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On the Foundations of Approximate Algebra: Axioms, Extensions, and Geometric Structures

Dang Vo Phuc

TL;DR

This work develops a general axiomatic theory of approximate algebra by introducing an algebra-compatible closure $Φ^{\!*}$ satisfying $(C1)$–$(C4a)$ and $(C4b)$ with absorption restricted to ideals. It builds robust foundations for approximate rings and modules, defines an approximate prime spectrum with a Zariski-type topology, and establishes localization and extension–contraction correspondences that mirror classical commutative algebra. The paper proves structural results such as the equality of the approximate prime radical and nilradical and provides a concrete example with $\mathrm{Spec}_{Φ}(\mathbb{Z})$ under a modular closure, highlighting a notable departure from classical geometry. It also outlines a pathway to an Approximate Nullstellensatz and discusses model closures (pointwise, metric-tolerance, sampling-based) that satisfy evaluation-separation properties, emphasizing potential applications in settings with uncertainty and approximation.

Abstract

Building on the recent works of Inan [4] and Almahareeq-Peters-Vergili [1], we develop a rigorous axiomatic foundation for approximate algebra via an algebra-compatible closure operator $Φ^{\!*}$ satisfying (C1)-(C4a) together with the balanced multiplicativity axiom (C4b) (and absorption required only for ideals). Our framework encompasses a theory of approximate modules with their isomorphism theorems, the construction of an approximate Zariski topology on the prime spectrum, and a compatible theory of localization. Key results include a $\mathrm{T}_0$ property and a $\mathrm{T}_1$-criterion for the spectrum, an extension-contraction bijection for approximate prime ideals in localizations, and the equality of the approximate prime radical and the nilradical. The theory's utility is illustrated by computing $\mathrm{Spec}_{\!Φ}(\mathbb{Z})$ for the modular closure $Φ^{\!*}(A)=\langle A\rangle+m\mathbb{Z}$, which yields a finite discrete space -- in stark contrast to the classical $\mathrm{Spec}(\mathbb{Z})$, which is infinite and not even $\mathrm{T}_1$. We also outline a pathway toward an Approximate Nullstellensatz and record model classes of closures that ensure its evaluation-separation hypothesis.

On the Foundations of Approximate Algebra: Axioms, Extensions, and Geometric Structures

TL;DR

This work develops a general axiomatic theory of approximate algebra by introducing an algebra-compatible closure satisfying – and with absorption restricted to ideals. It builds robust foundations for approximate rings and modules, defines an approximate prime spectrum with a Zariski-type topology, and establishes localization and extension–contraction correspondences that mirror classical commutative algebra. The paper proves structural results such as the equality of the approximate prime radical and nilradical and provides a concrete example with under a modular closure, highlighting a notable departure from classical geometry. It also outlines a pathway to an Approximate Nullstellensatz and discusses model closures (pointwise, metric-tolerance, sampling-based) that satisfy evaluation-separation properties, emphasizing potential applications in settings with uncertainty and approximation.

Abstract

Building on the recent works of Inan [4] and Almahareeq-Peters-Vergili [1], we develop a rigorous axiomatic foundation for approximate algebra via an algebra-compatible closure operator satisfying (C1)-(C4a) together with the balanced multiplicativity axiom (C4b) (and absorption required only for ideals). Our framework encompasses a theory of approximate modules with their isomorphism theorems, the construction of an approximate Zariski topology on the prime spectrum, and a compatible theory of localization. Key results include a property and a -criterion for the spectrum, an extension-contraction bijection for approximate prime ideals in localizations, and the equality of the approximate prime radical and the nilradical. The theory's utility is illustrated by computing for the modular closure , which yields a finite discrete space -- in stark contrast to the classical , which is infinite and not even . We also outline a pathway toward an Approximate Nullstellensatz and record model classes of closures that ensure its evaluation-separation hypothesis.

Paper Structure

This paper contains 21 sections, 22 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Foundations of Approximate Ring and Ideal Theory
  3. Prior Concepts and Motivation for an Axiomatic Framework
  4. The Algebra-Compatible Closure Axioms
  5. Rigorous Definitions of Ideal Operations and Primeness
  6. Rigorous Definitions of Ideal Operations and Primeness
  7. Stability under Ring Homomorphisms
  8. Theory of Approximate Modules
  9. Module-closure axioms (CM1--CM4)
  10. Approximate morphisms and functoriality
  11. Isomorphism theorems for approximate modules
  12. The Approximate Zariski Topology
  13. The Spectrum and the Zariski Topology
  14. Fundamental Topological Properties
  15. A Deep Example: $\mathrm{Spec}_{\!\Phi}(\mathbb{Z})$ under Modular Closure
  16. ...and 6 more sections

Key Result

Proposition 2.6

The operations on $R/I$ are well-defined, and $R/I$ is a ring.

Theorems & Definitions (68)

  • Definition 2.1: Algebra-Compatible Closure
  • Example 2.2: The Proximity-Based Closure Model
  • Example 2.3: Modeling Similarity in Image Processing
  • Example 2.4: Handling Noise and Errors in Signals
  • Definition 2.5: Approximate Product and Quotient Ring
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Theorem 2.8: Refined and Strengthened Factorization Property
  • Remark 2.9: Refinement of Theorem 6 in APV25
  • ...and 58 more