Monoids

It's a things.

Axiomatic Satisfaction

Associativity

(x \cdot y) \cdot z = x \cdot (y \cdot z)

(x \cdot y) \cdot z = x \cdot (y \cdot z)

a \space b \space c \in S

a b c \in S

(a \cdot b) \cdot c = a \cdot (b \cdot c)

(a \cdot b) \cdot c = a \cdot (b \cdot c)

\cdot : S \times S \rightarrow S

\cdot : S \times S \to S

∀

Identity

a \cdot e = a = e \cdot a

a \cdot e = a = e \cdot a

a \in S

a \in S

a \cdot e = a = e \cdot a

a \cdot e = a = e \cdot a

\cdot : S \times S \rightarrow S

\cdot : S \times S \to S

∀

e \in S

e \in S

Monoidal Objects

Integers are a Monoid

with + and 0

(a + b) + c = a + (b + c)

(a + b) + c = a + (b + c)

a + 0 = a = 0 + a

a + 0 = a = 0 + a

Integers are a Monoid

with * and 1

(a * b) * c = a * (b * c)

(a * b) * c = a * (b * c)

a * 1 = a = 1 * a

a * 1 = a = 1 * a

Lists are a Monoid

with concat and []

(a \space \space \space \space b) \space \space \space \space c = a \space \space \space \space (b \space \space \space \space c)

(a b) c = a (b c)

a \space \space \space \space [] = a = [] \space \space \space \space a

a [] = a = [] a

⧺

Booleans are a Monoid

with (AND) and True

(a \wedge b) \wedge c = a \wedge (b \wedge c)

(a \land b) \land c = a \land (b \land c)

a \wedge true = a = true \wedge a

a \land t r u e = a = t r u e \land a

∧

Booleans are a Monoid

with (OR) and False

(a \vee b) \vee c = a \vee (b \vee c)

(a \lor b) \lor c = a \lor (b \lor c)

a \vee false = a = false \vee a

a \lor f a l s e = a = f a l s e \lor a

∨

Booleans are a Monoid

with (XOR) and False

(a \space \oplus \space b) \space \oplus \space c = a \space \oplus \space (b \space \oplus \space c)

(a \oplus b) \oplus c = a \oplus (b \oplus c)

a \space \oplus \space false = a = false \space \oplus \space a

a \oplus f a l s e = a = f a l s e \oplus a

⊕

Sets are a Monoid

with (union) and {}

(a \cup b) \cup c = a \cup (b \cup c)

(a \cup b) \cup c = a \cup (b \cup c)

a \cup \{\} = a = \{\} \cup a

a \cup {} = a = {} \cup a

∪

Function Composition

add5 : Number \rightarrow Number

a d d 5 : N u m b e r \to N u m b e r

add5 \space x = x + 5

a d d 5 x = x + 5

div2 : Number \rightarrow Number

d i v 2 : N u m b e r \to N u m b e r

div2 \space x = x / 2

d i v 2 x = x / 2

a : Number

a : N u m b e r

a = 3

a = 3

add5 \space 6 = 6 + 5 = 11

a d d 56 = 6 + 5 = 11

div2 \space 12 = 12 / 2 = 6

d i v 212 = 12 / 2 = 6

div2 \space (add5 \space 8) = 4

d i v 2 (a d d 58) = 4

ad5dv2 \space x = div2 \space (add5 \space x)

a d 5 d v 2 x = d i v 2 (a d d 5 x)

ad5dv2 \space 7 = 6

a d 5 d v 27 = 6

ad5dv2 = div2 \space \space \space \space \space \space \space add5

a d 5 d v 2 = d i v 2 a d d 5

⋘

foo : a \rightarrow b

f o o : a \to b

bar : b \rightarrow c

b a r : b \to c

baz = bar \space \space \space \space \space \space \space foo

b a z = b a r f o o

baz : a \rightarrow c

b a z : a \to c

⋘

Functions are a Monoid

with and id

(a \space \space \space \space \space b) \space \space \space \space \space c = a \space \space \space \space \space (b \space \space \space \space \space c)

(a b) c = a (b c)

a \space \space \space \space \space id = a = id \space \space \space \space \space a

a i d = a = i d a

⋘

id : a \rightarrow a

i d : a \to a

id \space x = x

i d x = x

Other Monoids

Observables (Subscriptions)
Publishing
Promises
Any Functor containing a Monoid is a Monoid

Example Functors

Lists
Trees
Sets

mconcat

mconcat [[0,1,2], [3], [], [4,5]] = [0, 1, 2, 3, 4, 5]

mconcat ["hi", " ", "there", " ", "folks"] = "hi there folks"

mconcat [0, 1, 2, 4, 5] = 12

mconcat [{a, b}, {}, {c, d}] = {a, b, c, d}

mconcat : Monoid \space a \space \space \space \space \space \space \space [a] \rightarrow a

m c o n c a t : M o n o i d a [a] \to a

⇒