Download Cardinality and Algebraic Structures

Cardinality and
Algebraic Structures
Dr Tijl De Bie
Dept. Eng. Maths.
email:
[email protected]
Contents
Part I (weeks 1-7)
• 1 Introduction
• 2 Combinatorics, permutations and
combinations.
• 3 Algebraic Structures and matrices:
Homomorphism, isomorphism, group,
semigroup, monoid, rings, fields
• 4 Lattices and Boolean algebras
• If time remains: some illustrations of the
use of group theory in cryptography
Part II (weeks 8-12)
• Vector spaces
Introduction
• Computer programs frequently handle real world
data.
• This data might be financial e.g. processing the
accounts of a company.
• It may be engineering data e.g. from sensors or
actuators in a robotic system.
• It may be scientific data e.g. weather data or
geological data concerning rock strata.
• In all these cases data typically consists of a set of
discrete elements.
• Furthermore there may exist orderings or
relationships among elements or objects.
• It may be meaningful to combine objects in some
way using operators.
• We hope to clarify our concepts of orderings and
relationships among elements or objects
• We look at the idea of formal structures such as
groups, rings and and formal systems such as
lattices and Boolean algebras
Number Systems
• The set of natural numbers is the infinite set
of the positive integers. It is denoted N and
can have different representations:
{1,2,3,4,........}
{1,10,11,100,101,.....}
are alternative representations of the same
set expressed in different bases. Nm is the set
of the first m positive numbers i.e. {1,2,3,4,
......,m}. N0 is the set of natural numbers
including 0 i.e. {0,1,2,3,5,....}
• Q denotes the set of rational numbers i.e.
signed integers and fractions
{0,1,-1,2,-2,3,-3,....,1/2,-1/2,3/2,-3/2,5/2,
-5/2,....,1/3,-1/3,2/3,-2/3,........}
• R is the set of real numbers i.e. the
coordinates of all the points on a line.
• Z is the set of all integers, both positive and
negative {0,1,-1,2,-2,3,-3,......}
2 Combinatorics: Permutations
• A permutation of the elements of a set A is
a bijection from A onto itself.
• If A is finite we can calculate the number of
different permutations. Suppose
A={a1,...,an}
n
choices
a1
n-1
choices
a2
1
choice
an
total number of ways of filling the n boxes
n x (n-1)x(n-2)x(n-3)..............x1=n!
nPn=n!
eg a possible permutation of {1,2,3,4,5,6} is

1 2 3 4 5 6
5 6 3 1 4 2
Composition of Permutations
• If :A A and :A A are permutations of A
then the composition or product .of  and
satisfies for all x in A
.x)= (x))
Notice that since both and are bijections
from A into A so is . In other words . is a
permutation of A.
• Example: Let A={1,2,3,4,5,6} then two
possible permutations are
1 2 3 4 5 6



5 6 3 1 4 2
1 2 3 4 5



3 2 6 1 4
For . we have that
1  5  4,2  6  5,3  3  6
4  1 3,5  4 1,6  2  2
1 2 3 4 5 6


. 
4 5 6 3 1 2
6
5
Cyclic Permutations
A cyclic permutation on a set A of n elements has
the form where k  n :
a1



a 2
a2
a3
a k-1
ak
ak
a1
a k +1
a k +1
a n 
a n 
For shorthand we often write a 1 a 2
ak 
 is said to be a k cycle
Example

6 1
1 4
4 2 3 5
6 2 3 5 or (6 1 4) is a cyclic
permutation
Two cyclic permutations a 1 a 2
ak 
b t  are said to be disjoint if
and b1 b2
a1,
,a k  b1 ,
, bt  
e.g. (4 5 2) and (3 1 6) are disjoint
Notice that
1 2 3 4 5 6



 1 5
5 6 3 1 4 2
4 2,6 3
Other examples are

1 2
4 2
3 4 5 
1 2


5 3 1 4 2
3 4 5
 3 5 1 4 2
5 3 1
or

1 2 3 4
2 3 1 5
5 6
 1 2 3 4
4 6
5 6
Can you spot a product of disjoint cyclic
permutations equivalent to the following
permutation ?

1 2
1 7
3 4
4 6
5 6 7
2 3 5
• Theorem: Every permutation of a finite set A
can be expressed as a combination of disjoint
cycles.
Structure underlying permutations
Note that the following hold:
(1) The product of two permutations is a uniquely
determined permutation of the same set.
(2) The composition of permutations is
associative.
(3) The permutation
a1 a2
a n 


I=
a 1 a 2
a n 
is called the identity permutation and has the
property that I. = .I = 
(4) For every permutation
a1


=
b1
b1


 =
a 1
1
a2
b2
b2
a2
a n 
there is an inverse
b n 
bn 
a n  such that
1
1
.   .  I
Combinations
• When we think about combinations we do not
allow repeats and unlike permutations we do
not consider order.
• Combinations look at the number of different
ways of picking a subset of k elements from a
set of n elements.
• Think of the number of ways of picking a list
of k distinct elements of n
no. of choices
n
n-1
n-k-2 n-k-1
places
= n(n-1)(n-2) ........... (n-k-1) = n!/(n-k)!
For each possible list there are k! permutations
so since we are not interested in order we
should divide the above by k!.
C(n,k) = Cnk = n!/(n-k)!k!
• Example: Choosing 2 elements from
{a,b,c,d}
{a,b},{a,c},{a,d},
{b,c},{b d},{c,d}
C(4,2)= 4!/(2! 2!) =6
Combinations with Repetitions
We could also consider combinations with
repetitions. With repetitions the number of
distinct combinations of k elements chosen
from n is:
C(n+k-1,k)= (n+k-1)!/k!(n-1)!
Number of different throws of 2 identical
dice
(1 1)(2 2)(3 3)(4 4)(5 5)(6 6)
(1 2)(1 3)(1 4)(1 5)(1 6)
(2 3)(2 4)(2 5)(2 6)
(3 4)(3 5)(3 6)(4 5)(4 6)(5 6)
C(7,2)=21
Algebraic Structures
• When we consider the behaviour of
permutations under the composition operation
we noticed certain underlying structures.
• Permutations are closed under this operation,
they exhibit associativity, an identity element
exists and an inverse exists for each
permutation
• These properties define a general type of
algebraic structure called a group.
• In this section we shall look at groups in more
detail as well as other similar algebraic
structures such as semigroups and monoids.
• Later we will progress to consider more
complex algebraic structures such as rings,
integral domains and fields.
• We will see that many real life situations are
examples of these algebraic structures
Groups
A group G,  or G,  is a set G with binary
operation  which satisfies the following properties
1.  is a closed operation i.e. if a  G and
b G then a  b G
2. a,b,c G a  b  c   a  b  c this is the
associative law
3. G has an element e, called the identity, such that
a G a  e = e  a = a
4. a G there corresponds an element
a-1 G such that a a-1  a-1 a = e
Example:
The set of all permutations of a set A
onto itself is group (called the symmetric group Sn
for n elements).
Group of Symmetries of a
Triangle
Consider the triangle
l
X
O
Y
Z
n
m
We can perform the following transformations
on the triangle
1=identity mapping from the plane to itself
p=rotation anticlockwise about O through 120
degrees
q=rotation clockwise about O through 120 degrees
a=reflection in l
b=reflection in m
c=reflection in n
Let x  y denote transformation y followed by
transformation x for x and y in {1,p,q,a,b,c}
So for example p a = c
l
l
X
X
a
O
mY
l
Y
p
O
O
Z n mZ
m X
Y n
 1 p q a
b c
1 1 p q a
b c
p p q 1
c a
b
q q 1 p b c
a
a a
b c 1 p q
b b c
a q 1 p
c c
b p q 1
a
Notice the table is not symmetric
Zn
Other examples of a group
• The set of all permutations onto itself is a group
(called the symmetric group Sn)
• The sets of all invertible nxn matrices forms a
group under ordinary matrix multiplication (called
GL(n), the general linear group)
• The quaternion group:
Let G={I,-I,J,-J,K,-K,L,-L} where
I=
[ ] [ ] [ ] [ ]
1 0
0 1
j 0
, J= 0 -j
0 1
, K= -1 0
0 j
, L= j 0
Order of a group
• A finite group is a group where G is
finite
• The order of a finite group is |G|
• For example if G is the set of
permutations of a set A with n elements
then the order of G is n!
Abelian Groups
If G,  is a group and  is also commutative
then G,  is referred to as an Abelian group
(the name is taken from the 19’th century
mathematician N.H. Abel)
 is commutative means that
a,b G, a  b = b  a
Examples: R,+ , Z , and R - 0, are
abelian groups.
Why is R, not a group at all?
Modular arithmetic
• Recall a=b mod p iff p|a-b
• Notice a=b mod p iff a=kp+b for
some integer k
•  a=b mod p implies p|a-b implies
a-b=kp implies a=kp+b
•  a=kp+b implies a-b=kp implies
p|a-b implies a=b mod p
Modular addition
• Modular addition mod 6:
+
0
1
2
3
4
5
0
0
1
2
3
4
5
1
1
2
3
4
5
0
2
2
3
4
5
0
1
3
3
4
5
0
1
2
4
4
5
0
1
2
3
5
5
0
1
2
3
4
Modular multiplication
• Modular multiplication mod 7:
x
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
4
6
1
3
5
3
3
6
2
5
1
4
4
4
1
5
2
6
3
5
5
3
1
6
4
2
6
6
5
4
3
2
1
Modular multiplication
• Modular multiplication mod 6:
x
1
2
3
4
5
1
1
2
3
4
5
2
2
4
0
2
4
3
3
0
3
0
3
4
4
2
0
4
2
5
5
4
3
2
1
Modular multiplication
• Not a group! (Why not?)
• Which subset of {1,2,3,4,5} does form a group?
x
1
2
3
4
5
1
1
2
3
4
5
2
2
4
0
2
4
3
3
0
3
0
3
4
4
2
0
4
2
5
5
4
3
2
1
Modular multiplication
Theorem: If n>=2 and n|p then n has no
inverse under multiplication mod p
Prove it!
The subset of {1,…,p-1} relatively
prime to p is a group under
multiplication mod p denoted Zp*
We will clarify this on the next slides…
Modular arithmetic
• Recall Euclid’s algorithm to find the
gcd of x and y:
x=k1y+r1
The old remainder is divided
y=k2r1+r2
by the new one repeatedly
until the remainder is 0
r1=k3r2+r3
The gcd is the last non zero
…
remainder
rn-2 =kn-1rn-1+rn
rn-1=knrn
From this… Theorem:
There exist integer a and b such
ax+by=gcd(x,y)
Modular arithmetic
• An element n has an inverse n-1
under multiplication mod p for
which
n. n-1 =1 mod p
if and only if (iff) n is relatively
prime to p.
• Prove this!
• Clearly then if p is prime then every
element will have an inverse.
Groups in logic
Consider exclusive or defined by
A⊕ B ≡(¬A∧ B)∨ (A∧ ¬B)
• {t,f} is an abelian group under
exclusive or.
• What is the identity?
• What is the inverse of t (and f)?
Two show that an algebraic system is a group we
must show that it satisfies all the axioms of a group.
Question: Let A,,, be a Boolean algebra
so that A is a set of propositional elements,  is
like ‘or’,  is like ‘and’ and
is like ‘not’. Show
that A, is an abelian group where
a,b  A a  b = a  b  a  b 
Answer:
(1) Associative since a  b   c = a  b  c
prove this ?
(2) Has an identity element 0 (false) since
a a  0 = a  0 a  0  a  1  0
 a 0 = a
(3) Each element is its own inverse
a  a = a  a   a  a   0  0  0
(4) The operation commutes a  b = b  a
prove this ?
Iterated operations
• a=a1
• a◦a=a1
• a◦a◦a=a2
• a◦a…◦a=ak
• (Why is this unambiguously
defined?)
Cyclic groups
A group G is cyclic if there exists a∈G such
that for any b∈G there is an integer k≥0
such that ak=b.
I.e. Every element of G is some power of a.
Element a is called the generator of G denoted
G=<a>
Example:
• <{1,-1},×>=<-1> since –12=1, -13=-1
Order of a cyclic
permutation group
• (1 2 … p)
• Show that the order is equal to p
• [Show by making a drawing…]
Weaker structures
• An Abelian group is a strengthening of the
notion of group (i.e. requires more axioms
to be satisfied)
• We might also look at those algebraic
structures corresponding to a weakening of
the group axioms
Semigroup ⊆ monoid ⊆ group ⊆ Abelian Group
Semigroup
A,
is a semigroup if the following conditions
are satisfied:
1.  is a closed operation i.e. if a A and
b G then a  b A
2.  is associative
Example: The set of positive even integers
{2,4,6,.....} under the operation of ordinary addition
since
• The sum of two even numbers is an even number
• + is associative
The reals or integers are not semigroups under why?
Monoid
A,
is a monoid if the following conditions
are satisfied:
1. is a closed operation i.e. if a A and
b G then a  b A
2.  is associative
3. There is an identity element
Examples: Let A be a finite set of heights. Let
 be a binary operation such that a  b
is equal to the taller of a and b. Then A,
is a monoid where the identity is the shortest
person in A
true, false , is a monoid: is associative,
true is the identity, but false has no inverse
true, false , is a monoid: is associative
false is the identity, but true has no inverse
Properties of Algebraic
Structures
properties
Semigroup  monoid  group  Abelian Group
Theorem: (unique identity) Suppose that A,
is a monoid then the identity element is unique
Proof: Suppose there exist two identity elements
e and f. [We shall prove that e=f]
e = e  f since f is an identity 
= f since e is an identity

Theorem: (unique inverse) Suppose that A,
is a monoid and the element x in A has an inverse.
Then this inverse is unique.
Proof: ??
Properties of Groups
Theorem (The cancellation laws): Let G , be
a group then a,x, y G
(i) a  x = a  y  x = y
(ii) x  a = y  a  x = y
Proof: (i) Suppose that a x = a y then by axiom 3
-1
a
a has an identity
and we have that
a -1  a  x   a -1  a  y
 a -1  a  x = a -1  a  y associativity
 e  x = e  y a -1 is the inverse
 x  y identity


(ii) is proved similarly
Theorem (The division laws): Let
a group then a,x, y G
(i) a x = b  x = a-1  b
(ii) x  a = b  x = b -1  a
Proof ??

G , be
Theorem (double inverse) :If x is an element of
the group G , then
x 
-1 -1
=x
Proof:
x  is inverse of x 
 x   x  x = e  x = x
x 
-1 -1
-1
x = e
-1 -1
 x
-1 -1
-1
-1
  x  x  x associativity 
 x   e = x x is inverse of x 
 x  = x identity 
-1 -1
-1
-1 -1
-1
-1 -1
Theorem (reversal rule)
If x and y are elements of the group G , then
x y1  y-1 x -1
Proof ??
For an arbitrary element of a group G , we
can define functions  a : G  G and a : G  G
such that
x  G  a x  a  x and a x   x  a
Theorem:  a : G  G and a : G  G
are permutations of G
Proof: Consider  a
[prove 1-1] suppose for x,y in G
 a x   a y
 a  x = a  y  x = y (cancellation laws)
[Prove onto] For any y in G
 a a -1  y  a  a -1  y 
 a  a -1  y (associativity)
= e  y (a -1 is inverse of a)
= y (identity)
Corollary: In every row or column of the
multiplication table of G each element of G appears
exactly once.
Subgroups
H,  is a subgroup of the group G,  if H  G
and H,  is also a group
Examples: Q - 0, is a subgroup of R - 0,
1,1, i, -i, is a subgroup of C - 0,
Test for a subgroup
Let H be a subset of G. Then H,  is a subgroup
of G,  iff the following conditions all hold:
(1) H  
(2) H is closed under multiplication
-1
x
H

x
H
(3)
For every group G,  , G,  and e, are
subgroups
e, is called the trivial subgroup of G, 
a proper subgroup of G,  is a subgroup
different from G
A non-trivial proper subgroup is a subgroup
equal neither to G,  or to e,
Cosets
Consider a set A with a subset H. Let a A .
Then the left coset of H with respect to a is
the set of elements:
a x x H
This is denoted by a  H
Similarly the right coset of H with respect to a is
x  a x H
and is denoted by H  a
Example: Let A be the set of rotations
0 ,60 ,120 ,180 ,240 ,300  and
{0º,120º,240º}
H 
60 ,120 ,240 . Let a = 60 then
x  a x  H 60 ,180
,300

which is the right coset with respect to
60
Normal Subgroups
Let H,  be a subgroup of G,  . Then H, 
is a normal subgroup if, for any a  G , the left
coset a H is equal to the right coset H  a
      
      
      
      
      
      
      
H,  is a normal subgroup where H =  , , 
e.g.   H =  , ,    ,,
H     , ,   ,,
Theorem: In an Abelian group, every subgroup
is a normal subgroup
Coset cardinality
Theorem: For any H subset of G and any a in
G |a•H|=|H|
Proof:
By definition of Coset |a•H|≤|H|
Now suppose |a•H|<|H| then there must exist
b and c distinct elements of H such that
a•b=a•c.
But by the cancellation law this implies that
b=c which is a contradiction.
Hence |a•H|=|H|
Coset partitioning
Theorem: Let a,b∈G and let H be a
subgroup of G then either:
a•H=b•H or: a•H∩ b•H=∅
Proof:
Suppose a•H∩ b•H≠∅ then there exist s and t in
H such that a•s=b•t.
In this case a= b•t•s-1 and for an arbitrary x in H
a•x= b•t•s-1•x
Now by the inverse axiom and closure,
t•s-1•x∈H and hence b•t•s-1•x∈b•H, therefore
a•x∈b•H so that a•H⊆b•H
Similarly we can show that b•H⊆a•H
Hence if the two cosets are not disjoint then
b•H=a•H
LeGrange’s theorem
Theorem: Let H be a subgroup of finite
group G, then the cardinality of H evenly
divides the cardinality of G (i.e |H| | |G|)
Proof
Let |G|. Now for each element ai of G we can
generate a coset ai•H.
Now notice that ai∈ai•H because since H is a
subgroup, e∈H and ai•e= ai
Suppose there are m distinct cosets of H then
picking one representative ai from each this
means that:
G= a1•H∪ a2•H ∪ a3•H … ∪ am•H
LeGrange’s theorem
Now by the previous theorem it follows that
since these m cosets are distinct then they
must be disjoint.
Hence,
|G|=|a1•H|+ |a2•H| + |a3•H| … + |am•H|
Also by the cardinality theorem for cosets
they all have the same cardinality, namely
|H|. Hence, |G|=m.|H| as required
Order of an element
• Let i be the smallest integer such that
ai=e where a is an element of group
G and e is the identity element.
• If i exists we call it the order of a.
• Otherwise we say that a has infinite
order.
Subgroup generated by
an element
Theorem: For any element a of G with
finite order the set:
H={aj: for some integer j}
is a subgroup of G.
Notice: if i is the order of element a then
• ai=e
• ai+1=e•a=a1
• ai+2= a •a =a2
• ai+n=an
Example
• Let σ=(1 2 3 4), a permutation
of 4 elements
• Then {σ, σ2, σ3, σ4} is a
subgroup of the group of
permutations of {1,2,3,4}
• The order of σ is 4
• [Work it out!]
Order of elements in
finite groups
• If the group G is finite then all
elements of G have finite order:
• For any a∈G, since G is finite there
must exist i<j such that ai=aj
• a•ai-1=a•aj-1 cancellation law implies
ai-1=aj-1
• Repeated application of the
cancellation law gives a=aj-i+1
• a•e=a•aj-i implies e=aj-i
Corollary of LeGrange
Theorem: The order of every element
of a finite group G, divides the order
of G
Proof...
Every element of G has finite order n
and hence generates a subgroup of
order n.
Hence by LeGrange’s theorem n
divides |G|
Isomorphism
• Two groups are isomorphic if there is a
bijection of one onto the other which preserves
the group operations i.e.
if G1 , and G2 , are groups then a bijection
f : G1  G2 is an isomorphism provided
x, y G1 f x  y   f x  f y 
Example: Consider the group of matrices
1 t 
of the form   where t R under matrix
0 1
multiplication. This is isomorphic to the group
R,+
The mapping is
1 t 
t


0
1


An isomorphism from a group onto itself is
called an automorphism.
Homomorphisms
The idea of isomorphic algebraic structures
can be readily generalised by dropping the
requirement that the functional mapping be
a bijection.
Let A,  and B, be two algebraic systems
then a homomorphism from A,  to B,
is a functional mapping f : A  B
such that
x, y A f x  y  f x f y
Example: consider the two structures














  
  
  
  
  
  
  















1
0
1
1 0 1
1 1 0
1 0 1
0 1 1
then f such that f  = 1, f = 1,f    1,f   0
f    0,f   1 is a homomorphism between
 ,,  ,,,, and 1,0,1,
Algebraic Structures with two
Operations
• So far we have studied algebraic systems with
one binary operation. We now consider systems
with two binary operations.
• In such a system a natural way in which two
operations can be related is through the property
of distributivity;
Let A, , be an algebraic system with two
binary operations  and  . Then the operation
 is said to distribute over the operation  if
x, y, z A x  y  z  x y  x  z
and
y  z  x = y x  z  x
Example:  distributes over +
 distributes over 
 distributes over
Ring
An algebraic system A, , is called a ring if
the following conditions are satisfied:
(1) A,  is an Abelian group
(2) A,  is a semigroup
(3) The operation  is distributive over the
operation 
Example: Z, +, is a ring since
Z, + is an Abelian group
Z,  is a semigroup
 distributes over +
Examples of rings
<Z, +, ×> is a ring because:
• <Z, +> is an Abelian group.
• <Z, ×> is a semigroup.
• × distributes over +
The set
{[ ],a,b є R}
0 a
0 b
is a ring under matrix addition and
multiplication
{0,1,…,n-1} is a ring under addition
and multiplication mod n
Rings of polynomials
• Let the set R[x] be the set of all
polynomial of the form:
anxn+…+ a2x2+ a1x1+a0
for some n, where an,…,a0 єR
• Then R[x] is a ring under addition
and multiplication of polynomials
• In fact for any ring R you can
construct a ring of polynomials R[x]
over R
Special types of ring
A commutative ring is a ring in which  is
commutative
A ring with unity contains an element 1 such
that x A x 1 = 1 x = x where 1  0
(0 is the identity of A,  )
Example: the ring of 2x2 matrices under matrix
addition and multiplication is a ring with unity.
1 0
The element 1=I= 0 1
Division rings
• A division ring is a (not
necessarily commutative) ring
with unity, in which every
element a not equal to 0 has an
inverse a-1 such that
a•a-1= a-1•a=1
• The ring of complex matrices of
the form:
[ ]
a b
-b a
Integral Domains and Fields
A, ,
is an integral domain if it is a commutative
ring with unity that also satisfies the following
property;
x, y A x  y = 0  x = 0 or y = 0
Z, +, is also an integral domain
A, ,
is a field if:
(1) A,  is an Abelian group
(2) A - 0, is an Abelian group
(3) The operation  is distributive over the
operation 
Example:The set of real numbers with respect to
+ and  is a field.
Z, +,
is not a field. Why?
Galois fields
• For a prime number p the set
{0,1,…,p-1} is a field under
modular addition and
multiplication mod p
• A field (like this one) with finite
number of elements is called a
Galois field.
A Field is an Integral Domain
Let A, , be a field then certainly A, ,
is a commutative ring with unity. Hence, it only
remains to prove that
x, y A x  y = 0  x = 0 or y = 0
Now suppose x  y = 0 then if x=0 the above
holds. Consider the case then where x  0
Since A - 0, is an Abelian group then it
-1
must contain an inverse to x, x , for which the
following holds
y = 1 y = x  x y  x  x  y  x  0
-1
-1
-1
Now
a  0  0  a  0
 a  0  a  0 = a  0 (distributivity)
 a  0  a  0 = a  0  0 0 is identity
 a  0 = 0 cancellation laws for
Therefore y=0 as required


Properties of a ring
Theorem: if A, , is a ring. Then
x A 0  x = x  0 = 0
Proof: as for previous argument
Let -x denote the inverse of x under 
Theorem: if A, , is a ring then the following
hold
(i) -x  y = x  -y  -x  y
(ii) -x  -y  x  y
Proof: (i)
x  -x  y = 0  y (additive inverse)
 0 (by above theorem)
 x  y  -x  y = 0 (distributivity)
 -x y = -x  y  0 (division laws for )
= -x  y (additive identity)
(ii) -x  -y   x  -y  (part(i))


= --x  y  (part(i))
= x  y (double inverse)
for both (i) and (ii) the symmetric cases are
proved similarly
Property of an integral
domain
Theorem: suppose that elements a,b and c of
an integral domain satisfya  b = a  c and a  0
then b=c.
Proof:
a  b  -a  c   a  c  -a  c   0 (additive inverse)
Now - a  c   a  -c (prev. theorem)
a  b  c   0 (distributivity)
by defn. of integer domain 


 b  c  0
since a  0

 b = 0  --c (by devision law for )
 b = c double inverse 
Subrings and subfield
Subring
• If (A,⊕,•) is a ring then (H,⊕,•) is a
subring if H⊆A and
• (H,⊕,•) is a ring
Subfield
• If (A,⊕,•) is a field then (H,⊕,•) is a
subfield if H⊆A and
• (H,⊕,•) is a field
Examples: Z is a subring of R, R is a
subfield of C
Ring morphisms
A morphism between rings (A,⊕,•)
and (B,*,⊗) is a function f:A→B
such that: ∀x,y∈A
• f(x⊕y)=f(x)*f(y) and
• f (x•y)=f(x)•f(y)
From these we have that
• f(0)=0′ where 0′ is the zero of
(B,*,⊗)
• Also f(-x)=-f(x)
Special morphisms
1. An injective ring morphism is
called a monomorphism
2. A surjective ring morphism is called
an epimorphism
3. A bijective ring morphism is called a
isomorphism
Examples of morphisms
• f(a) = a mod n, is an epimorphism
(surjective ring morphism) between
Z and {0,1,…,n-1}
• For the ring of polynomials R(x),
f(p)=p(j) is an epimorphism into C,
where p(j) is obtained by substituting
j for x in the polynomial p
Galois theorem
• For every prime power pk (k=1,2,…)
there is a unique (upto isomorphism)
finite field containing pk elements
denoted by GF(pk)
• All finite fields have cardinality pk
Galois theorem:
examples
• GF(2)
+ | 0 1
--+---0 | 0 1
1 | 1 0
• GF(3)
+ | 0 1 2
--+-----0 | 0 1 2
1 | 1 2 0
2 | 2 0 1
· | 0 1
--+---0 | 0 0
1 | 0 1
· | 0 1 2
--+-----0 | 0 0 0
1 | 0 1 2
2 | 0 2 1
Partial Orderings
• We have introduced formal structure governing
the properties of various sets of elements under
one or two binary operations.
These elements can also be ordered and restricted
by binary relations.
• In this section we revise our understanding of
binary relations in a set and also introduce a
graphical notation for binary relations.
A relation R on a set A is a partial order if it
satisfies;
(1) R is reflexive x A R(x, x)
(2) R is antisymmetric
x, y  A Rx, y and Ry, x   x = y
(3) R is transitive
x, y, z A R x, y and R y, z   Rx, z
The pair (A,R) is called a partially ordered set
or poset
Example: Set of reals R with the relation 
Example: The relation  can be defined on a
Boolean algebra by;
x  y iff x  y = y ( is the logical or)
(1) Thus from the idempotent law x x = x
we find that x  x and hence the relation is
reflexive.
(2) If x  y and y  x then x y = y and y x = x
From the commutative law x y = y  x
and hence the relation is antisymmetric
(3) If x  y and y  z then
x  y = y and y  z = z
z = x  y  z
 x  y  z  (associative law)
= x z
x  z
We can think of a relation as being represented by
the set of pairs of elements which satisfy the
relation.
In this case a partial ordering on A corresponds
to a subset B of AxA satisfying
x  A x, x B
x, y A x, y  B and y, x  B  x = y
x, y, z A x, y B and y, z B  x, z B
Other examples of partial orderings:
Divisibility on N: We say that a divides b iff there
is some x in Z such that ax=b. If this divisibility
exists we write a|b. Divisibility is a partial order
on N.
Inclusion  on a set of sets X
Graphical Representations
We can represent partial orderings graphically
by means of a directed graph where the nodes
are elements of A and the directed edges give
the partial order relations.
e.g. the graph
a
b
c
d
Denotes the partial ordering on {a,b,c,d}
where
a  a, b  b, c  c, d  d
b  a, c  a, d  a, d  b
Graphical Representations of
the Axioms
Reflexive:
a
Antisymmetric: the following does not occur
a
b
Transitive:
a
b
c
Example: Divisibility relation on{2, 3, 4, 6, 8, 9, 18}
2|4 4|8 2|8 2|6 3|6 3|9 9|18 6|18 3|18 2|18
2
4
8
6
3
9
18
Example: The collection of all subsets of {a,b,c}
{a,b,c}
{a,b}
{a,c}
{b,c}
{a}
{b}
{c}

Hasse Diagrams
Notice that some of the diagrams in the previous
examples were messy and difficult to read having
many links.
We can simplify these diagrams by introducing
certain conventions.
The Hasse diagram of a partially order set is a
drawing of the points in the set (as nodes) and
some of the links of the graph of the order relation.
The rules for drawing the Hasse diagram of a partial
order are:
(1) Omit all links that can be inferred from
transitivity.
(2) Omit all loops
(3) Draw links without arrow heads
(4) Understand that all arrows would
point upwards
Here are Hasse diagrams for the two examples
given previously:
Divisibility:
18
8
4
2
6
9
3
Example: subsets
{a,b,c}
{a,b}
{a}
{a,c}
{b}

{b,c}
{c}
Incomparable Elements
Consider the Hasse diagram for divisibility on
{2,3,....,10}
10
8
6
4
2
3
9
5
7
Notice that 5 and 6 are not related in either direction
Similarly for 2 and 3
If neither R(a,b) or R(b,a) then a and b are
incomparable or not comparable
Linear or Total Order
A linear or total order on a set A is a partial order
on A in which every two elements are comparable
a,b A either R a, b or Rb,a 
5
4
3
2
Maximal and Minimal
Elements
A maximal element of A is any element t of
A such that x A R(t, x)  x = t
A minimal element of A is any element
b of A such that x A R(x, b)  x = b
Example:For the subset ordering {a,b,c} is the
maximal element and  is the minimal element
For divisibility on {2,.....,10} the maximal
elements are 6, 7, 8, 9 and 10
and the minimal elements
are 2, 3, 5 and 7
The element 4 is neither maximal nor minimal
Upper Bounds and
Lower Bounds
Let S be a subset of A then x in A is an upper
bound of S if y S Ry, x
Similarly z in A is a lower bound of S if
y S Rz, y
An element u is the least upper bound of S if
u is an upper bound of S and for every x an
upper bound of S R(u,x)
An element l is the greatest lower bound of S if
l is an upper bound of S and for every z a
lower bound of S R(z,l)
The least upper bound (lub) of S is sometimes
referred to as the supremum of S (sup S)
The greatest lower bound (glb) of F is sometimes
referred to as the infimum of S (inf S)
Lattices
A partially ordered set in which every pair of
elements has a least upper bound and a greatest
lower bound is called a lattice.
a
b
c
d
e
f
This is not a lattice since {c,d} has no lub or
glb.
A lattice in which every subset has a lub and glb
is called complete.
Every finite lattice is complete.
For a complete lattice the lub of the whole lattice
is call top and the greatest lower bound bottom
Example: Consider elements of the form (a,b,c)
where a,b and c can take the values 0 or 1. For
two such elements f and g we say that f  g
if each coefficient of f is less than or equal to
the corresponding coefficient of g
e.g. 001  011 but not 001  010 
(111)
(011)
(001)
(101)
(110)
(010)
(000)
(100)
Meet and Join
In a lattice A,  the following equations
define binary operations on A
x  y = lub x, y
x  y = glb x, y
 is called the meet operation and  is
called the join operation.
They have the following properties
Commutativity: a  b = b  a
Associativity:
a  b  c = a  b  c
Since a  b is an upper bound of a and b
a ab
Similarly for the meet
aba
Theorem
If a  b and c  d then
a  c  b d
a  c  b d
Proof
b  b  d (by definition)
d  b  d (similarly)
a  b and c  d (given)
 b  d is an upper bound of a and c
a  c is the least upper bound of a and c
a  c  b  d
Let 1 denote the lub of the whole lattice and 0
denote the glb of the whole lattice. Then
x  L 0  x  1
x 1 = 1 x  1 = x
x 0 = 0 x 0 = x
Example
Let us order the following set of numbers with the
operation “is a factor of”. A={3,9,12,15,36,45}
45
36
9
15
12
3
The join operation  is the least common multiple
The meet operation  is the greatest common divisor
Complemented Lattice
For a complemented lattice we have that for
x L there exists x  L such that:
x  x = 0 and x  x = 1
e.g.
1
a
b
a
c
b
c
0
Distributive Lattice
A lattice is distributive if:
x  y  z   x  y  x  z
x  y  z   x  y  x  z
e.g. the following lattice is not distributive
e
b
c
d
a
Since
b  d  c  b  e = b
b  d   b  c  a  a = a
Boolean Algebra
A Boolean Algebra consists of two binary
operations  and  and the unary operation
on a set B with distinct elements 0 and 1 such
that the following hold.
(1) The commutative laws:
x y = y x
x  y= y x
(2) The associative laws:
x  y   z = x  y  z 
x  y  z = x  y  z
(3) The Distributive laws:
x  y  z   x  y  x  z
x  y  z   x  y  x  z
(4) The Identity Laws:
x 0 = x
x 1 = x
(5) The Complementation Laws:
x  x =1
xx=0
Theorem
If
L, 
is a complemented distributive lattice then
L, ,, is a Boolean algebra where , and
correspond to the meet, join and complement
operations on L respectively
Proof ?
(6) The following Idempotent Laws can be derived:
x x = x
x  x= x
Proof
x  x = x  x  1 identity law (4)

 x  x  x  x (complementation law (5))
 x  x  x (distributive law (3))
= x  0 (complementation law (5))
=x
(7) The following Identity Laws can also be derived
x 1 = 1,x  0 = 0
Proof
x 1 = x  x  x  (complementation law (5))
= x  x  x (associative law (2) )
= x  x (idempotent law (6))
= 1 (complementation law (5))