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An Introduction to Algebra
Tom Bourne
University of St Andrews
PIMS, 26th January 2015
Tom Bourne
An Introduction to Algebra
Semigroups, Monoids and Groups
Definition
A set X equipped with a binary operation ◦ : X × X → X is a
semigroup if:
x ◦ (y ◦ z) = (x ◦ y ) ◦ z ∀x, y , z ∈ X ;
Tom Bourne
An Introduction to Algebra
Semigroups, Monoids and Groups
Definition
A set X equipped with a binary operation ◦ : X × X → X is a
monoid if:
x ◦ (y ◦ z) = (x ◦ y ) ◦ z ∀x, y , z ∈ X ;
∃1X ∈ X such that 1X ◦ x = x = x ◦ 1X ∀x ∈ X ;
Tom Bourne
An Introduction to Algebra
Semigroups, Monoids and Groups
Definition
A set X equipped with a binary operation ◦ : X × X → X is a
group if:
x ◦ (y ◦ z) = (x ◦ y ) ◦ z ∀x, y , z ∈ X ;
∃1X ∈ X such that 1X ◦ x = x = x ◦ 1X ∀x ∈ X ;
∀x ∈ X ∃y ∈ X such that x ◦ y = 1X = y ◦ x.
Tom Bourne
An Introduction to Algebra
Semigroups, Monoids and Groups
Definition
A set X equipped with a binary operation ◦ : X × X → X is a
group if:
x ◦ (y ◦ z) = (x ◦ y ) ◦ z ∀x, y , z ∈ X ;
∃1X ∈ X such that 1X ◦ x = x = x ◦ 1X ∀x ∈ X ;
∀x ∈ X ∃y ∈ X such that x ◦ y = 1X = y ◦ x.
We have the following inclusions: groups ⊂ monoids ⊂ semigroups.
Between us we work with the above structures in a variety of
different settings, and, generally, we do not consider more
complicated examples such as rings and fields.
Tom Bourne
An Introduction to Algebra
Equivalence Relations and Congruences
Definition
A binary relation ∼ on a set X is an equivalence relation if and
only if for all x, y , z ∈ X we have:
x ∼ x (reflexive);
if x ∼ y then y ∼ x (symmetric);
if x ∼ y and y ∼ z then x ∼ z (transitive).
Definition
A congruence is an equivalence relation on an algebraic structure
that is compatible with the multiplication defined on that structure.
For example, if (G , ◦) is a group then a congruence on G is an
equivalence relation ≡ satisfying
g1 ≡ h1 and g2 ≡ h2 ⇒ g1 ◦ g2 ≡ h1 ◦ h2 ∀g1 , g2 , h1 , h2 ∈ G
.
Tom Bourne
An Introduction to Algebra
Green’s Relations
Green’s relations, written as L, R, J , H and D, are semigroup
equivalences defined in terms of principal ideals. We define the
relations over the monoid S 1 , which is the semigroup S with an
identity adjoined if necessary.
For elements a, b in a semigroup S, we define Green’s relations by:
a L b ⇔ S 1 a = S 1 b;
a R b ⇔ aS 1 = bS 1 ;
a J b ⇔ S 1 aS 1 = S 1 bS 1 ;
a H b ⇔ a L b and a R b;
a D b ⇔ ∃c ∈ S such that a L c and c R b.
Unfortunately, Green’s relations tell us nothing useful when dealing
with groups.
Tom Bourne
An Introduction to Algebra
Subgroups, Cosets and Normality
Definition
A subset H of a group G is a subgroup if H is a group in its own
right with binary operation inherited from G . We write H ≤ G .
Let G be a group and H ≤ G . For g ∈ G , the left coset of H in G
with respect to g is the set gH = {gh | h ∈ H}. We define right
cosets similarly.
Definition
A subgroup N of G is normal if and only if gN = Ng ∀g ∈ G ; that
is, the sets of left and right cosets coincide. We write N / G .
Tom Bourne
An Introduction to Algebra
Simplicity and Quotient Groups
Definition
A group G is simple if its only normal subgroups are the trivial
group and G itself.
Let G be a group and N / G . Let G /N denote the set of cosets of
N in G . Define a binary operation on G /N by (gN)(hN) = (gh)N.
Definition
The set G /N together with the binary operation defined above is
the quotient or factor group of N in G .
Essentially, we have placed an equivalence relation on G and
partitioned the set into its equivalence classes to form G /N.
Tom Bourne
An Introduction to Algebra
Free Objects I
Let A be an alphabet; that is, a non-empty finite set of letters. A
word (over A) is a finite sequence of letters from A.
The set of all non-empty words equipped with the operation of
word concatenation is the free semigroup on A, denoted A+ .
If we also include the empty word of length zero then we have the
free monoid on A, and we denote this by A∗ .
For the free group, we take an alphabet A and the corresponding
set of ‘inverse’ symbols A−1 . Let B = A ∪ A−1 , and define a word
to be a finite sequence of letters from B. If a letter and its inverse
appear adjacent in a word then we identify them with the empty
word to form an equivalent reduced word. The set of all reduced
words (including the empty word) equipped with the operation of
word concatenation is the free group on A.
Tom Bourne
An Introduction to Algebra
Free Objects II
In general, we refer to an arbitrary structure as free if it is
isomorphic to the corresponding free structure.
The free semigroup/monoid forms the basis of formal language
theory as we define a language over an alphabet A to be a subset
of the free semigroup/monoid on A.
For example, if A = {a, b} then the set of all words starting with
an a and ending with a b, denoted aA∗ b, and the set
{an b n | n ≥ 0} are both languages over A.
Tom Bourne
An Introduction to Algebra
Graphs
Definition
A graph is an ordered pair G = (V , E ), where V is a set of vertices
and E is a set of unordered pairs from V × V . The elements of E
are edges.
[Too lazy to code and too forgetful to remember: draw graph now.]
If we insist that the elements of E are ordered; that is, the edge
(1, 2) is not the same as (2, 1), then the graph is directed.
We use (directed) graphs to represent other structures...
Tom Bourne
An Introduction to Algebra
Automata
Definition
The common components of an automaton, denoted
A = (S, A, δ, . . .), are:
a finite set of states S;
an input alphabet A;
a transition function δ.
The addition of other features, such as start/final states, output
alphabets, and stacks, together with a defined domain and range
for δ allow the machine to process different information.
For example, finite state automata correspond to regular languages
whereas pushdown automata correspond to context-free languages.
The most well-known type of automaton is a Turing machine.
Tom Bourne
An Introduction to Algebra
Thompson’s Groups
Thompson’s groups F , T and V were introduced in the 1960s as a
potential counter-example to the von Neumann conjecture. All
three groups are infinite but finitely presented, and satisfy the
inclusions F ⊆ T ⊆ V .
The group F can be realised in terms of operations on ordered
rooted binary trees, which are special types of undirected graphs.
It can also be found in topological settings.
Tom Bourne
An Introduction to Algebra
Computational Algebra and GAP
Computational algebra, or symbolic computation, is the area of
study dedicated to the development of algorithms and software,
primarily for manipulating mathematical objects and expressions.
Such software includes Maple and Mathematica.
Another system is GAP (Groups, Algorithms and Programming),
which the University of St Andrews is a development centre for. It
is a system for computational discrete algebra with particular
emphasis on group theory, though numerous packages can be
added-on to the main software.
One such package is ‘Semigroups’, which is co-authored by a
selection of PhD students.
Tom Bourne
An Introduction to Algebra