Varieties of Metric and Quantitative Algebras

Abstract

Metric algebras are metric variants of Σ-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.

Continuous Varieties of Metric and Quantitative Algebras

“…However a metric version of the variety theorem has been missing for long. We give a very straightforward version in [10], and Mardare et al [19] give the characterization theorem for κ-variety, where κ is a cardinal, which generalizes our result in [10].…”

“…• We prove the variety theorem for classes of metric (or quantitative) algebras defined by metric equations. This is proved in our previous work [10]. Here we give a more concise proof by congruential pseudometrics.…”

“…Corollary 6. 10. A class K of quantitative algebras is defined by a continuous family of basic quantitative inferences in Q if and only if K is closed under products, subalgebras, ultraproducts and reflexive Q-quotients.…”

Metric monads

“…ℵ 1 -equational theories need not have only finite but also countable arities. The special case when arities X are finite discrete metric spaces (all distances are ∞) and arities Y are finite metric spaces was considered in [34], or [16] where the resulting categories Alg(E) were called varieties of metric algebras. Here, equations are replaced by "metric equations" s = ε t. If arities Y are also discrete, usual equations are used.…”