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{dX(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.
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BibTeX
@misc{pandey2026metric,
title = {{Metric preservation under approximate algebraic laws}},
author = {Daksh Pandey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24927},
eprint = {2610.24927},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24927},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Metric preservation under approximate algebraic laws
AU - Daksh Pandey
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.24927
UR - https://arxiv.org/abs/2610.24927
AB - 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.
ER -