Skip to content
Home » Morphisms

Morphisms

Below is a simple, clear, and exam-oriented explanation of Morphisms — with full detail, covering semigroups, monoids, groups, and rings.
This will help you in Discrete Mathematics, Abstract Algebra, and theoretical CS.

⭐ MORPHISMS – INTRODUCTION

A morphism is a structure-preserving map between two algebraic structures.

In simple words:

A morphism is a function that keeps the operation and structure intact.

Morphisms are generalizations of:

  • Homomorphisms (groups, rings, semigroups)
  • Isomorphisms
  • Automorphisms
  • Endomorphisms

In algebra, the term morphism commonly means homomorphism.

⭐ 1. HOMOMORPHISM (CORE IDEA OF MORPHISMS)

Let (A) and (B) be algebraic structures with the same type of operation (e.g., +, ×, concatenation).

A function
[
f : A \to B
]
is a homomorphism if it preserves the operation.

For binary operation (*),

[
f(a * b) = f(a) * f(b)
]

This is the most important property.

⭐ 2. MORPHISMS IN DIFFERENT STRUCTURES

A. Morphisms of Semigroups

Let ( (S, \cdot) ) and ( (T, \cdot) ) be two semigroups.
A map ( f : S \to T ) is a semigroup homomorphism if:

[
f(ab) = f(a)f(b)
]

Identity not required since semigroups may not have identity.

✔ Example:
Strings under concatenation
[
f(a^n) = b^n
]

B. Morphisms of Monoids

Let (M) and (N) be monoids with identity elements (e_M) and (e_N).
A map ( f : M \to N ) is a monoid homomorphism if:

[
f(ab) = f(a)f(b)
]

AND preserves identity:

[
f(e_M) = e_N
]

✔ Example
[
(M, +) = (\mathbb{N}, +),\ \ (N, +) = (\mathbb{Z}, +)
]

Define ( f(n) = n ).
Identity: f(0)=0 → OK.

C. Morphisms of Groups

Let ( (G, *) ) and ( (H, *) ) be groups.
A map ( f : G \to H ) is a group homomorphism if:

[
f(a * b) = f(a) * f(b)
]

Properties preserved:

1️⃣ Identity preserved:

[
f(e_G) = e_H
]

2️⃣ Inverse preserved:

[
f(a^{-1}) = (f(a))^{-1}
]

✔ Example
( f:\mathbb{R} \to \mathbb{R}^+ ) defined as
[
f(x) = e^x
]
is a homomorphism because:
[
f(x+y)=e^{x+y}=e^xe^y=f(x)f(y)
]

D. Morphisms of Rings

For rings (R) and (S),
(f:R \to S) is a ring homomorphism if:

[
f(a+b) = f(a) + f(b)
]
[
f(ab) = f(a) f(b)
]
[
f(1_R) = 1_S \quad \text{(for unital rings)}
]

✔ Example
[
f:\mathbb{Z} \to \mathbb{Z}_n,\ \ f(k) = k \mod n
]

⭐ 3. TYPES OF MORPHISMS

Morphisms are classified based on properties of the function f.

⭐ 1. Monomorphism (Injective Homomorphism)

One-to-one (no two elements map to same element).

[
f(a) = f(b) \Rightarrow a = b
]

⭐ 2. Epimorphism (Surjective Homomorphism)

Onto (covers all target elements).

⭐ 3. Isomorphism (Bijective Homomorphism)

One-to-one and onto.
It indicates the two structures are identical in structure.

[
G \cong H
]

⭐ 4. Automorphism

Isomorphism from a structure to itself.

[
f : G \to G
]

Represents symmetries of the structure.

⭐ 5. Endomorphism

Homomorphism from a structure to itself (not necessarily bijective).

[
f : G \to G
]

⭐ 4. KERNEL OF A MORPHISM (VERY IMPORTANT)

For homomorphism ( f : S \to T ):

[
\ker(f) = { a \in S : f(a) = e_T }
]

Properties:

  • Kernel is a congruence relation in semigroups.
  • Kernel is a normal subgroup in groups.
  • Kernel is an ideal in rings.

⭐ 5. IMAGE OF A MORPHISM

[
\text{Im}(f) = { f(a) : a \in S }
]

This is always a:

  • subsemigroup in semigroups
  • submonoid in monoids
  • subgroup in groups
  • subring in rings

⭐ 6. FIRST ISOMORPHISM THEOREM (Linked to morphisms)

For semigroups, groups, rings:

[
S/\ker(f) \cong \text{Im}(f)
]

This shows:

  • Congruences arise from morphisms
  • Quotient structures are controlled by kernels

⭐ 7. Examples in Detail

✔ Example 1: Semigroup homomorphism

Strings over {a} → strings over {b}:

[
f(a^n) = b^n
]

Preserves concatenation.

✔ Example 2: Group homomorphism

[
f:\mathbb{Z} \to \mathbb{Z}_6,\ f(n)=n\mod 6
]

  • f(a+b)=f(a)+f(b)
  • Kernel = multiples of 6
  • Image = all of (\mathbb{Z}_6)

✔ Example 3: Ring homomorphism

[
f:\mathbb{Z}[x] \to \mathbb{Z},\ f(p(x)) = p(1)
]

Evaluation map.

⭐ 8. Why Morphisms Are Important?

✔ They preserve structure
✔ They classify algebraic systems
✔ They help build quotient semigroups/groups/rings
✔ They connect different algebraic systems
✔ Essential in automata theory, logic, and computer science
✔ Used in cryptography (e.g., RSA map)

⭐ Exam-Oriented Summary

Morphism:

Structure-preserving map between algebraic structures.

Homomorphism:

[
f(a*b)=f(a)*f(b)
]

Types:

  • Monomorphism (injective)
  • Epimorphism (surjective)
  • Isomorphism (bijective)
  • Automorphism (self-isomorphism)
  • Endomorphism (self-homomorphism)

Kernel:

Set of elements mapped to identity.

Image:

Set of outputs of the morphism.

Key theorem:

[
S/\ker(f) \cong \text{Im}(f)
]