Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Similarity Algebra: A Framework for Approximate Algebraic and Lie Structures with Collapse to Classical Algebra

Benyamin Ghojogh, Golbahar Amanpour

Abstract

Classical algebraic structures require exact satisfaction of their defining axioms. We propose similarity algebra, a framework extending algebraic and Lie structures to settings where operations satisfy quantitative bounds up to a tolerance $\varepsilon$. Instead of strict associativity, inverses, or distributivity, we study families of operations controlled by explicit $\varepsilon$-estimates and analyze their behavior under limit collapse. Under uniform error control and $C^1_{\mathrm{loc}}$ convergence of the structure maps, we prove a general collapse theorem showing that similarity structures converge to classical algebraic objects, as $\varepsilon \rightarrow 0$. We develop a hierarchy of approximate structures, including similarity groups, rings, fields, vector spaces, and Lie groups, formalized through axioms satisfied within metric distance $\varepsilon$. We further define a category of similarity algebras governing morphisms between approximate systems. Moreover, we clarify the relationship between similarity algebra and fuzzy algebra, showing that the former generalizes the latter. The proposed similarity algebra can be useful to model real-world phenomena where operations or relations are inherently approximate.

Similarity Algebra: A Framework for Approximate Algebraic and Lie Structures with Collapse to Classical Algebra

Abstract

Classical algebraic structures require exact satisfaction of their defining axioms. We propose similarity algebra, a framework extending algebraic and Lie structures to settings where operations satisfy quantitative bounds up to a tolerance . Instead of strict associativity, inverses, or distributivity, we study families of operations controlled by explicit -estimates and analyze their behavior under limit collapse. Under uniform error control and convergence of the structure maps, we prove a general collapse theorem showing that similarity structures converge to classical algebraic objects, as . We develop a hierarchy of approximate structures, including similarity groups, rings, fields, vector spaces, and Lie groups, formalized through axioms satisfied within metric distance . We further define a category of similarity algebras governing morphisms between approximate systems. Moreover, we clarify the relationship between similarity algebra and fuzzy algebra, showing that the former generalizes the latter. The proposed similarity algebra can be useful to model real-world phenomena where operations or relations are inherently approximate.
Paper Structure (10 sections, 21 theorems, 148 equations)

This paper contains 10 sections, 21 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Similarity Algebra and Its Structures
  3. Similarity Lie Algebra and Similarity Lie Group
  4. Collapse Theorems
  5. Practical Examples for Similarity Algebra
  6. A Practical Example of a Similarity Field
  7. A Practical Example of Perturbed Matrix Lie Groups
  8. Category of Similarity Algebra
  9. Comparison of Similarity Algebra and Fuzzy Algebra
  10. Conclusion

Key Result

Lemma 1

Let $(X, d)$ be a metric space and $\varepsilon \geq 0$. All operations are uniformly Lipschitz and uniformly convergent. If for all $\varepsilon \geq 0$, and uniformly, as $\varepsilon \rightarrow 0$, then:

Theorems & Definitions (58)

  • Definition 1: Similarity Algebra
  • Remark 1
  • Definition 2: Similarity Semigroup
  • Definition 3: Similarity Monoid
  • Definition 4: Similarity Abelian/Commutative Monoid
  • Definition 5: Similarity Group
  • Definition 6: Similarity Abelian/Commutative Group
  • Definition 7: Similarity Semiring
  • Definition 8: Similarity Ring
  • Definition 9: Similarity Field
  • ...and 48 more