Magma → Semigroup → Monoid

We build algebraic structures by stacking axioms: each level adds a property to the previous one. This produces the magma hierarchy (notes p.124-126, slides Chapter 8 Example 1.7-1.10).

Magma $\langle G, \circ \rangle$ (Def 1.6):
- A set $G$ with a binary operation $\circ : G \times G \to G$ (closure).
- Just: $a, b \in G \Rightarrow a \circ b \in G$.

Semigroup (Def 1.7): a magma + associativity:

Monoid (Def 1.8): a semigroup + an identity element $e$:

Each level inherits everything from the level below. So every monoid is a semigroup; every semigroup is a magma. The arrows in a theory graph go "upward" — adding axioms.

Examples:
- $\langle \mathbb{N}, + \rangle$ is a monoid with identity 0. (Not a group because subtraction can leave $\mathbb{N}$.)
- $\langle \mathbb{N}, \cdot \rangle$ is a monoid with identity 1.
- String concatenation $\langle \Sigma^*, \cdot, \varepsilon \rangle$ is a monoid with empty string as identity.

Theory graph direction (notes p.196): arrows go from the GENERAL (magma) at the TOP to the SPECIFIC (ring) at the BOTTOM, with each arrow ADDING an axiom. So Magma → Semigroup → Monoid → Group → AbelGroup → Ring.