Monoidal multiplexing

Abstract

Given a classical algebraic structure—e.g. a monoid or group—with carrier set X, and given a positive integer n, there is a canonical way of obtaining the same structure on carrier set Xn by defining the required operations “pointwise”. For resource-sensitive algebra (i.e. based on mere symmetric monoidal, not cartesian structure), similar “pointwise” operations are usually defined as a kind of syntactic sugar: for example, given a comonoid structure on X, one obtains a comultiplication on X⊗X by tensoring two comultiplications and composing with an appropriate permutation. This is a specific example of a general construction that we identify and refer to as multiplexing. We obtain a general theorem that guarantees that any equation that holds in the base case will hold also for the multiplexed operations, thus generalising the “pointwise” definitions of classical universal algebra.

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