Quantitative Algebraic Reasoning

We develop a quantitative analogue of equational reasoning which we call quantitative algebra. We define an equality relation indexed a = ε$b$ which we think of as saying that "$a$ is approximately equal to $b$ up to an error of $\varepsilon $". We have 4 interesting examples where we have a quantitative equational theory whose free algebras correspond to well known structures. In each case we have finitary and continuous versions. The four cases are: Hausdorff metrics from quantitive semilattices; $p - $Wasserstein metrics (hence also the Kantorovich metric) from barycentric algebras and also from pointed barycentric algebras and the total variation metric from a variant of barycentric algebras.