Algebra 1—Closure to Commutativity

I. Conceptual ladder for (V,★)

A. Magma
- Begin with closure under composition ★.
- A set V equipped with at least one binary operation (★) under which V is closed: ( V , ★ ).
  - Formally, ★: V × V → V.
  - Informally, closure under ★ means any time ★ operates on inputs a, b ∈ V, the output must also be an element of V.
B. Semigroup
- Adds associativity to the magma’s binary operation ★.
- a ★ ( b ★ c ) = ( a ★ b ) ★ c for all a,b,c ∈ V.
C. Monoid
- Add neutral element α (aka identity element) to semigroup.
- Left identity: α ★ a = a for all a ∈ V.
- Right identity: a ★ α = a for all a ∈ V.
D. Group
- Adds inverse a^-1.
- For every a ∈ V, there exists an inverse element a^-1 ∈ V such that a ★ a^-1 = a^-1 ★ a = α.
E. Abelian Group
- Adds commutativity to binary operation ★.
- a ★ b = b ★ a for all a,b ∈ V.
- Algebraists commonly use the plus symbol to denote commutative composition: ( V , + ).

abelian groups ⊂ groups ⊂ monoids ⊂ semigroups ⊂ magmas.
In general, each of the rules are independent, except for inverses which are dependent on the neutral (identity) element.
- Associativity is independent (B. = semigroup).
- Neutral element α is independent (C. = monoid).
- Commutativity is independent (E. = abelian group).
- Inverses reference neutral element (D. = group).
For example, one may have abelian semigroups, that have no neutral elements (C.) and no inverses (D.); but has a composition ★ that is both associative and commutative.
For further thought:
- Monoids = accumulating process
- Groups (i.e., commutation) = reversible process