Combining Algebraic Effect Descriptions Using the Tensor of Complete Lattices

Algebras can be used to interpret the behaviour of effectful programs. In particular, we use Eilenberg-Moore algebras given over a complete lattices of truth values, which specify answers to queries about programs. The algebras can be used to formulate a quantitative logic of behavioural properties, specifying a congruent notion of program equivalence coinciding with a notion of applicative bisimilarity. Many combinations of effects can be interpreted using these algebras. In this paper, we specify a method of generically combining effects and the algebras used to interpret them. At the core of this method is the tensor of complete lattices, which combines the carrier sets of the algebras. We show that this tensor preserves complete distributivity of complete lattices. Moreover, the universal properties of this tensor can then be used to properly combine the Eilenberg-Moore algebras. We will apply this method to combine the effects of probability, global store, cost, nondeterminism, and error effects. We will then compare this method of combining effects with the more traditional method of combining equational theories using interaction laws.