Varieties of Metric and Quantitative Algebras

TL;DR

This paper explores the structure of metric and quantitative algebras, showing that classes defined by metric equations are limited and proposing the need for broader logical frameworks in metric algebra theory.

Contribution

It characterizes varieties of metric algebras as classes closed under subalgebras, products, and quotients, and highlights limitations of metric equations in capturing important classes like normed vector spaces.

Findings

Varieties of metric algebras are exactly classes closed under subalgebras, products, and quotients.

The class of normed vector spaces cannot be defined by metric equations.

Metric equations are insufficient to express many natural classes of metric algebras.

Abstract

Metric algebras are metric variants of $\Sigma$-algebras. They are first introduced in the field of universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. Recently a similar notion of quantitative algebra is used in computer science to formulate computational effects of probabilistic programs. In this paper we show that varieties of metric algebras (classes defined by a set of metric equations) are exactly classes closed under (metric versions of) subalgebras, products and quotients. This result implies the class of normed vector spaces cannot be defined by metric equations, differently from the classical case where the class of vector spaces is equational. This phenomenon suggests that metric equations are not a very natural class of formulas since they cannot express such a typical class of metric algebras. Therefore we need a broader class of formulas to acquire an appropriate metric counterpart of the classical variety theory.