Metric preservation under approximate algebraic laws

Abstract

We investigate which metric information can be retained when an initial metric space is realized in a complete algebra satisfying quantitative laws. Using free metric realizations, we express preservation of designated distances as constraints on the permitted errors. In a fixed weighted-path presentation, the associated feasible error budgets form a closed, convex, upward-closed region with finite-path certificates of failure, and free term distances depend continuously on the budgets. For an idempotent binary operation with positive coordinate weights $a+b=1$, we establish sharp associativity-defect bounds and an exact formula for the generator distances in the free realization with associativity tolerance $α$: $\min\{d_X(x,y),α/(ab)\}$. This yields explicit injectivity and isometry thresholds. We also give a necessary diameter bound when both idempotence and associativity are approximate, and a countably presented example whose free distance is continuous but not Lipschitz at zero.