group-like-structures.md

Totality

all operations are 'T -> 'T
counter example: for rational numbers sqrt(2) produces an irrational number

Associativity

(a * b) * c = a * (b * c)
counter example: (a - b) - c <> a - (b - c)

Identity

there exists some data where the operation does nothing
counter example: 
addition for the natural numbers. 
There is no zero for natural numbers so there is no identity for addition there.

Divisibility

there exists some opration and data that can "undo" a previous operation
(a + b) - b = a

Commutativity

a * b = b * a

graph 
    pm[partial magma]
    sgd(semigroupoid)
    c(category)
    gd(groupoid)
    m(magma)
    qg(quasigroup)
    um(unital magma)
    l(loop)
    sg(semigroup)
    aq(associative quasigroup)
    mo(monoid)
    cm(commutative monoid)
    g(group)
    ag(abelian group)
    pm -- associativity --> sgd
    sgd -- identity --> c
    c -- divisibility --> gd
    pm -- totality --> m 
    m -- divisibility --> qg
    qg -- identity --> l
    m -- identity --> um
    m -- associativity --> sg
    sg -- divisibility --> aq
    sg -- identity --> mo
    mo -- commutativity --> cm
    mo -- divisibility --> g
    g -- commutativity --> ag
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