Discrete Lawvere Theories

We introduce the notion of discrete countable Lawvere V-theory and study constructions that may be made on it. The notion of discrete countable Lawvere V-theory extends that of ordinary countable Lawvere theory by allowing the homsets of an ordinary countable Lawvere theory to become homobjects of a well-behaved axiomatically defined category such as that of ω-cpo's. Every discrete countable Lawvere V-theory induces a V-enriched monad, equivalently a strong monad, on V. We show that discrete countable Lawvere V-theories allow us to model all the leading examples of computational effects other than continuations, and that they are closed under constructions of sum, tensor and distributive tensor, which are the fundamental ways in which one combines such effects. We also show that discrete countable Lawvere V-theories are closed under taking an image, allowing one to treat observation as a mathematical primitive in modelling effects.