Algebraic Laws
For a set \(M\) with operation \(*\), the most common laws are:
- Closure: \(\forall a,b\in M: a*b\in M\)
- Associativity: \(\forall x,y,z\in M: x*(y*z)=(x*y)*z\)
- Commutativity: \(\forall x,y\in M: x*y=y*x\)
- Neutral element: \(\exists e\in M:\forall x\in M: x*e=e*x=x\)
- Inverse element: \(\forall x\in M\,\exists y\in M: x*y=y*x=e\)
- Distributivity (for two operations): \(x*(y+z)=x*y+x*z\)
Structures with One Operation
| Structure | Data | Axioms | Example |
|---|---|---|---|
| Semigroup | \((M,*)\) | Closure, associativity | \((\mathbb{N}_{\ge 1},\cdot)\) |
| Monoid | \((M,*,e)\) | Semigroup + neutral element | \((\mathbb{N}_0,+,0)\) |
| Commutative monoid | \((M,*,e)\) | Monoid + commutativity | \((\mathbb{N}_{\ge 1},\cdot,1)\) |
| Group | \((M,*,e,{}^{-1})\) | Monoid + inverses | \((\mathbb{Z},+,0,-x)\) |
| Abelian group | \((M,*,e,{}^{-1})\) | Group + commutativity | \((\mathbb{Q}\setminus\{0\},\cdot,1,x^{-1})\) |
Note that \((\mathbb{Z},\cdot)\) is not a group under multiplication, since most integers have no multiplicative inverse in \(\mathbb{Z}\).
Structures with Two Operations
| Structure | Data | Axioms | Example |
|---|---|---|---|
| Ring | \((M,+,\cdot,0,1)\) | \((M,+)\) abelian group, \((M,\cdot)\) monoid, distributive laws | \((\mathbb{Z},+,\cdot,0,1)\) |
| Commutative ring | \((M,+,\cdot,0,1)\) | Ring + commutative multiplication | \((\mathbb{Q},+,\cdot,0,1)\) |
| Integral domain | \((M,+,\cdot,0,1)\) | Commutative ring + no zero divisors | \((\mathbb{Z},+,\cdot,0,1)\) |
| Field | \((M,+,\cdot,0,1)\) | Commutative ring + \((M\setminus\{0\},\cdot)\) abelian group | \((\mathbb{R},+,\cdot,0,1)\) |
These structures form a hierarchy: every field is a commutative ring, every commutative ring is a ring, and every ring's additive part alone is already an abelian group. Recognizing which structure a set and its operations form determines which algebraic manipulations — cancellation, factoring, division — are actually valid.