Below is a clear, simple, and exam-oriented explanation of Cyclic Semigroups and Submonoids, perfect for BCA/MCA/Engineering/Discrete Mathematics.

⭐ CYCLIC SEMIGROUPS

A cyclic semigroup is a semigroup that can be generated from one single element by repeated application of the operation.

Just like a cyclic group is generated by powers of one element,
a cyclic semigroup is generated by repeated products (or operation) of a single element.

⭐ Definition

A semigroup ( S ) is cyclic if:

[
S = \langle a \rangle = {a, a^2, a^3, \ldots}
]

for some element ( a \in S ).

Here:

[
a^1 = a, \quad a^2 = aa, \quad a^3 = aa*a, \ldots
]

⭐ Properties of a Cyclic Semigroup

1️⃣ It has at most one generator (only one element produces all others).
2️⃣ It has at most one idempotent element (element e such that ( e^2 = e )).
3️⃣ Eventually, powers of (a) repeat → leads to a pattern (useful in automata theory).
4️⃣ Can be finite or infinite.

⭐ Examples of Cyclic Semigroups

✔ Example 1: Natural numbers under addition

Take element 1:

[
\langle 1 \rangle = {1,2,3,\ldots}
]

Operation is +, so:

[
1, 1+1 = 2, 1+1+1 = 3, \ldots
]

→ Infinite cyclic semigroup.

✔ Example 2: Positive powers of a matrix

Let
[
A = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}
]

Then
[
A, A^2, A^3, \ldots
]

forms a semigroup.

✔ Example 3: Strings under concatenation

Alphabet Σ = {a}

[
\langle a \rangle = {a, aa, aaa, \ldots}
]

→ Cyclic semigroup generated by “a”

✔ Example 4: Modulo arithmetic

⟨2⟩ in semigroup ((\mathbb{Z}_7, \times)):

[
2, 2^2 = 4, 2^3 = 8 \equiv 1,\ 2^4 = 2,\ldots
]

⭐ Structure of Finite Cyclic Semigroups

If the sequence of powers starts repeating after some point:

[
a^k = a^m
]

we get:

an initial chain (non-repeating part)
then a cycle (repeating loop)

This is known as the index-period decomposition:

Index (λ) = number of distinct powers before repetition
Period (μ) = length of repeating cycle

This structure is extremely important in automata theory and computation.

⭐ SUBMONOIDS

A submonoid is a subset of a monoid that is itself a monoid.

⭐ Definition of Submonoid

Let ( (M, *) ) be a monoid with identity ( e ).

A subset ( N \subseteq M ) is a submonoid if:

1️⃣ Closure:
[
a, b \in N \Rightarrow a * b \in N
]

2️⃣ Identity:
[
e \in N
]

Unlike groups, inverses are NOT required.

⭐ Examples of Submonoids

✔ Example 1: Even natural numbers

Monoid: ( (\mathbb{N}, +) ) with identity 0
Submonoid:
[
2\mathbb{N} = {0, 2, 4, 6, \dots}
]

Closed under +
Contains identity 0
→ Submonoid

✔ Example 2: Strings (Σ*, concatenation)

Let Σ = {a, b}.
Monoid: Σ* under concatenation with identity = ε (empty string).

Submonoid:
[
{a^n : n \ge 0} = {\varepsilon, a, aa, aaa, \ldots}
]

→ Closed under concatenation
→ Contains ε
→ Submonoid
→ Also cyclic (generated by “a”)

✔ Example 3: Multiples of a number

Monoid: (ℕ, ×) with identity = 1

Submonoid generated by 3:

[
{1, 3, 9, 27, \ldots}
]

⭐ How to Test a Submonoid (Submonoid Test)

A non-empty subset ( N ) of monoid ( M ) is a submonoid if:

1️⃣ ( e \in N ) (identity is included)

2️⃣ Closed under operation
[
a,b \in N \Rightarrow a*b \in N
]

No need to check for inverses or associativity
(because those are inherited).

⭐ Relation between Cyclic Semigroups and Submonoids

If M is a monoid and a ∈ M, then:

[
\langle a \rangle = {e, a, a^2, a^3, \ldots}
]

is always a submonoid of M.

Thus:

👉 Every cyclic semigroup inside a monoid is a cyclic submonoid.

⭐ Exam-Oriented Summary