Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group
Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group
Contents [hide]
- 1 Discrete Mathematics – Groupoid, Semigroup, Monoid, Group & Abelian Group
- 2 Groupoid
- 3 Semigroup
- 4 Monoid
- 5 Group
- 6 Abelian Group (Commutative Group)
- 7 Summary of Algebraic Structures
- 8 Day 06Part 04 – Discrete Mathematics – Groupoid , Semi group , monoid , Group and abelian group
- 9 Semigroup, monoid and group models of groupoid identities
- 10 Section I.1. Semigroups, Monoids, and Groups
- 11 Discrete Mathematical Structures Unit-1
Discrete Mathematics – Groupoid, Semigroup, Monoid, Group & Abelian Group
In Discrete Mathematics & Abstract Algebra, different algebraic structures help in understanding mathematical operations and their properties. Let’s go step by step!
Groupoid
A groupoid is a non-empty set with a binary operation (a rule that combines two elements to form another element in the set).
Definition: A set G with a binary operation (*) is called a groupoid if:
∀a,b∈G,a∗b is also in G.\forall a, b \in G, \quad a * b \text{ is also in } G.
Example: The set of natural numbers (N, +) is a groupoid since the sum of any two natural numbers is also a natural number.
Semigroup
A semigroup is a groupoid with the associative property.
Definition: A set S with a binary operation (*) is a semigroup if:
∀a,b,c∈S,(a∗b)∗c=a∗(b∗c).\forall a, b, c \in S, \quad (a * b) * c = a * (b * c).
Example: The set of natural numbers (N, +) is a semigroup because addition is associative:
(2+3)+4=2+(3+4).(2 + 3) + 4 = 2 + (3 + 4).
Monoid
A monoid is a semigroup with an identity element.
Definition: A set M with a binary operation (*) is a monoid if:
- Associativity: (a∗b)∗c=a∗(b∗c)(a * b) * c = a * (b * c) for all a,b,c∈Ma, b, c \in M.
- Existence of an Identity Element (e): There exists an element e∈Me \in M such that a∗e=e∗a=a,∀a∈M.a * e = e * a = a, \quad \forall a \in M.
Example: The set of natural numbers (N, ×) is a monoid because multiplication is associative and has an identity element 1.
2×(3×4)=(2×3)×4,2×1=2.2 \times (3 \times 4) = (2 \times 3) \times 4, \quad 2 \times 1 = 2.
Group
A group is a monoid in which every element has an inverse.
Definition: A set G with a binary operation (*) is a group if:
- Associativity: (a∗b)∗c=a∗(b∗c)(a * b) * c = a * (b * c), for all a,b,c∈Ga, b, c \in G.
- Identity Element (e): There exists an element e∈Ge \in G such that a∗e=e∗a=a,∀a∈G.a * e = e * a = a, \quad \forall a \in G.
- Inverse Element: For every a∈Ga \in G, there exists an element a−1a^{-1} such that a∗a−1=a−1∗a=e.a * a^{-1} = a^{-1} * a = e.
Example: The set of integers (Z, +) forms a group because:
- It is associative.
- The identity element is 0.
- Each element has an inverse: 5+(−5)=05 + (-5) = 0.
However, (N, +) is not a group because natural numbers do not have an inverse in N (e.g., 5 has no negative number in N to make the sum 0).
Abelian Group (Commutative Group)
A group is called an Abelian (or commutative) group if the binary operation is also commutative.
Definition: A group G is Abelian if:
a∗b=b∗a,∀a,b∈G.a * b = b * a, \quad \forall a, b \in G.
Example: The set of integers (Z, +) is an Abelian group because addition is commutative:
2+3=3+2.2 + 3 = 3 + 2.
However, (Matrix multiplication) is not always commutative, so the set of matrices under multiplication does not always form an Abelian group.
Summary of Algebraic Structures
| Structure | Associativity | Identity Element | Inverse Element | Commutativity |
|---|---|---|---|---|
| Groupoid | (Not necessarily) | |||
| Semigroup | ||||
| Monoid | ||||
| Group | ||||
| Abelian Group |
Would you like examples or proofs for any of these concepts? Let me know!