Below is a clear, simple, and exam-oriented explanation of Congruence Relations on Semigroups — perfect for BCA/MCA/Engineering/Discrete Structures or Abstract Algebra exams.
⭐ CONGRUENCE RELATIONS ON SEMIGROUPS
A congruence relation on a semigroup is a type of equivalence relation that is compatible with the semigroup operation.
It allows us to partition a semigroup into blocks (classes) where each block behaves like an element of a new, simpler semigroup.
Congruences lead to the construction of quotient semigroups, just like in groups or rings.
⭐ 1. Semigroup Review
A semigroup is a set ( S ) with:
✔ a binary operation
✔ associativity
No identity or inverses required.
⭐ 2. What Is a Congruence Relation?
A congruence relation on a semigroup ( (S, *) ) is a relation ( \sim ) on S such that:
1️⃣ ( \sim ) is an equivalence relation
● Reflexive
● Symmetric
● Transitive
2️⃣ It is compatible with the semigroup operation:
[
a \sim b \quad \Rightarrow \quad
ac \sim bc \quad \text{and} \quad ca \sim cb \qquad \forall c \in S
]
This is called left and right compatibility.
⭐ Example: why compatibility is needed
Suppose we have an equivalence relation, but:
[
a \sim b
]
But multiplying by some (c):
[
ac \not\sim bc
]
→ The structure breaks.
→ Classes cannot act like elements of a new semigroup.
Thus compatibility ensures semigroup structure is preserved.
⭐ 3. Formal Definition
A relation ( \theta ) on semigroup (S, *) is a congruence if:
- ( \theta ) is an equivalence relation
- For all (a, b, c \in S):
[
(a, b) \in \theta \Rightarrow (ac, bc) \in \theta
]
[
(a, b) \in \theta \Rightarrow (ca, cb) \in \theta
]
⭐ 4. Examples of Congruence Relations
✔ Example 1: Modular Congruence on Natural Numbers
Semigroup: ( (\mathbb{N}, +) ).
Define relation:
[
a \sim b \quad \Leftrightarrow \quad a \equiv b \pmod{n}
]
This is a congruence because:
- If (a \equiv b), then
[
(a+c) \equiv (b+c) \pmod{n}
] - Compatible with addition (left and right same)
Thus modular arithmetic gives a semigroup congruence.
✔ Example 2: Last Digit Relation
Semigroup: ( (\mathbb{N}, \times) )
Define:
[
a \sim b \iff a \mod 10 = b \mod 10
]
Multiplication preserves last digit relation:
If:
[
a \sim b
]
Then for any c:
[
ac \mod 10 = bc \mod 10
]
Thus this is a congruence.
✔ Example 3: Strings under concatenation
Semigroup: ( (\Sigma^*, \text{concat}) )
Define relation:
[
u \sim v \iff \text{length}(u) = \text{length}(v)
]
Check compatibility:
If length(u)=length(v), then:
- concat with same string c will increase both lengths equally.
Thus:
[
u \sim v \Rightarrow uc \sim vc
]
So this is a semigroup congruence.
(This is important in automata theory.)
⭐ 5. Examples of Relations That Are NOT Congruences
✘ Example: “Have the same first letter”
Not preserved under concatenation.
✘ Example: “Strings that end with same character”
Not preserved if you add a new letter in front.
Thus compatibility is essential.
⭐ 6. Congruence Classes
Given a congruence ( \theta ):
A congruence class of (a) is:
[
[a]_\theta = { b \in S : a \sim b }
]
These classes partition the semigroup.
⭐ 7. Quotient Semigroup
Congruence relations are used to build quotient semigroups, similar to quotient groups or quotient rings.
If ( \theta ) is a congruence on S:
[
S / \theta = { [a]_\theta : a \in S }
]
Operation:
[
[a]\theta \cdot [b]\theta = [ab]_\theta
]
This is well-defined because of the compatibility condition.
⭐ 8. Trivial Congruences
There are always two extreme congruences:
✔ Identity Congruence (smallest)
[
a \sim b \iff a = b
]
Quotient = original semigroup.
✔ Universal Congruence (largest)
[
a \sim b \quad \forall a, b \in S
]
Quotient semigroup has only one element.
⭐ 9. Congruences in Finite Semigroups (Important)
In finite semigroups:
- Congruences correspond to certain partitions
- Must respect multiplication
- Useful in DFA (Deterministic Finite Automata) minimization
- Related to Green’s relations
⭐ 10. Applications of Congruence Relations
✔ Automata theory
- Myhill–Nerode theorem uses congruences
- DFA minimization is based on semigroup congruences
✔ Formal languages
- Equivalent strings under congruence identify equivalent states
✔ Computer Algebra
- Quotient semigroups
- Algebraic rewriting systems
✔ Number theory
- Modular arithmetic = congruence
✔ Coding theory
- Partitioning codes using congruences
⭐ Quick Exam-Oriented Summary
Congruence Relation on Semigroup:
An equivalence relation that satisfies:
[
a \sim b \Rightarrow ac \sim bc \quad \text{and} \quad ca \sim cb
]
Properties:
• Compatibility with operation
• Partitions the semigroup
• Leads to quotient semigroups
Examples:
• Modular congruence on (ℕ, +)
• Last-digit congruence on (ℕ, ×)
• Length-based congruence on strings under concatenation
Non-examples:
• First-letter relation
• Last-character relation
Quotient Semigroup:
[
S/\theta = \text{set of equivalence classes}
]
Operation is well-defined due to compatibility.