Exploring the Boundaries of Monad Tensorability on Set

TL;DR

This paper investigates the existence of tensor operations on monads, providing counterexamples to their general existence and identifying conditions under which they do exist, with implications for programming language semantics.

Contribution

It demonstrates that tensor of two monads may not always exist, solving a long-standing open problem, and shows tensors with bounded powerset monads do exist under certain conditions.

Findings

Counterexamples show tensor of two monads may not exist.

Tensors with bounded powerset monads exist from countable upwards.

Contrasts with previous assumptions about tensor existence.

Abstract

We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the component theories. As such, they extend the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several contexts; in particular, in the semantics of programming languages, the monad transformer for global state is given by a tensor. We present two main results: we show that the tensor of two monads need not in general exist by presenting two counterexamples, one of them involving finite powerset (i.e. the theory of join semilattices); this solves a somewhat long-standing open problem, and contrasts with recent results that had ruled out previously expected counterexamples. On the other hand, we show that tensors with bounded powerset monads do exist from countable powerset upwards.