Tree ❡

Menge / Set

• no operators

Ring

• Addition

Gruppe / Group

• Multiplikation - not commutative, Order DOES matter

Abelsche Gruppe / Abelian Group

• ist kommutativ == Multiplikations-Ordnung ist egal

Körper / Field

• Division

• forget Division

Ring

• forget Multiplication

Set
 │
 ├──> Pos (Adds <= relation)
 ├──> Top (Adds open sets)
 ├──> Met (Adds distance function)
 │
 └──> Mon (Adds associative '*' and '1')
       │
       └──> Grp (Adds 'inverse')
             │
             └──> Ab (Adds commutative rule)
                   │
                   ├──> Ring (Mixes Ab for '+' and Mon for '*')
                   │     │
                   │     └──> CRing (Multiplication is commutative)
                   │           │
                   │           └──> Fld (Division is possible)
                   │
                   └──> RMod<R: Ring> (Ab + Scalar Multiplication by R)
                         │
                         ├──> Vect<K: Fld> (RMod where R is a Field)
                         │
                         └──> Alg<R: CRing> (RMod that is ALSO a Ring)

// ==========================================
// 1. THE ROOT
// ==========================================

interface Set<T> {
    elements: T[];
    contains(element: T): boolean;
}

// ==========================================
// 2. ALGEBRA (The "Operators" Branch)
// ==========================================

interface Group<T> extends Set<T> {
    operate(a: T, b: T): T;       // 1. Closure
    identity(): T;                // 2. Neutral element (e.g., 0 or 1)
    inverse(a: T): T;             // 3. Reversibility
}

// Abelian means the order doesn't matter (Commutative)
interface AbelianGroup<T> extends Group<T> {
    // Axiom: operate(a,b) === operate(b,a)
}

interface Ring<T> extends AbelianGroup<T> { // Abelian for Addition (+)
    multiply(a: T, b: T): T;                // Adds a 2nd operator
    // Axiom: Distributive law connects (+) and (*)
}

// "Körper" (e.g., Real numbers, Complex numbers)
interface Field<T> extends Ring<T> {
    divide(a: T, b: T): T;        // Now multiplication is fully reversible too!
}

// "Vektorraum" - Combines an AbelianGroup (Vectors) with a Field (Scalars)
interface VectorSpace<V, S extends Field<S>> extends AbelianGroup<V> {
    scale(scalar: S, vector: V): V;
}

// ==========================================
// 3. TOPOLOGY & GEOMETRY (The "Proximity" Branch)
// ==========================================

// "Topologischer Raum" - The most abstract definition of "Shape" and "Limits"
interface TopologicalSpace<T> extends Set<T> {
    isOpen(subset: Set<T>): boolean; // Defines what is "near" what, WITHOUT numbers
}

// "Metrischer Raum" - Gives Topology a strict numerical ruler
interface MetricSpace<T> extends TopologicalSpace<T> {
    distance(a: T, b: T): number;    // Introduces the ruler: d(x,y)
    // Axiom: Triangle Inequality d(x,z) <= d(x,y) + d(y,z)
}

// "Mannigfaltigkeit" - Curved space that pretends to be flat
interface Manifold<T> extends TopologicalSpace<T> {
    // If you zoom in infinitely close to ANY point, it looks like a flat VectorSpace (R^n)
    tangentSpaceAt(point: T): VectorSpace<T, RealNumbers>;
    chart(localSubset: Set<T>): FlatCoordinates;
}

// "Riemannsche Mannigfaltigkeit" (General Relativity lives here)
interface RiemannianManifold<T> extends Manifold<T>, MetricSpace<T> {
    // Allows you to measure angles and distances on a CURVED surface
    metricTensor(point: T, v1: Vector, v2: Vector): number;
}

// ==========================================
// 4. MEASURE THEORY (The "Integration" Branch)
// ==========================================

// "Messraum" - Defines WHAT can be measured (Sigma-Algebra)
interface MeasurableSpace<T> extends Set<T> {
    isMeasurable(subset: Set<T>): boolean; // Filters out paradoxes like Banach-Tarski
}

// "Maßraum" - Actually assigns a size/volume/probability to the subsets
interface MeasureSpace<T> extends MeasurableSpace<T> {
    measure(subset: Set<T>): number; // Returns "Volume" or "Probability"
    integrate(func: Function): number; // Lebesgue Integration (Superior to Riemann!)
}

// ==========================================
// 5. FUNCTIONAL ANALYSIS (The "God-Tier" Branch)
// Where Algebra and Topology implement Multiple Inheritance
// ==========================================

// "Normierter Raum" - A VectorSpace that is also a MetricSpace
interface NormedSpace<V, S> extends VectorSpace<V, S>, MetricSpace<V> {
    norm(vector: V): number; // Every vector has an absolute length ||v||
}

// "Banachraum" - A Normed Space with NO HOLES
interface BanachSpace<V, S> extends NormedSpace<V, S> {
    isComplete(): boolean; // Axiom: If a sequence looks like it's converging, the limit actually exists inside the space.
}

// "Hilbertraum" - The framework for Quantum Mechanics and infinite-dimensional geometry
interface HilbertSpace<V, S> extends BanachSpace<V, S> {
    innerProduct(v1: V, v2: V): S; // Introduces Angles and Orthogonality (Dot Product)
    project(vector: V, ontoSubspace: Set<V>): V;
}