2 a 2i b 2 a a 2i+1 b 2 a 2i+1 b 2 a 0

62 Cayley Graphs: Monoids vs. Groups

63 Cayley Graphs: Monoids vs. Groups Cayley graph is essentially a monoid concept. (Compare with: transformations/permutations; words/reduced words.)

64 Cayley Graphs: Monoids vs. Groups Cayley graph is essentially a monoid concept. (Compare with: transformations/permutations; words/reduced words.) The Cayley graph of a general monoid possesses group-theoretic and order-theoretic aspects.

65 Cayley Graphs: Monoids vs. Groups Cayley graph is essentially a monoid concept. (Compare with: transformations/permutations; words/reduced words.) The Cayley graph of a general monoid possesses group-theoretic and order-theoretic aspects. Main difference between groups and monoids: poset aspect, direction, (ir)reversibility.

66 Cayley Graphs: Monoids vs. Groups Cayley graph is essentially a monoid concept. (Compare with: transformations/permutations; words/reduced words.) The Cayley graph of a general monoid possesses group-theoretic and order-theoretic aspects. Main difference between groups and monoids: poset aspect, direction, (ir)reversibility. Many concepts/viewpoints of combinatorial/geometrical/computational group theory have natural extensions in the world of monoids.

67 Cayley Graphs: Monoids vs. Groups For instance:

68 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,...

69 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,...

70 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,... Todd Coxeter, Knuth Bendix procedures: Neumann, Jura, St Andrews group,...

71 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,... Todd Coxeter, Knuth Bendix procedures: Neumann, Jura, St Andrews group,... Automatic structures: St Andrews group, Thomas, Otto, Kambites, Duncan,...

72 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,... Todd Coxeter, Knuth Bendix procedures: Neumann, Jura, St Andrews group,... Automatic structures: St Andrews group, Thomas, Otto, Kambites, Duncan,... Topology: Stephen, Margolis, Meakin, Steinberg,...

73 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,... Todd Coxeter, Knuth Bendix procedures: Neumann, Jura, St Andrews group,... Automatic structures: St Andrews group, Thomas, Otto, Kambites, Duncan,... Topology: Stephen, Margolis, Meakin, Steinberg,... Categories, groupoids: Lawson, Gilbert,...

74 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,... Todd Coxeter, Knuth Bendix procedures: Neumann, Jura, St Andrews group,... Automatic structures: St Andrews group, Thomas, Otto, Kambites, Duncan,... Topology: Stephen, Margolis, Meakin, Steinberg,... Categories, groupoids: Lawson, Gilbert,... Ends: Jackson, Kilibarda.

75 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,... Todd Coxeter, Knuth Bendix procedures: Neumann, Jura, St Andrews group,... Automatic structures: St Andrews group, Thomas, Otto, Kambites, Duncan,... Topology: Stephen, Margolis, Meakin, Steinberg,... Categories, groupoids: Lawson, Gilbert,... Ends: Jackson, Kilibarda. Combinatorial theory: the main topic of this talk.

76 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,... Todd Coxeter, Knuth Bendix procedures: Neumann, Jura, St Andrews group,... Automatic structures: St Andrews group, Thomas, Otto, Kambites, Duncan,... Topology: Stephen, Margolis, Meakin, Steinberg,... Categories, groupoids: Lawson, Gilbert,... Ends: Jackson, Kilibarda. Combinatorial theory: the main topic of this talk....

77 Cayley Graphs: Monoids vs. Groups For instance: Word problems, decidability: Markov, Post, Adjan,... Diagrams, pictures: Remmers, Guba, Howie, Pride,... Todd Coxeter, Knuth Bendix procedures: Neumann, Jura, St Andrews group,... Automatic structures: St Andrews group, Thomas, Otto, Kambites, Duncan,... Topology: Stephen, Margolis, Meakin, Steinberg,... Categories, groupoids: Lawson, Gilbert,... Ends: Jackson, Kilibarda. Combinatorial theory: the main topic of this talk.... Even if considering Cayley graphs and related topics in this more general setting of monoids may not yield solutions of the hard group-theoretic problems, it may aid understanding of the fundamental concepts involved.

78 Generators M a monoid; H an H-class; R the R-class of H. R H

79 Generators M a monoid; H an H-class; R the R-class of H. H = H 1 H 2 H i R H i (i I ) the H-classes in R (H 1 = H).

80 Generators M a monoid; H an H-class; R the R-class of H. R H = H 1 H 2 r i H i H i (i I ) the H-classes in R (H 1 = H). r i M (i I ) representatives : Hr i = H i.

81 Generators M a monoid; H an H-class; R the R-class of H. R H = H 1 H 2 r i r i H i H i (i I ) the H-classes in R (H 1 = H). r i M (i I ) representatives : Hr i = H i. r i (i I ) their inverses : hr i r i = h (h H).

82 Generators M a monoid; H an H-class; R the R-class of H. R H = H 1 H 2 r i H i a H j H i (i I ) the H-classes in R (H 1 = H). r i M (i I ) representatives : Hr i = H i. r i (i I ) their inverses : hr i r i = h (h H).

83 Generators M a monoid; H an H-class; R the R-class of H. R H = H 1 H 2 r i H i a H j r j H i (i I ) the H-classes in R (H 1 = H). r i M (i I ) representatives : Hr i = H i. r i (i I ) their inverses : hr i r i = h (h H).

84 Generators M a monoid; H an H-class; R the R-class of H. R H = H 1 H 2 r i H i a H j r j H i (i I ) the H-classes in R (H 1 = H). r i M (i I ) representatives : Hr i = H i. r i (i I ) their inverses : hr i r i = h (h H). Theorem If M is generated by A then the Schützenberger group Σ(H) is generated by the elements r i ar j (i, j I, a A, Hr i a = H j ).

85 Generators M a monoid; H an H-class; R the R-class of H. R H = H 1 H 2 r i H i a H j r j H i (i I ) the H-classes in R (H 1 = H). r i M (i I ) representatives : Hr i = H i. r i (i I ) their inverses : hr i r i = h (h H). Theorem If M is generated by A then the Schützenberger group Σ(H) is generated by the elements r i ar j (i, j I, a A, Hr i a = H j ). Remark Observe a close analogy with the Schreier generating set for a subgroup of a group.

86 Finite Generation (1)

87 Finite Generation (1) Definition The index of an H-class is the number of H-classes in its R-class.

88 Finite Generation (1) Definition The index of an H-class is the number of H-classes in its R-class. Corollary The Schützenberger group of an H-class of finite index in a finitely generated monoid is itself finitely generated.

89 Finite Generation (1) Definition The index of an H-class is the number of H-classes in its R-class. Corollary The Schützenberger group of an H-class of finite index in a finitely generated monoid is itself finitely generated. Corollary Let M be a monoid with finitely many H-classes (i.e. finitely many left and right ideals). Then M is finitely generated if and only if all its Schützenberger groups are finitely generated.

90 Finite Generation (2) Definition A monoid is regular if for every x M there exists y M such that xyx = x.

91 Finite Generation (2) Definition A monoid is regular if for every x M there exists y M such that xyx = x. Corollary Let M be a regular monoid with finitely many idempotents. M is finitely generated if and only if all its maximal subgroups are finitely generated.

92 Presentations: Maximal Subgroups

93 Presentations: Maximal Subgroups Theorem A maximal subgroup of finite index in a finitely presented monoid is itself finitely presented.

94 Presentations: Maximal Subgroups Theorem A maximal subgroup of finite index in a finitely presented monoid is itself finitely presented. Proof is a straightforward modification of the Reidemeister Schreier Theorem for subgroups of groups, building on the Schreier-type generating set from above.

95 Presentations: Maximal Subgroups Theorem A maximal subgroup of finite index in a finitely presented monoid is itself finitely presented. Proof is a straightforward modification of the Reidemeister Schreier Theorem for subgroups of groups, building on the Schreier-type generating set from above. Corollary A regular monoid with finitely many idempotents is finitely presented if and only if all its maximal subgroups are finitely presented.

96 Presentations: Schützenberger Groups an Example

97 Presentations: Schützenberger Groups an Example G the group a 1, a 2, a 3, a 4 a 1 a 2 = a 3 a 4 (free of rank 3).

98 Presentations: Schützenberger Groups an Example G the group a 1, a 2, a 3, a 4 a 1 a 2 = a 3 a 4 (free of rank 3). M = G, b, c, d a j b = ba 2 j, cb2 = cb, a j d = da j, cbda j = a j cbd.

99 Presentations: Schützenberger Groups an Example G the group a 1, a 2, a 3, a 4 a 1 a 2 = a 3 a 4 (free of rank 3). M = G, b, c, d a j b = ba 2 j, cb2 = cb, a j d = da j, cbda j = a j cbd. The left Cayley graph between 1 and H = H cbd : d a j bd H 1 b a j a j H1 = F 3 d d H1 a j a j b 2 dh b 1 d b b 3 dh 1 c d c c a j cbdh 1 b c c d b

100 Presentations: Schützenberger Groups an Example G the group a 1, a 2, a 3, a 4 a 1 a 2 = a 3 a 4 (free of rank 3). M = G, b, c, d a j b = ba 2 j, cb2 = cb, a j d = da j, cbda j = a j cbd. The left Cayley graph between 1 and H = H cbd : Theorem H has index 1; its Schützenberger group is d a j bd H 1 b a j a j H1 = F 3 d d H1 a j a j b 2 dh b 1 d b b 3 dh 1 c d c c cbdh 1 a 1, a 2, a 3, a 4 a 2i 1 a 2i 2 = a 2i 3 a 2i 4 (i = 0, 1, 2,...). a j b c c d b

101 Presentations: Schützenberger Groups

102 Presentations: Schützenberger Groups Theorem Let M be a monoid with finitely many left and right ideals. Then M is finitely presented if and only if all its Schützenberger groups are finitely presented.

103 Other Properties Theorem (Gray, NR) Let M be a monoid with finitely many left and right ideals. Then M is residually finite if and only if all its Schützenberger groups are residually finite.

104 Other Properties Theorem (Gray, NR) Let M be a monoid with finitely many left and right ideals. Then M is residually finite if and only if all its Schützenberger groups are residually finite. F= a,b Example (Campbell, Robertson, Thomas, NR) There exists a monoid which is a disjoint union of two copies of the free group of rank 2 which is not automatic. φ 2 2 a,b a =b =1

105 Other Properties Theorem (Gray, NR) Let M be a monoid with finitely many left and right ideals. Then M is residually finite if and only if all its Schützenberger groups are residually finite. F= a,b Example (Campbell, Robertson, Thomas, NR) There exists a monoid which is a disjoint union of two copies of the free group of rank 2 which is not automatic. φ Question a,b a =b =1 Suppose M is a monoid with finitely many left and right ideals. 2 2

106 Other Properties Theorem (Gray, NR) Let M be a monoid with finitely many left and right ideals. Then M is residually finite if and only if all its Schützenberger groups are residually finite. F= a,b Example (Campbell, Robertson, Thomas, NR) There exists a monoid which is a disjoint union of two copies of the free group of rank 2 which is not automatic. φ Question a,b a =b =1 Suppose M is a monoid with finitely many left and right ideals. If M is automatic, are all its Schützenberger groups automatic? 2 2

107 Other Properties Theorem (Gray, NR) Let M be a monoid with finitely many left and right ideals. Then M is residually finite if and only if all its Schützenberger groups are residually finite. F= a,b Example (Campbell, Robertson, Thomas, NR) There exists a monoid which is a disjoint union of two copies of the free group of rank 2 which is not automatic. φ Question a,b a =b =1 Suppose M is a monoid with finitely many left and right ideals. If M is automatic, are all its Schützenberger groups automatic? If M is hyperbolic, are all its Schützenberger groups hyperbolic? 2 2

108 Finite Complement (Rees Index)

109 Finite Complement (Rees Index) Theorem Let M be a monoid, and let N M be such that M \ N is finite. Then M is finitely generated (resp. finitely presented) if and only if N is finitely generated (resp. finitely presented).

110 Finite Complement (Rees Index) Theorem Let M be a monoid, and let N M be such that M \ N is finite. Then M is finitely generated (resp. finitely presented) if and only if N is finitely generated (resp. finitely presented). Remarks Analogous to Schreier/ Reidemeister Schreier theorems for groups.

111 Finite Complement (Rees Index) Theorem Let M be a monoid, and let N M be such that M \ N is finite. Then M is finitely generated (resp. finitely presented) if and only if N is finitely generated (resp. finitely presented). Remarks Analogous to Schreier/ Reidemeister Schreier theorems for groups. Proofs conceptually similar, but very different in details.

112 Finite Complement (Rees Index) Theorem Let M be a monoid, and let N M be such that M \ N is finite. Then M is finitely generated (resp. finitely presented) if and only if N is finitely generated (resp. finitely presented). Remarks Analogous to Schreier/ Reidemeister Schreier theorems for groups. Proofs conceptually similar, but very different in details. Proof of finite presentability surprisingly hard.

113 Finite Complement (Rees Index) Theorem Let M be a monoid, and let N M be such that M \ N is finite. Then M is finitely generated (resp. finitely presented) if and only if N is finitely generated (resp. finitely presented). Remarks Analogous to Schreier/ Reidemeister Schreier theorems for groups. Proofs conceptually similar, but very different in details. Proof of finite presentability surprisingly hard. Analogous theorems for many other finiteness conditions, including residual finiteness (Thomas, NR), automaticity (Hoffmann, Thomas, NR), and many conditions to do with ideals (Malcev, Mitchell, NR).

114 Finite Complement (Rees Index) Theorem Let M be a monoid, and let N M be such that M \ N is finite. Then M is finitely generated (resp. finitely presented) if and only if N is finitely generated (resp. finitely presented). Remarks Analogous to Schreier/ Reidemeister Schreier theorems for groups. Proofs conceptually similar, but very different in details. Proof of finite presentability surprisingly hard. Analogous theorems for many other finiteness conditions, including residual finiteness (Thomas, NR), automaticity (Hoffmann, Thomas, NR), and many conditions to do with ideals (Malcev, Mitchell, NR). Open: finite complete rewriting systems, FDT.

115 Boundaries Joint work with R. Gray.

116 Boundaries Joint work with R. Gray. M = A a monoid; N M.

117 Boundaries Joint work with R. Gray. M = A a monoid; N M. The right boundary of N: Elements of N which, considered as points in Γ(M, A) receive an edge from the outside.

118 Boundaries Joint work with R. Gray. M = A a monoid; N M. The right boundary of N: Elements of N which, considered as points in Γ(M, A) receive an edge from the outside. Γ r ( S) T U

119 Boundaries Joint work with R. Gray. M = A a monoid; N M. The right boundary of N: Elements of N which, considered as points in Γ(M, A) receive an edge from the outside. Γ r ( S) T U Boundary := left boundary right boundary.

120 Boundaries: Example S = {a, b}, T = as.

121 Boundaries: Example S = {a, b}, T = as. Right boundary: a b aa ab ba bb aaa aab aba abb baa bab bba bbb

122 Boundaries: Example S = {a, b}, T = as. Left boundary: a b aa ba ab bb aaa baa aba bba aab bab abb bbb

123 Finite Boundaries

124 Finite Boundaries Proposition N M has a finite boundary if any of the following hold:

125 Finite Boundaries Proposition N M has a finite boundary if any of the following hold: M \ N is finite;

126 Finite Boundaries Proposition N M has a finite boundary if any of the following hold: M \ N is finite; M \ N is an ideal;

127 Finite Boundaries Proposition N M has a finite boundary if any of the following hold: M \ N is finite; M \ N is an ideal; N is an ideal, M \ N is a submonoid and the orbits of N under the action of M \ N are finite.

128 Finite Boundaries Proposition N M has a finite boundary if any of the following hold: M \ N is finite; M \ N is an ideal; N is an ideal, M \ N is a submonoid and the orbits of N under the action of M \ N are finite. Proposition Having a finite boundary is independent of the choice of (finite) generating set.

129 Finite Generation and Presentability Theorem Let M be a finitely generated monoid, and let N M have finite boundary.

130 Finite Generation and Presentability Theorem Let M be a finitely generated monoid, and let N M have finite boundary. N is finitely generated.

131 Finite Generation and Presentability Theorem Let M be a finitely generated monoid, and let N M have finite boundary. N is finitely generated. If M is finitely presented then N is finitely presented.

132 Finite Generation and Presentability Theorem Let M be a finitely generated monoid, and let N M have finite boundary. N is finitely generated. If M is finitely presented then N is finitely presented. Questions How about other properties: FDT, automaticity, etc.?

133 One-sided Boundaries

134 One-sided Boundaries Theorem Let F be a finitely generated free monoid, and let N F. If N is finitely generated and has a finite right boundary, then N is finitely presented.

135 One-sided Boundaries Theorem Let F be a finitely generated free monoid, and let N F. If N is finitely generated and has a finite right boundary, then N is finitely presented. Theorem Let M = i I S i be a finite disjoint union of subsemigroups. If each S i is finitely presented and has a finite boundary in M then M is finitely presented as well.

136 A Strange Union Proposition (Araujo, NR) If S is the four element semigroup S a b c 0 a a a c 0 b b b c 0 c then S C is not finitely presented, but is a disjoint union of two finitely presented subsemigroups {a} C ( = C ) and {b, c, 0} C.

137 Green Index: a Preview

138 Green Index: a Preview Boundaries do not bring the Rees and group indices closer together.

139 Green Index: a Preview Boundaries do not bring the Rees and group indices closer together. A new development: Green index. (Also with R. Gray.)

140 Green Index: a Preview Boundaries do not bring the Rees and group indices closer together. A new development: Green index. (Also with R. Gray.) Unifies group index, Rees index and Schützenberger groups.

141 Green Index: a Preview Boundaries do not bring the Rees and group indices closer together. A new development: Green index. (Also with R. Gray.) Unifies group index, Rees index and Schützenberger groups. For finite presentability it features a curious two-directional Reidemeister Schreier type rewriting.

142 Project: Monoids, Groups, Posets, Actions

143 Project: Monoids, Groups, Posets, Actions Haven t mentioned: M acts on the set of R-classes, and on the set of L-classes

144 Project: Monoids, Groups, Posets, Actions Haven t mentioned: M acts on the set of R-classes, and on the set of L-classes (on the left and right respectively :-( ).

145 Project: Monoids, Groups, Posets, Actions Haven t mentioned: M acts on the set of R-classes, and on the set of L-classes (on the left and right respectively :-( ). One would like to prove results of the following type for some properties P:

146 Project: Monoids, Groups, Posets, Actions Haven t mentioned: M acts on the set of R-classes, and on the set of L-classes (on the left and right respectively :-( ). One would like to prove results of the following type for some properties P: A monoid M satisfies P if and only if:

147 Project: Monoids, Groups, Posets, Actions Haven t mentioned: M acts on the set of R-classes, and on the set of L-classes (on the left and right respectively :-( ). One would like to prove results of the following type for some properties P: A monoid M satisfies P if and only if: All Schützenberger groups satisfy P;

148 Project: Monoids, Groups, Posets, Actions Haven t mentioned: M acts on the set of R-classes, and on the set of L-classes (on the left and right respectively :-( ). One would like to prove results of the following type for some properties P: A monoid M satisfies P if and only if: All Schützenberger groups satisfy P; The actions of M on M/R and M/L satisfy a related property P ;

149 Project: Monoids, Groups, Posets, Actions Haven t mentioned: M acts on the set of R-classes, and on the set of L-classes (on the left and right respectively :-( ). One would like to prove results of the following type for some properties P: A monoid M satisfies P if and only if: All Schützenberger groups satisfy P; The actions of M on M/R and M/L satisfy a related property P ; The posets M/R and M/L satisfy another related property P.

150 Project: Monoids, Groups, Posets, Actions Haven t mentioned: M acts on the set of R-classes, and on the set of L-classes (on the left and right respectively :-( ). One would like to prove results of the following type for some properties P: A monoid M satisfies P if and only if: All Schützenberger groups satisfy P; The actions of M on M/R and M/L satisfy a related property P ; The posets M/R and M/L satisfy another related property P. A promising candidate for P: residual finiteness.