Unbounded nested number sequences

UNNS: Formal Algebraic Structure of Nests

🏛️ UNNS: Formal Algebraic Structure

Mathematical Framework for Nested Number Sequence Algebras

📐 Fundamental Algebraic Definitions

UNNS nests form algebraic structures with well-defined operations, identities, and morphisms. Each nest N induces an algebra over the domain of modulus values.

Definition 1: UNNS Nest Algebra
Let 𝒩ₙ = (𝔽ₙ, ∘, e) where:
• 𝔽ₙ = {f(m,n) : m ∈ ℤ⁺} (the nest function space)
• ∘ : 𝔽ₙ × 𝔽ₙ → 𝔽ₙ (composition operation)
• e ∈ 𝔽ₙ (identity element)

Definition 2: Nest Homomorphism
φ : 𝒩ₘ → 𝒩ₙ is a nest homomorphism if:
φ(a ∘ᵐ b) = φ(a) ∘ⁿ φ(b) for all a, b ∈ 𝒩ₘ

Definition 3: Integer-Preserving Subgroup
𝔦ₙ = {f(kn,n) : k ∈ ℤ⁺} ⊆ 𝒩ₙ
Forms a subgroup under UNNS composition

∘

Group Structure

UNNS nests exhibit group-like properties under specific operations, with well-defined closure, associativity, and identity elements.

Closure: f(a,n) ∘ f(b,n) ∈ 𝒩ₙ
Associativity: (a ∘ b) ∘ c = a ∘ (b ∘ c)
Identity: ∃e ∈ 𝒩ₙ : a ∘ e = e ∘ a = a
Inverses: ∀a ∈ 𝒩ₙ, ∃a⁻¹ : a ∘ a⁻¹ = e

⊕

Ring Properties

UNNS algebras form ring structures with addition and multiplication operations that preserve nest relationships.

(𝒩ₙ, +, ·) forms a ring where:
• (𝒩ₙ, +) is an abelian group
• (𝒩ₙ, ·) is a monoid
• Distributivity: a·(b+c) = a·b + a·c

≅

Isomorphisms

Structural equivalences between different nests reveal deep mathematical relationships and classification theorems.

𝒩ₚ ≅ 𝒩ₑ iff p ≡ q (mod φ(lcm(p,q)))
where φ is Euler's totient function

⟨⟩

Generated Substructures

Prime nest values generate maximal substructures, while composite nests decompose into product structures.

⟨f(p,p)⟩ generates 𝒩ₚ when p is prime
𝒩ₚₑ ≅ 𝒩ₚ × 𝒩ₑ when gcd(p,q) = 1

🧮 Interactive Algebraic Structure Explorer

Nest Configuration

Primary Nest (N₁): 𝒩₃ (Prime) 𝒩₄ (Composite) 𝒩₅ (Prime) 𝒩₆ (Composite) 𝒩₇ (Prime) 𝒩₈ (Power of 2) 𝒩₉ (Perfect Square)

Secondary Nest (N₂): 𝒩₃ 𝒩₄ 𝒩₅ 𝒩₇ 𝒩₁₁

Structure Analysis

Analysis Depth:

Operation Type: Function Composition Nest Addition Scalar Multiplication Modular Operations

Morphism Testing

Morphism Type: Homomorphism Isomorphism Endomorphism Automorphism

⊗

Closure Property

Operations remain within the nest algebra

((∘))

Associativity

Grouping doesn't affect results

𝐞

Identity Element

Neutral element for operations

⁻¹

Inverse Elements

Undoing operations

⇄

Commutativity

Order independence

∇

Distributivity

Distribution over addition

🔄 Structural Isomorphisms and Classifications

𝒩₃

φ

𝒩₅

ψ

𝒩₇

Morphism chain: Prime nests under cross-validation mappings

📏 Structural Invariants

Order: |𝒩ₙ| = ∞ (infinite nest algebras)

Rank: rank(𝒩ₙ) = ⌊log₂(n)⌋ + 1 (generator complexity)

Integer Density: δ(𝒩ₙ) = 1/n (density of integer-preserving elements)

Cross-Nest Degree: deg(𝒩ₘ, 𝒩ₙ) = gcd(m,n) (structural connection strength)

🎯 Classification Theorem

Theorem (UNNS Classification):
Every UNNS nest algebra 𝒩ₙ is isomorphic to exactly one of:

Type I (Prime Nests): 𝒩ₚ ≅ ℤₚ[x]/(x² - (p+1)²x + p²)
• Simple structure, no proper subalgebras
• Maximal integer-preserving density

Type II (Prime Power Nests): 𝒩₍ₚᵏ₎ ≅ ∏ᵢ₌₁ᵏ 𝒩ₚ
• Direct product of prime nest algebras
• Nilpotent elements present

Type III (Composite Nests): 𝒩₍ₘₙ₎ ≅ 𝒩ₘ ⊗ 𝒩ₙ when gcd(m,n) = 1
• Tensor product structure
• Decomposable into coprime factors

Proof Strategy:
1. Show UNNS formula respects prime factorization
2. Establish Chinese Remainder Theorem for nests
3. Prove uniqueness via structural invariants

UNNS — Formal Algebraic Toolkit

We formalize nests as algebraic objects, produce provable decomposition theorems (via CRT), and provide interactive examples and tables to validate statements.

Quick controls

Pick modulus n (integer ≥ 2)

Choose elements a, b ∈ End(Z_n) (viewed as multiplication-by-k maps)

Theory: We model "nest algebra" as the ring of Z-linear endomorphisms of the cyclic Z-module Z/nZ. Each Z-linear endomorphism is multiplication by some residue k (mod n). Hence End_Z(Z_n) ≅ Z_n (as rings). Use the controls to test.

Terminology

Definition (Ring of nests). For n ≥ 2 define N_n = End_Z(Z/nZ), the ring of Z-linear endomorphisms of the cyclic Z-module Z/nZ. Each φ ∈ N_n is φ(x)=k·x (mod n) for a unique k ∈ Z/nZ. So N_n ≅ Z/nZ by k ↦ (x↦k·x).
Classification: By the Chinese Remainder Theorem, if n = ∏ p_i^{e_i} then Z/nZ ≅ ∏ Z/p_i^{e_i} and hence N_n ≅ ∏ N_{p_i^{e_i}} ≅ ∏ Z/p_i^{e_i}.

1. Rigorous definitions

We keep definitions minimal and precise so theorems directly follow from classical algebra.

1.1 The algebraic model

Let n ≥ 2.
- Z_n denotes the ring Z / nZ (integers modulo n).
- Z_n is a Z-module and is cyclic, generated by [1].
- N_n := End_Z(Z_n) = {Z-linear maps φ: Z_n → Z_n }.
Claim: Every φ ∈ N_n is multiplication by some k ∈ Z_n:
        φ([x]) = [k·x]  (for all x), uniquely determined by k = φ([1]).
Therefore N_n ≅ Z_n as rings via k ↦ (x ↦ k·x).
    

1.2 Types (algebraic classification)

1. Type I: n is prime p. Then Z_n ≅ F_p field; N_n ≅ F_p (a field).
2. Type II: n = p^e (prime power). Then Z_n is a local ring; N_n ≅ Z/p^eZ is local (has unique maximal ideal pZ/p^eZ).
3. Type III: n composite with ≥2 distinct prime factors. By CRT, N_n ≅ ∏ Z/p_i^{e_i} (product of smaller rings).

2. Theorems & proof sketches

Theorem (Endomorphism identification)

Statement: N_n = End_Z(Z_n) ≅ Z_n via k ↦ (x ↦ kx).

Proof sketch: Z_n as Z-module is generated by [1]. Any Z-linear map φ is determined by φ([1]) = k ∈ Z_n. For any x ∈ Z_n, x = m·[1], so φ(x)=m·φ([1]) = m·k = k·x. Uniqueness: different k's give different maps. Composition corresponds to multiplication of residues: (k∘ℓ)(x) = k(ℓ x)=kℓ x. Thus ring isomorphism.

Theorem (CRT decomposition)

Statement: If n = ∏ p_i^{e_i}, then Z_n ≅ ∏ Z_{p_i^{e_i}} and hence N_n ≅ ∏ N_{p_i^{e_i}} ≅ ∏ Z_{p_i^{e_i}}.

Proof sketch: The classical Chinese Remainder Theorem gives an explicit isomorphism Z/nZ → ∏ Z/p_i^{e_i} by reduction maps x ↦ (x mod p_i^{e_i}). Passing to End_Z(−) and using functoriality (End respects finite products), End(Z/nZ) ≅ End(∏ Z/p_i^{e_i}) ≅ ∏ End(Z/p_i^{e_i}). By the previous theorem End(Z/p^e Z) ≅ Z/p^e Z. Combine to obtain the product decomposition.

3. Worked examples (interactive)

Enter n and click Analyze n. The page factors n, shows the CRT decomposition, a representative isomorphism, and the multiplication (composition) table of N_n.

4. Concrete example: n = 12 (worked through)

n = 12 = 2^2 · 3. CRT: Z/12Z ≅ Z/4Z × Z/3Z. Therefore N_12 ≅ Z/4Z × Z/3Z, i.e. endomorphisms correspond to pairs (a mod 4, b mod 3) and composition is pairwise multiplication.

5. Visual proof element

Below: interactive circle representing residues mod n; colors mark CRT components. Click residues to see their component residues and see composition act as multiplication.

6. Composition / Morphism tests

Pick a, b then compute a∘b (composition) which corresponds to ab (mod n). The interactive table below shows the multiplication table in N_n (same as Z_n multiplication).

7. Further remarks (UNNS-specific ideas & next steps)

If your original UNNS usage envisaged non-Z-linear maps (e.g., shift/rotate/chunk operations), we can extend the algebra beyond N_n to the full monoid of functions Func(Z_n,Z_n). That monoid is far larger (n^n elements) and decompositions require different techniques (Young subgroups, permutation module theory). The core formalization above shows a minimal and provable foundation; extensions can be layered on top with explicit examples.