Inverse Monoids, Thompson's Group V and Generalisations - CAUL

Inverse Monoids, Thompson’s Group V and
Generalisations
John Fountain
25 July 2011
Lisbon - Groups and Semigroups

About 1965
F < T < V

Properties
1. V contains every finite group
2. V is simple
3. V is finitely presented
4. V has type F P ∞
5. V has solvable word problem
6. V has solvable conjugacy problem
7. V has a subgroup isomorphic to F 2 × F 2
8. The generalised word problem for V is undecidable

Inverse Monoids
◮ A monoid M is an inverse monoid if
∀a ∈ M, ∃ unique b ∈ M with aba = a and bab = b.
b is the inverse of a and is denoted by a −1 .

Inverse Monoids
◮ A monoid M is an inverse monoid if
∀a ∈ M, ∃ unique b ∈ M with aba = a and bab = b.
b is the inverse of a and is denoted by a −1 .
◮ Fact. A monoid M is inverse iff
(i) ∀a ∈ M, ∃ b ∈ M such that aba = a and
(ii) e 2 = e, f 2 = f ∈ M ⇒ ef = fe.

Inverse Monoids
◮ A monoid M is an inverse monoid if
∀a ∈ M, ∃ unique b ∈ M with aba = a and bab = b.
b is the inverse of a and is denoted by a −1 .
◮ Fact. A monoid M is inverse iff
(i) ∀a ∈ M, ∃ b ∈ M such that aba = a and
(ii) e 2 = e, f 2 = f ∈ M ⇒ ef = fe.
◮ Symmetric Inverse Monoid
For a set X,
I X
:= set of all bijections between subsets of X.
Inverse monoid under composition of partial functions.

Inverse Monoids: minimum group congruence
◮ M inverse: on M, define σ by
aσb ⇔ ae = be for some e 2 = e.
◮ σ is a congruence and M/σ is a group.
◮ If ρ is a congruence on M and M/ρ is a group, then ρ ⊆ σ.
◮ Thus σ is the minimum group congruence on M and M/σ
is the maximum group homomorphic image of M.

LCM monoids
C is a right LCM monoid if C is left cancellative and for
a, b ∈ C, aC ∩ bC = ∅ or is principal.
Artin monoids (in particular, free monoids).

LCM monoids
C is a right LCM monoid if C is left cancellative and for
a, b ∈ C, aC ∩ bC = ∅ or is principal.
Artin monoids (in particular, free monoids).
A projective right ideal of C is a disjoint union of principal
right ideals. If
P = a 1 C ⊔ · · · ⊔ a t C
is a projective right ideal, say {a 1 , . . . , a t } is a basis for P .

LCM monoids
C is a right LCM monoid if C is left cancellative and for
a, b ∈ C, aC ∩ bC = ∅ or is principal.
Artin monoids (in particular, free monoids).
A projective right ideal of C is a disjoint union of principal
right ideals. If
P = a 1 C ⊔ · · · ⊔ a t C
is a projective right ideal, say {a 1 , . . . , a t } is a basis for P .
A right ideal R of C is essential if R ∩ I ≠ ∅ for every right ideal
I of A ∗ .
Assumption C has finitely generated essential projective right
ideals. Holds if C is a finitely generated monoid.

LCM monoids
C is a right LCM monoid if C is left cancellative and for
a, b ∈ C, aC ∩ bC = ∅ or is principal.
Artin monoids (in particular, free monoids).
A projective right ideal of C is a disjoint union of principal
right ideals. If
P = a 1 C ⊔ · · · ⊔ a t C
is a projective right ideal, say {a 1 , . . . , a t } is a basis for P .
A right ideal R of C is essential if R ∩ I ≠ ∅ for every right ideal
I of A ∗ .
Assumption C has finitely generated essential projective right
ideals. Holds if C is a finitely generated monoid.
R e fp
(C):= set of all C-isomorphisms between finitely generated
essential projective right ideals of C. Inverse submonoid of I C
.

C = A ∗
◮ A = {a 1 , . . . , a k
}. Every right ideal of A ∗ is projective.

C = A ∗
◮ A = {a 1 , . . . , a k
}. Every right ideal of A ∗ is projective.
◮ A basis for a right ideal of A ∗ is a prefix code P over A:
P ⊆ A ∗ and uA ∗ ∩ vA ∗ = ∅ ∀u, v ∈ P .

C = A ∗
◮ A = {a 1 , . . . , a k
}. Every right ideal of A ∗ is projective.
◮ A basis for a right ideal of A ∗ is a prefix code P over A:
P ⊆ A ∗ and uA ∗ ∩ vA ∗ = ∅ ∀u, v ∈ P .
◮ P is maximal if for a prefix code Q over A,
P ⊆ Q ⇒ P = Q.

C = A ∗
◮ A = {a 1 , . . . , a k
}. Every right ideal of A ∗ is projective.
◮ A basis for a right ideal of A ∗ is a prefix code P over A:
P ⊆ A ∗ and uA ∗ ∩ vA ∗ = ∅ ∀u, v ∈ P .
◮ P is maximal if for a prefix code Q over A,
P ⊆ Q ⇒ P = Q.
◮ If R a right ideal of A ∗ , then
(i) R = P A ∗ for a uniquely determined prefix code P ;
(ii) P is the unique minimal set of generators for R;
(iii) R is essential if and only if P maximal.

C = A ∗
◮ A = {a 1 , . . . , a k
}. Every right ideal of A ∗ is projective.
◮ A basis for a right ideal of A ∗ is a prefix code P over A:
P ⊆ A ∗ and uA ∗ ∩ vA ∗ = ∅ ∀u, v ∈ P .
◮ P is maximal if for a prefix code Q over A,
P ⊆ Q ⇒ P = Q.
◮ If R a right ideal of A ∗ , then
(i) R = P A ∗ for a uniquely determined prefix code P ;
(ii) P is the unique minimal set of generators for R;
(iii) R is essential if and only if P maximal.
◮ V k,1 = V k := R e fp (A∗ )/σ. V = V 2 .

Left Rees Monoids
Left Rees monoid M: left cancellative; all right ideals projective;
∀a ∈ M, {bM : aM ⊆ bM} is finite.
G, C monoids.Actions: G on C: (g, c) ↦→ g · c; C on G: (g, c) ↦→ g| c .
On C × G define
(c, g)(d, h) = (c(g · d), g| d h).
With appropriate conditions on the actions, get a monoid
C ⊲⊳ G, the Zappa-Szép product of C and G.

Left Rees Monoids
Left Rees monoid M: left cancellative; all right ideals projective;
∀a ∈ M, {bM : aM ⊆ bM} is finite.
G, C monoids.Actions: G on C: (g, c) ↦→ g · c; C on G: (g, c) ↦→ g| c .
On C × G define
(c, g)(d, h) = (c(g · d), g| d h).
With appropriate conditions on the actions, get a monoid
C ⊲⊳ G, the Zappa-Szép product of C and G.
Theorem (Lawson).
M is a left Rees monoid iff ∃ set A, group G with M ∼ = A ∗ ⊲⊳ G.
Action of G on A ∗ is self-similar, i.e.,
∀g ∈ G, a ∈ A, ∃ unique b ∈ A, h ∈ G such that
g · (aw) = b(h · w) for all w ∈ A ∗ . (b = g · a and h = g| a .)
For A = {a 1 , . . . , a k }, R e fp (A∗ ⊲⊳ G)/σ ∼ = V k (G). (Scott, Röver,
Nekrashevych)

Alternative view of R e fp (C)
Three stage construction from C.
Step 1: C left cancellative, a ∈ C. Define λ a : C → C by
λ a (c) = ac.
is one-one with domain C. IH(C) = Inv〈λ a : a ∈ C〉 ⊆ I C is
the inverse hull of C.
{
IH 0 IH(C) if 0 ∈ IH(C)
(C) =
IH(C) ∪ {0} otherwise.

Alternative view of R e fp (C)
Three stage construction from C.
Step 1: C left cancellative, a ∈ C. Define λ a : C → C by
λ a (c) = ac.
is one-one with domain C. IH(C) = Inv〈λ a : a ∈ C〉 ⊆ I C is
the inverse hull of C.
{
IH 0 IH(C) if 0 ∈ IH(C)
(C) =
IH(C) ∪ {0} otherwise.
C is a right LCM monoid iff every non-zero element of IH 0 (C)
can be written as λ a λ −1
b
for some a, b ∈ C.
Part of a theorem due to McAlister and independently
Nivat/Perrot.

Alternative view of Rfp e (C): Step 2
S inverse monoid with zero. a, b ∈ S are orthogonal (a ⊥ b) if
a −1 b = 0 = ab −1 .
Clearly, a ⊥ b iff aa −1 ⊥ bb −1 and a −1 a ⊥ b −1 b.
A ⊆ S is orthogonal if a ⊥ b for all distinct a, b ∈ A.

Alternative view of Rfp e (C): Step 2
S inverse monoid with zero. a, b ∈ S are orthogonal (a ⊥ b) if
a −1 b = 0 = ab −1 .
Clearly, a ⊥ b iff aa −1 ⊥ bb −1 and a −1 a ⊥ b −1 b.
A ⊆ S is orthogonal if a ⊥ b for all distinct a, b ∈ A.
Define
D(S) = {A ⊆ S : 0 ∈ A, |A| < ∞, A is orthogonal}.
Then D(S) is an inverse submonoid of P (S) and the map given
by a ↦→ {0, a} embeds S in D(S).
Theorem
If C is a right LCM monoid, then D(IH 0 (C)) ∼ = R fp (C) where
R fp (C) is the inverse monoid of all C-isomorphisms between
finitely generated projective right ideals of C.

Alternative view of Rfp e (C): Step 3
S inverse monoid with zero.
S e := {a ∈ S : Sa and aS are essential}.
S e is an inverse submonoid of S called the essential part of S.
An idempotent e is essential if e ∈ S e . This is true if and only if
ef ≠ 0 for all non-zero idempotents f of S.
a ∈ S e if and only if aa −1 and a −1 a are essential idempotents

Alternative view of Rfp e (C): Step 3
S inverse monoid with zero.
S e := {a ∈ S : Sa and aS are essential}.
S e is an inverse submonoid of S called the essential part of S.
An idempotent e is essential if e ∈ S e . This is true if and only if
ef ≠ 0 for all non-zero idempotents f of S.
a ∈ S e if and only if aa −1 and a −1 a are essential idempotents
The isomorphism D(IH 0 (C)) ∼ = R fp (C) restricts to an
isomorphism
D e (IH 0 (C)) ∼ = R e fp (C).

Right Ore Right LCM Monoids
Let C be right Ore and right LCM. Then every projective right
ideal is principal. (Two principal right ideals cannot be
disjoint).
So, all orthogonal subsets of IH 0 (C) have the form {λ a λ −1
b
, 0};
hence the embedding of IH 0 (C) into D(IH 0 (C)) is surjective.
Every nonzero idempotent of IH 0 (C) is essential. So
D e (IH 0 (C)) = IH(C).
For cancellative right Ore C it is well known that the group
G(C) of right fractions of C is isomorphic to IH(C)/σ, so in
this case, R e fp (C) ∼ = G(C).

More on Zappa-Szép Products I
Let C be a right LCM monoid with trivial group of units, and
G be a group. Suppose we have actions so that we can form
D = C ⊲⊳ G. Then
1. D is left cancellative;
2. D is right LCM;
3. the group of units of D is {(1, g) : g ∈ G};
4. the partially ordered set of principal right ideals of D is
order-isomorphic to the partially ordered set of principal
right ideals of C.

More on Zappa-Szép Products II
Remember that for any right LCM monoid B,
λ a λ −1
b
IH 0 (B) = {λ a λ −1
b
: a, b ∈ B} ∪ {0}.
= λ c λ −1
d
⇔ ∃ unit u ∈ B such that au = c, bu = d.
Write elements of IH 0 (B) as ∼-equivalence classes [a, b] where
(a, b) ∼ (c, d) ⇔ ∃ unit u ∈ B such that au = c, bu = d.

More on Zappa-Szép Products II
Remember that for any right LCM monoid B,
λ a λ −1
b
IH 0 (B) = {λ a λ −1
b
: a, b ∈ B} ∪ {0}.
= λ c λ −1
d
⇔ ∃ unit u ∈ B such that au = c, bu = d.
Write elements of IH 0 (B) as ∼-equivalence classes [a, b] where
(a, b) ∼ (c, d) ⇔ ∃ unit u ∈ B such that au = c, bu = d.
Now consider IH 0 (D) where D = C ⊲⊳ G. Elements are:
[(a, g), (b, h)] = [(a, gh −1 ), (b, 1)]
so we can represent elements by triples (a, g, b) ∈ C × G × C.

More on Zappa-Szép Products III: Orthogonal Subsets
The following are equivalent:
1. X = {(a 1 , g 1 , b 1 ), . . . , (a t , g t , b t )} ∪ {0} is an orthogonal
subset of IH 0 (D);
2. X = {λ a1
λ −1
b
, . . . , λ at
λ −1
1 b
} ∪ {0} is an orthogonal subset of
t
IH 0 (C);
3. A = {a 1 , . . . , a t } and B = {b 1 , . . . , b t } are bases for
projective right ideals of C.

More on Zappa-Szép Products III: Orthogonal Subsets
The following are equivalent:
1. X = {(a 1 , g 1 , b 1 ), . . . , (a t , g t , b t )} ∪ {0} is an orthogonal
subset of IH 0 (D);
2. X = {λ a1
λ −1
b
, . . . , λ at
λ −1
1 b
} ∪ {0} is an orthogonal subset of
t
IH 0 (C);
3. A = {a 1 , . . . , a t } and B = {b 1 , . . . , b t } are bases for
projective right ideals of C.
Consequently,
D e (IH 0 (D)) = {X : X ∈ D e (IH 0 (C))}
= {X : A, B are bases for essential projective right ideals}.

Brin’s groups
Brin studied groups nV described as subgroups of the
homeomorphism group of C × · · · × C (n factors) where C is the
Cantor set.
Also describes groups nV k .

Brin’s groups
Brin studied groups nV described as subgroups of the
homeomorphism group of C × · · · × C (n factors) where C is the
Cantor set.
Also describes groups nV k .
Properties
◮ nV is simple.
◮ nV is finitely presented.
◮ nV contains a copy of every finite group.
◮ 2V and 3V are of type F ∞ (so of type F P ∞ ).
(Kochloukova, Martínez-Pérez, Nucinkis)

Brin’s groups
Brin studied groups nV described as subgroups of the
homeomorphism group of C × · · · × C (n factors) where C is the
Cantor set.
Also describes groups nV k .
Properties
◮ nV is simple.
◮ nV is finitely presented.
◮ nV contains a copy of every finite group.
◮ 2V and 3V are of type F ∞ (so of type F P ∞ ).
(Kochloukova, Martínez-Pérez, Nucinkis)
If |A| = k and C = A ∗ × · · · × A ∗ (n factors), then
R e fp (C)/σ ∼ = nV k .

Higman algebras (1974)
◮ An algebra U with k (k 2) unary operations α 1 , . . . , α k
and one k-ary operation λ satisfying
(aα 1 , . . . , aα k )λ = a
for all a, a 1 , . . . , a k
∈ U
(a 1 , . . . , a k )λα i = a i (i = 1, . . . , k)

Higman algebras (1974)
◮ An algebra U with k (k 2) unary operations α 1 , . . . , α k
and one k-ary operation λ satisfying
(aα 1 , . . . , aα k )λ = a
for all a, a 1 , . . . , a k
∈ U
(a 1 , . . . , a k )λα i = a i (i = 1, . . . , k)
◮ The free Higman k-algebra F k (x) on {x} has bases of every
finite non-zero cardinality.
◮ k = 2 (Jónsson-Tarski (1961).

Higman algebras (1974)
◮ An algebra U with k (k 2) unary operations α 1 , . . . , α k
and one k-ary operation λ satisfying
(aα 1 , . . . , aα k )λ = a
for all a, a 1 , . . . , a k
∈ U
(a 1 , . . . , a k )λα i = a i (i = 1, . . . , k)
◮ The free Higman k-algebra F k (x) on {x} has bases of every
finite non-zero cardinality.
◮ k = 2 (Jónsson-Tarski (1961).
◮ V k
∼ = the automorphism group of Fk (x).

Generalised Higman algebras 2010
Given S = {1, . . . , s} and an integer k i > 1 for each i ∈ S,
an algebra U with, for each i ∈ S, k i unary operations
αi 1, . . . , αk i
i
and one k i -ary operation λ i satisfying:
(aαi 1 , . . . , aα k i
i )λ i = a
(a 1 , . . . , a ki
)λ i α j i = a j (j = 1, . . . , k i )
for all a, a 1 , . . . , a ki
∈ U and i ∈ S.
Also have a set Σ of laws (possibly empty) given by equations
between words in the α j i

Generalised Higman algebras 2010
Given S = {1, . . . , s} and an integer k i > 1 for each i ∈ S,
an algebra U with, for each i ∈ S, k i unary operations
αi 1, . . . , αk i
i
and one k i -ary operation λ i satisfying:
(aαi 1 , . . . , aα k i
i )λ i = a
(a 1 , . . . , a ki
)λ i α j i = a j (j = 1, . . . , k i )
for all a, a 1 , . . . , a ki
∈ U and i ∈ S.
Also have a set Σ of laws (possibly empty) given by equations
between words in the α j i
Kochloukova, Martínez-Pérez and Nucinkis use free generalised
Higman algebras to get their results on 2V and 3V .

Generalised Higman algebras 2010
Given S = {1, . . . , s} and an integer k i > 1 for each i ∈ S,
an algebra U with, for each i ∈ S, k i unary operations
αi 1, . . . , αk i
i
and one k i -ary operation λ i satisfying:
(aαi 1 , . . . , aα k i
i )λ i = a
(a 1 , . . . , a ki
)λ i α j i = a j (j = 1, . . . , k i )
for all a, a 1 , . . . , a ki
∈ U and i ∈ S.
Also have a set Σ of laws (possibly empty) given by equations
between words in the α j i
Kochloukova, Martínez-Pérez and Nucinkis use free generalised
Higman algebras to get their results on 2V and 3V .
Associated monoid presentation 〈{α j i : 1 i s; 1 j k i}|Σ〉.
In particular, the set of laws for the generalised Higman algebra
with automorphism group nV gives the presentation for
A ∗ × · · · × A ∗ (n factors) with |A| = 2.

Brown-Stein groups
Let P ⊆ R >0 be a finitely generated multiplicative group.
Let A be a ZP -submodule of R with P A = A.
Let l ∈ A, l > 0.
G(l, A, P ) is the group of right continuous bijections of [0, l)
that are piecewise linear with a finite number of discontinuities
and singularities, all in A, slopes in P , and mapping A ∩ [0, l)
to itself.
Theorem (Stein)
G(1, Z[ 1 k ], 〈k〉) ∼ = V k
The commutator subgroup of G(l, A, P ) is simple.

Brown-Stein groups II
Stein shows that we can take P to have a basis (as a free
abelian group) {k 1 , . . . , k s } with the k i being positive integers
and A = Z[ 1 k 1
, . . . , 1 k s
].
For G = G(l, A, P ) with the above P and A, Martínez-Pérez
and Nucinkis give a set of laws for a generalised Higman algebra
with automorphism group isomorphic to G. Here we have
S = {1, . . . , s} and associated integers k i , i = 1, . . . , s. The laws
are:
(α 1 i α1 j , . . . , α1 i αk j
j , α2 i α1 j , . . . , α2 i αk j
j , . . . , αk i
i α1 j , . . . , αk i
i αk j
j ) =
(α 1 j α1 i , . . . , α1 j αk i
i , α2 j α1 i , . . . , α2 j αk i
i , . . . , αk j
j α1 i , . . . , αk j
j αk i
i )
where i ≠ j and i, j ∈ {1, . . . , s}.