Definition 1 (homomorphism Semigroup)

Let (S, ∗) and (T, ⋆) be semigroup. Mapping f: S ⟶ T is called a homomorphism Semigroup if for every x, y in S, satisfies

f (x ∗ y) = f (x) ⋆ f (y)

Example 1:

The set of all enumeration numbers ℕ + = {0, 1, 2, 3, …} is reviewed. For example

S = (ℕ +, +) and T = (ℕ +, +)

It is easy to show that both are semigroups.

Define the mapping h: S ⟶ T, where h (n) = 4n, for each n in S.

h (a + b) = 4 (a + b) = 4a + 4b = h (a) + h (b)

so f is a homomorphism semigroup

Theorem 1

Let (S, ∗) and (T, ⋆) be semigroup. If 𝛼: S ⟶ T is a semigroup homomorphism, then

𝛼 (x ^ n) = (𝛼 (x)) ^ n,

for every x in S and n = 1.

Aspects related to homomorphism semigroup

Review the semigroup (S, ∗) and (T, ⋆) and homomorphism f: S ⟶ T. Let X ⊆ S, for mapping f: S ⟶ T then

f (X) = {f (x) | x ∊ X}

Lemma 2

For example f: S ⟶ T homomorphism semigroup. If X ⊆ S, then

f ([X]S) = [f (X)]T

as a result:

If A ≤ S and an f: S ⟶ T homomorphism semigroup, then (A) ≤ T.

Lemma 4

If f: S ⟶ T and g: T ⟶ U are homomorphisms semigroup, then

gf: S ⟶ U

is also a homomorphism semigroup.

Restrictions on a function

Let (S, ∗) and (T, ⋆) be semigroup. For functions f: S ⟶ T, denoted f⥔X is intended as a limitation of function f on X (with X subsets of S), i.e.

f⥔X: X ⟶ T,

which is defined by (f⥔X) (x) = f (x), for every x ∊ X.

Theorem 5

For example f, g: S ⟶ T two homomorphisms semigroup and X ⊆ S, then

f⥔X = g⥔X if and only if f⥔ [X] S = g⥔ [X] S

Types of homomorphisms semigroup

Let (S, ∗) and (T, ⋆) be semigroup. Mapping f: S ⟶ T is called a semigroup homomorphism if for every x, y in S, satisfies

f (x ∗ y) = f (x) ⋆ f (y)

1. Monomorphism, denoted f: S ↪ T, if f injective (1–1), i.e. if for any x, y ∊ S with f (x) = f (y) then x = y.
2. Epimorphism, denoted f: S ↠ T, if f is objective (on), i.e. for every y ∊ T, there is x ∊ S, such that y = f (x).
3. Isomorphism, denoted f: S ↣ T, if f is both a monomorphism and an epimorphism.
4. Endomorphism if T = S.
5. Automorphism if f is isomorphism and endomorphism.

Lemma 6

Let (S, ∗) and (T, ⋆) be semigroup.

If f: S ⟶ T is a isomorphism semigroup, then f ^ –1: T ⟶ S is also a isomorphism semigroup.

Definition 3

• An S semigroup is said to be embeddable on another T semigroup, if there is a monomorphism, 𝛼: S ↪ T.
• Semigroup S is said isomorphic to T, denoted by S ≅ T, if there is an isomorphism 𝛼: S ↣ T.

Theorem 7

suppose S is a semigroup then

• The set of all endomorphisms of S forms monoid against the operation of the functional composition.
• The set of all automorphisms of S forms a group against the operation of the composition of functions.