Module

Ch9

#Semigroup 

class Semigroup a  where

MAGMA

A Magma is a Set that is Closed under a Binary Operator (+, -, *, /, &&, ||)

Being "Closed" means that by applying a Binary operator to every combination on a Set, the result will always be a value of that Set

E.g: M = { true, false } and • = &&

The set "M" is the set of all "Booleans" and the Binary operator • is the "Logical And" operation "&&"

true && true = true 
true && false = false 
false && true = false
false && false = false

Formal Definition of a Magma ∀ a, b ∈ M ⇒ a • b ∈ M

 ∀ a, b ∈ M   (For all a, and b that is an Element of M)
 ⇒            (It follows that)
 a • b ∈ M    (the result of a • b is an Element of M)

SEMIGROUP

A Semigroup is a Magma where the Binary Operator is Associative

A Binary operator • is "Associative" if the following is true

a • (b • c) == (a • b) • c

E.g: S = { "a", "b", ..., "z", "aa", "ab", ...} and • = <>

  • S is the Set of NON-empty String
  • <> is the Concatenation (or Append) operator

("ab" <> "cd") <> "ef" == "ab" <> ("cd" <> "ef")

Formal definition of a Semigroup ∀ a, b, c ∈ S ⇒ a • (b • c) = (a • b) • c

∀ a, b, c ∈ S               (For all a, b and c that is an Element of S)
⇒                           (It follows that)
a • (b • c) = (a • b) • c   (the result of applying a to
                             the result of b • c is equal to
                             the result of applying a • b to
                             c)

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Instances

#(<>)

Operator alias for Ch9.append (right-associative / precedence 5)

#Monoid 

class (Semigroup a) <= Monoid a  where

MONOID

A Monoid is a Semigroup where the Set has a Neutral Element, e.

A formal definition of a Monoid e ∈ M, ∀ a ∈ M ⇒ a • e = e • a = a

Where M is the Monoid Set

e ∈ M                (e is an Element of M)
∀ a ∈ M              (and for all a that is an Element of M)
⇒                    (it follows that)
a • e = e • a = a    (applying a to e is equal to
                      applying e to a which is equal to ai)

The Neutral Elemnet (Identity) can be applied to any Element of the Set Including itself, and have no impact on that Element.

E.g:

M = { 0, 1, 2, 3, ...} and • = +, e = 0
  • M is first a Magma: 1 + 3 = 4 --> (Closed)
  • And then a Semigroup: 1 + (2 + 3) = (1 + 2) + 3 --> (Associative)
  • And then a Monoid: 0 + 1 = 1 + 0 = 0 --> (Identity)

E.g2:

M = { true, false } and • = ||, e = false

M is first a Magma then a Semigroup and then a Monoit.

true || false = true                                 (Closed)
true || (true || false) = (true || true) || false    (Associative)
false || true = true || false = false                (Identity)

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#Group 

class (Monoid a) <= Group a  where

Group

A Group is a Monoid where every element of the Set has an Unique Inverse

A formal definition of a Monoid ∀ a ∈ G, a' ∈ G, e ∈ G ⇒ a • a' = a' • a = e

Where M is the Monoid Set

∀ a ∈ G                (For all a that is an element of G)
a' ∈ G                 (and a' that is an element of G)
e ∈ G                  (and e that is an element of G)
⇒                      (it follows that)
a • a' = a' • a = e    (applying a to a' is equal to
                        applying a' to a which is equal to e)

Essentially, a' is the Unique Inverse of a, and e is the Empty element (Identity)

E.g:

G = { ..., -3, -3, -1, 0, 1, 2, 3, ...} and • = +, e = 0

E.g2:

G = { ..., 1/3, 1/2, 1, 2, 3, ... } and • = *, e = 1

1 is its own Inverse (1 * 1 = 1), and it's the identity for every element including itself.

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