# Continuous Varieties of Metric and Quantitative Algebras

## TL;DR

This paper extends universal algebra to metric structures, establishing variety theorems for classes of metric algebras characterized by metric equations and continuous inferences.

## Contribution

It introduces the concepts of congruential pseudometric and continuous varieties, providing a framework for classifying metric algebras via closure properties.

## Key findings

- Metric versions of the variety theorem are established.
- Introduction of congruential pseudometric as an analogue of classical congruence.
- Characterization of strict and continuous varieties through closure properties.

## Abstract

A metric algebra is a metric variant of the notion of $\Sigma$-algebra, first introduced in universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. In this paper, we showed metric versions of the variety theorem, which characterizes strict varieties (classes of metric algebras defined by metric equations) and continuous varieties (classes defined by a continuous family of basic quantitative inferences) by means of closure properties. To this aim, we introduce the notion of congruential pseudometric on a metric algebra, which corresponds to congruence in classical universal algebra, and we investigate its structure.