A mapping that preserves algebraic structure

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Presentation transcript:

A mapping that preserves algebraic structure HOMOMORPHISM A mapping that preserves algebraic structure

range domain x T(x) y T(y) x * y T(x) # T(y) Operation: * Operation: #

homomorphism range domain T( x*y ) = T(x) # T(y) x T(x) y T(y) x * y Operation: * Operation: # T( x*y ) = T(x) # T(y)

DOMAIN a d b e c f RANGE P Q R T(e*c)= T(e) #T(c)

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.

DOMAIN a d b e c f RANGE P Q R

kernel DOMAIN a d b e c f identity in range RANGE P Q R

DOMAIN a d b e c f RANGE P = identity Q R a d = kernel a+b= d+b= a+c= d+c= b e c f A coset of the kernel A coset of the kernel