The algebraic theory of partial symmetries
(or why groups aren’t enough)
Mark V Lawson
Heriot-Watt University
and the
Maxwell Institute for Mathematical Sciences
Edinburgh, Scotland
UK
May 2012
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1. Semigroups and monoids
A semigroup is a set equipped with an associative binary operation.
A monoid is a semigroup with an identity.
Subsemigroups and submonoids are defined in
the usual way.
I shall now describe a couple of examples of
semigroups.
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2. The natural numbers with respect to
addition
A simple question: describe all submonoids and
their structure.
A special case of this problem is the following
from recreational mathematics.
You have an unlimited supply of 3 cent stamps
and 5 cent stamps. By combining them, you
can make up various denominations and so can
send your letters to anywhere from Fiji to the
Faroes.
For example, 11 cents in value can be obtained
by taking two 3 cent stamps and one 5 cent
stamp. And so on.
Now the question: what is the largest value
you cannot make in this way?
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This is called the theory of numerical semigroups. In particular, there are applications to
algebraic geometry.
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3. Combinatorics on words
We now generalize the monoid of natural numbers to non-commutative monoids called free
monoids.
An alphabet is a finite set of symbols.
A string is a finite sequence of symbols from
an alphabet, with ε denoting the empty string.
If A is an alphabet then A∗ is the set of all
strings over A.
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Given x, y ∈ A∗ the string xy is the concatenation of x and y.
The set A∗ enjoys some additional properties.
Observe that (xy)z = x(yz). Thus concatenation is associative.
Also εx = x = xε. Thus the empty string is the
identity for the operation of concatenation.
The monoids of the form A∗ are called free
monoids.
If A consists of exactly one symbol then A∗ is
just N with respect to addition.
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Application 1: codes
For example, {a, ba, b2} and {a2, ab, b} are both
maximal prefix codes.
The theory of codes is the theory of free submonoids of free monoids.
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Application 2: syntactic monoids
A language is a set of strings over an alphabet.
Example: Welsh
Alphabet all words from your favourite Welsh
dictionary.
Language W all grammatically correct sentences
in Welsh constructed using this dictionary, such
as
iechyd da i chwi yn awr ac yn oesoedd
From languages we can construct monoids.
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Let L ⊆ A∗ be a language. Strings x, y ∈ A∗
belong to the same grammatical category with
respect to the language L, and we write x σL y,
if they occur in exactly the same contexts in
the language.
This means
uxv ∈ L ⇔ uyv ∈ L.
So, here, u v is the context in which the
string x occurs.
If L were a natural language, then grammatical
categories would be things like nouns (or possibly special kinds of nouns: say, animate nouns,
or feminine nouns etc), verbs, noun phrases
etc.
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If x ∈ A∗ we denote by [x]L the set of all strings
in A∗ that are in the same grammatical category as x. Thus [x]L should be regarded as a
grammatical category.
The set of all grammatical categories of L is
written A∗/σL
We can multiply grammatical categories by defining
[x]L[y]L = [xy]L.
For example, the grammatical category nounphrase might be obtained by multiplying the
grammatical categories determiner, adjective
and noun together in that order.
In this way, the set of all grammatical categories of L becomes a monoid called the syntactic monoid of the language L.
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There are many other applications of free monoids:
• Finite semigroup theory and its connections with finite state automata and regular languages.
• Symbolic dynamics and through that to
sundry mathematical fields.
See the work of J. Rhodes (who exists) and
M. Lothaire (who doesn’t).
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This talk will now deal with a third example:
the theory of inverse semigroups.
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4. Inverse semigroups
A semigroup S is said to be inverse when for
each element s ∈ S there exists a unique element s−1 such that the following two equations
hold
s = ss−1s and s−1 = s−1ss−1.
Two immediate examples.
All groups.
All meet semilattices.
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Basic theory:
(s−1)−1 = s and (st)−1 = t−1s−1
Elements of the form ss−1 and s−1s are idempotents; that is, equal to their square.
Idempotents commute with each other.
Groups are the inverse semigroups with exactly
one idempotent (and so groups are degenerate
inverse semigroups).
Meet semilattices are the inverse semigroups
in which every element is an idempotent.
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Inverse subsemigroups are defined in the usual
way as are homomorphisms and isomorphisms.
Ky motivating example.
Let X be a non-empty set. Denote by I(X) the
set of all partial bijections of X. Then I(X) is
an inverse monoid, called a symmetric inverse
monoid.
Theorem (Wagner-Preston) Every inverse semigroup is isomorphic to an inverse subsemigroup
of a symmetric inverse monoid.
Inverse semigroups are viewed as the theory
of partial symmetries just as groups are the
theory of symmetries.
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The goal of the rest of this talk is to describe
one interesting example of an inverse semigroup in depth.
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5. The polycyclic inverse monoid P2
I shall define this inverse monoid as a set of
partial symmetries of a particular fractal.
Fractals have become part of the wall-paper of
everyday life. The term was coined by Benoˆıt
Mandelbrot in 1975, but his examples were
drawn from developments in late 19th and early
20th century analysis.
A formal definition is difficult, but self-similarity
is an ingredient in what we regard as a fractal.
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I shall take one of the simplest fractals: the
Cantor set C.
This is constructed by starting with the closed
unit interval [0, 1] and succesively removing open
middle-thirds ad infinitum.
The self-similarity properties of this set are
manifested by two maps p, q: C → C given by
2+x
x
p(x) = and q(x) =
3
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and their respective ‘inverses’ p−1 and q −1.
The polycyclic monoid on two generators is
defined as the inverse submonoid of I(C) generated by p, q, p−1, q −1.
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More concretely, the elements of the Cantor
set may be identified with the right-infinite
strings over a two-letter alphabet (a + b)ω .
This leads to the following more convenient
description of P2.
Every non-zero element of P2 is of the form
yx−1 where x and y are elements of the free
monoid on {a, b}.
The product of two elements yx−1 and vu−1
is zero unless x and v are prefix-comparable in
which case
yx−1 · vu−1 =
(
yzu−1
if v = xz for some z
y(uz)−1 if x = vz for some z
The elements of P2 can be thought of as a
combination of two operations on a pushdown
stack: xy −1 means ‘pop y and push x’.
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I shall now explain how P2 arises in algebraic
linguistics.
I shall define a family of languages called parentheses languages.
Our alphabet An will consist of n matching
pairs of brackets: x1, x
¯1, . . . , xn, x
¯n.
Such languages arise naturally in many ways.
For small n, it is more convenient to use different kinds of brackets. For example, when
n = 2, we could use, say, (, ), [, ].
The language Ln is the set of all correct bracketing sequences. This can be made precise,
but I shall make do with an example:
( [ ( ) ] {, } [ ], ) ( )
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Parenthesis languages are of a type known as
context-free (CF), which are amongst the most
important classes of languages, and play a distinguished role in their theory.
Theorem [Chomsky-Sch¨
utzenberger] Every CF
language is an alphabet image of the intersection of a language of type Ln for some n and
a local language.
This tells us that parenthesis languages are the
templates from which all CF languages can be
constructed.
Local languages are very simple kinds of regular languages essentially arising from finite directed graphs.
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We are interested in the syntactic monoid of
L2.
This can be computed using the minimal automaton of L2.
This machine is an infinite binary tree (with an
additional dump state). The syntactic monoid
of L2 is just the transition monoid of this machine.
In particular, x1x
¯1 = 1 and x2x
¯2 = 1 and
x
¯2x1 = 0 and x
¯1x2, where 0 comes from the
dump state.
Theorem The syntactic monoid of L2 is P2.
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6. A completion of P2
The way I have defined P2 it comes equipped
with an action on (a + b)ω .
I am now going to write down three equations,
the Cantor relations, that summarize the important properties of P2 and its action.
(1). 1 = p−1p = q −1q
(2). 0 = p−1q = q −1p
(3). 1 = pp−1 + qq −1
where 1 means the identity function defined on
C.
I claim that where you see these equations,
you see self-similarity analogous to that of the
Cantor set.
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Examples
1. Let N be the set of natural numbers and
E and O the sets of even and odd numbers,
respectively. Let p: N → E be the doubling map,
and let q: N → O be the doubling-and-add-one
map. Cantor’s relations hold and tell us that
ℵ0 + ℵ0 = ℵ0.
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2. Let R be a unital ring that contains elements p, p−1, q, q −1 satisfying Cantor’s relations. Let M2(R) denote the set of all 2 × 2
matrices over R. We expect M2(R) to be ‘bigger than’ R.
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Define
v=
p−1
!
q −1
and h =
p
q
!
.
Observe that
vht =
1 0
0 1
!
and htv = (1)
where we have utilized all the Cantor relations.
In what follows, I identify (x) and x.
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Define
Φ: M2(R) → R
by
A 7→ htAv.
Φ is an injective homomorphism
Define
Ψ: R → M2(R)
by
r 7→ vrht.
Ψ is an injective homomorphism
Φ and Ψ are mutually inverse.
Conclusion: M2(R) is isomorphic to R.
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3. Banach-Tarski paradox. A closed ball B
in R3 can be partitioned into two pieces B1
and B2 in such a way that each piece Bi is
piecewise congruent to B.
Subsets X and Y of R3 are said to be piecewise
congruent if X can be partitioned into a finite
number of pieces that can be moved in space
using only translations and rotations and then
glued back together to yield Y .
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We may glue suitable elements of P2 together.
Here is an example.
Consider
α = a2a−1 + ab(ba)−1 + bb−2.
We have that
• a2a−1: aAω → a2Aω is given by aw 7→ a2w.
• ab(ba)−1: baAω → abAω is given by baw 7→
abw.
• bb−1: b2Aω → bAω is given by b2w 7→ bw.
The set {a, ba, b2} is a maximal prefix code as
is the set {a2, ab, b}.
We may draw a picture of what α is doing.
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A final theorem
By extending the previous example, the polycyclic inverse monoid P2 may be completed to
an inverse monoid C2 called the Cuntz inverse
monoid.
This inverse monoid is closely related to a C ∗algebra, the Cuntz C ∗-algebra O2.
The group of units of C2 is Thompson’s group
V, an infinite finitely presented simple group.
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Envoi
Its soon, no sense, that faddoms the herts o
men,
And by my sangs the rouch auld Scots I ken
Een herts that hae nae Scots’ll dirl richt thro
As nocht else could — for here’s a language
rings
Wi datchie sesames, and names for nameless
things.
Gairmscoile, by Hugh MacDiarmid
Tapadh leat!
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