Notes on algebra, probability theory, and linear algebra
| Structure | Operations | Main axiom added | Central intuition |
|---|---|---|---|
| Binary operation | one operation | closure | combine two elements |
| Semigroup | one operation | associativity | combine non-empty finite sequences |
| Monoid | one operation | identity | allow the empty composition |
| Group | one operation | inverses | reversible actions |
| Abelian group | one operation | commutativity | order-independent contributions |
| Ring | two operations | distributivity | addition plus compatible multiplication |
| Field | two operations | multiplicative inverses and commutativity | arithmetic of scalars |
| Vector space | addition plus scalar action | compatibility with a field | linear combinations |
| Algebra over a field | vector space plus internal product | bilinearity | linear space with multiplication |
| Sigma-algebra | set operations on subsets | countable closure | language of measurable events |
\((ab)c=a(bc)\) Means: parentheses do not matter.
It turns a binary operation into finite composition.
\(ea=ae=a\) Means: doing nothing is part of the structure.
It allows empty products and zero-step processes.
\(aa^{-1}=a^{-1}a=e\) Means: every action can be undone.
It gives cancellation and equation-solving.
\(ab=ba\) Means: order does not matter.
It turns sequences of operations into unordered contributions.
\(a(b+c)=ab+ac\) Means: multiplication respects additive decomposition.
It allows computation by parts, expansion, and factoring.
\(\lambda(u+v)=\lambda u+\lambda v\) Means: scaling respects vector addition.
It makes linear combinations coherent.
\((\alpha x+\beta y)z=\alpha xz+\beta yz\) Means: internal multiplication in an algebra respects linear combinations in each input.
A binary operation combines two elements.
Associativity lets it combine any finite sequence.
An identity element gives a neutral action.
Inverses make actions reversible.
Groups model reversible transformations.
Symmetry groups preserve structure.
Rings introduce addition and multiplication.
Distributivity makes them one coherent system rather than two unrelated operations.
Fields are rings where nonzero multiplication behaves like reversible scalar scaling.
A vector space is an additive world acted on by a field of scalars.
The central objects are linear combinations.
An algebra over a field is a vector space where vectors also multiply, and multiplication is compatible with linearity.
Sigma-algebras are different: they are not vector-space algebras.
They are collections of subsets closed under logical/countable operations, used to define measure and probability.