HOMOMORPHISM

1
HOMOMORPHISM

• A mapping that preserves algebraic structure

2
range
domain
x
T(x)
y
T(y)
x y
T(x) T(y)
Operation
Operation
3
homomorphism
range
domain
x
T(x)
y
T(y)
x y
T(x) T(y)
Operation
Operation
T( xy ) T(x) T(y)
4
DOMAIN
a d b e c f
RANGE
P Q R
T(ec) T(e) T(c)
5
If we rearrange the order of the elements of the
table on the left we can clearly see the effect
of the homomorphism - the preservation of the
structure of the domain.
6
DOMAIN
a d b e c f
RANGE
P Q R
7
kernel
DOMAIN
identity in range
a d b e c f
RANGE
P Q R
8
DOMAIN
a d b e c f
RANGE
P identity Q R
a d
kernel
ab db
ac dc
b e
c f
A coset of the kernel
A coset of the kernel