Notes on Discrete Mathematics: Algebraic 2_ Homomorphic Mappings (1)

Review: On mapping

In a mapping, every image must have a primitive

• Surjection: Any image can find a primal image. An image can have multiple primitives (i.e., many-to-one, and no remaining elements in the image set);
• Injective: An image has at most one primal image. If the images are the same, the primordia must be the same; if they are different, the primordia must be different. Both the image set and the preimage set can have remaining elements);
• Bijective: injective and surjective. One – to – one correspondence between the elements of the primal image set and the image set.

Homomorphic mapping

A homomorphic map is also a mapping from algebra to algebra — a homomorphic map establishes a mapping relationship from algebra to algebra. In this relation, naturally also includes the mapping relation between carriers, operations and constant elements in two algebras

For algebra A = < S, ∗, Δ, k > A = < S *, Δ, k > A = < S, ∗, Δ, k > and A ‘= < Δ ∗ S’, ‘, ‘, ‘k > A’ = < S ‘*’, Δ ‘k’ > A ‘= < Δ ∗ S’, ‘, ‘, ‘k >, if there is f: S – > S’ f: S – S ‘f: S→S ‘makes ∀ A, B ∈S∀ A, B ∈S∀ A, B ∈S∀ A, B ∈S, both satisfy:

• 
  $f(a*b) = f(a) *’ f(b)$
• 
  $F = Δ ‘f (Δ a) (a)$
• 
  $f(k) = k’$

FFF is A homomorphic mapping from AAA to A’A ‘A ‘. AAA is homomorphic to A’A ‘A ‘A ‘A ‘A ‘A ‘A ‘A ‘

The above three conditions can be simply stated as “operation before mapping, equal to mapping before operation” : F (a∗b)f(a*b) F (a∗b) is the f mapping of the value of operation A ∗ba*ba∗b, ∗ ‘f and f (a) (b) (a) f * f (b) (a) f ∗’ f (b) is a, ba, ba, b respectively after mapping the value of f (a), f (b) f (a), f (b) f (a), f (b) ∗ ‘*’ ∗ ‘operation. It should be noted that the operations used by elements before mapping are ∗, δ *, δ ∗, δ in AAA, while the operations used by elements after mapping are ∗ ‘*’∗’ and δ ‘δ’ IN A’A ‘A ‘

Pr oduct

Homomorphic image is also an algebraic system which is carried by the set of images of all elements in the original carrier. This algebra and the proimage algebra satisfy the relation of homomorphic mappings described in the previous paper

If the FFF from A = < S, ∗, Δ, k > A = < S, *, / Delta, k > A = < S, ∗, Δ, k > to A ‘= < Δ ∗ S’, ‘, ‘, ‘k > A’ = < S ‘*’, \ ‘k’ Delta > A ‘= < Δ ∗ S’, ‘, ‘, ‘k > homomorphic mapping, Said “f (S), ∗ ‘, Δ ‘, ‘k > < f (S), *’, \ ‘Delta, k’ > < f (S), ∗ ‘, Δ ‘, ‘k > for A homomorphic image under the FFF.

F (S) = {x ∣ x = f (a), a ∈ S} f (S) = \ {x | x = f (a), a \ ‘S \} in f (S) = {x ∣ x = f (a), a ∈ S}

Pr oduct < f (S), ∗ ‘, Δ ‘, ‘k > < f (S), *’, \ ‘Delta, k’ > < f (S), ∗ ‘, Δ ‘, ‘k >, Algebraic system is A ‘= < Δ ∗ S’, ‘, ‘, ‘k > A’ = < S ‘*’, \ ‘k’ Delta > A ‘= < Δ ∗ S’, ‘, ‘, ‘k > the subalgebra (which is A theorem, the following will have to prove)

case

set
$<I, \cdot>$ ，
$I$Is a set of integers,
$\cdot$It’s ordinary multiplication over a set of integers. If you only care about whether the result of the operation is positive, negative, or zero, you can use an algebraic system to characterize the result
$<B, \odot>$Represents: where,
$B = {+, -, 0}$.
$Even though$ 为
$B$The operation table is shown in the figure below. Please structure from
$<I, \cdot>$ 到
$<B, \odot>$Homomorphic mapping of (gray dotted box is solution and annotation/interpretation)

Homomorphisms can simplify complex algebraic systems to simplify complex problems — the problem is mapping an operation on an infinite set to an operation on a carrier with only three elements.

The key here is how do you construct a mapping that works

Classification of homomorphic maps

Before we begin, let’s review three more mappings

In a mapping, every image must have a primitive

• Surjection: Any image can find a primal image. An image can have multiple primitives (i.e., many-to-one, and no remaining elements in the image set);
• Injective: An image has at most one primal image. If the images are the same, the primordia must be the same; if they are different, the primordia must be different. Both the image set and the preimage set can have remaining elements);
• Bijective: singly set and surjective. One – to – one correspondence between the elements of the primal image set and the image set.

Set FFF from A = < S, ∗, Δ, k > A = < S, *, / Delta, k > A = < S, ∗, Δ, k > to A ‘= < Δ ∗ S’, ‘, ‘, ‘k > A’ = < S ‘*’, \ ‘k’ Delta > A ‘= < Δ ∗ S’, ‘, ‘, ‘k > A homomorphic mapping. According to the different types of homomorphic mappings, homomorphic mappings can be divided into the following types:

• Single homomorphism: FFF is injective;
• Epimorphism: FFF is surjective. In this case, A’A ‘A ‘is A homomorphic image of the epimorphism FFF;
• Isomorphism: FFF is bijective, so AAA is homomorphic to A’A ‘, denoted as A Low-power feature A’A \cong A’A ‘;
• A=A ‘A=A ‘A=A ‘, then FFF is an endomorphism on AAA;
• Automorphism: A=A ‘A=A ‘A=A ‘and FFF is bijective

Ex. :

1. 
   $N$Is the set of natural numbers,
   $+$ 是
   $N$Ordinary addition on. set
   $N_k={0, 1, ,2 … , k-1}$ ，
   $+_k$ 是
   $N$On the mold
   $k$Add operation
   
   设
   $f:N\rightarrow N_k$Defined as:
   $f(x)=x(mod\space k)$
   
   Please prove that:
   $f$from
   $<N, +, 0>$ 到
   $<N_k, +_k, 0>$A full homomorphic map of
2. set
   $A={A, b, c, d}$ ,
   $A$The ☆ operations on are shown in the following table
   $a$.
   $B={0,1,2,3}$ ，
   $B$On the
   $*$The operations are as follows
   $b$. Please structure from
   $< A, do things >$ 到
   $<B, *>$Is an isomorphic mapping of

theorem

set
$f$from
$A = <S, *, Δ, k>$ 到
$A’ = <S’, *’, δ ‘, k’>$A homomorphic map of, then the homomorphic image of A
$<f(S), *’, \Delta’, k’>$ 是
$A’$The number of offspring