Approimate satisfaction of identities

TL;DR

This paper introduces a metric-based measure for how closely continuous operations on a space satisfy a set of equations, providing a way to quantify approximate algebraic satisfaction.

Contribution

It proposes a new quantitative approach to evaluate the deviation of continuous operations from exact algebraic identities in metric spaces.

Findings

Defines a measure for approximate satisfaction of identities

Applies the measure to various metric spaces and equations

Provides insights into the stability of algebraic structures

Abstract

For a metric space $(A,d)$, and a set $\Sigma$ of equations, a quantity is introduced that measures how far continuous operations must deviate from satisfying $\Sigma$ on $(A,d)$. }