Constructive Lattice Theory

A notion of simulation of one datatype by another is deened as a constructive preorder. A calculus of datatype simulation is then developed by formulating constructive versions of least-xed-point theorems in lattice theory. The calculus is applied to the construction of several isomorphisms between classes of datatypes. In particular constructive adaptations of theorems in lattice theory about closure operators are shown to yield simulations and isomorphisms between monad structures, and constructive adaptations of theorems in regular algebra are shown to yield isomorphisms between list structures. A question to which any respectable theory of datatypes should provide immediate answers is when two datatypes are isomorphic, i.e. entirely equivalent modulo implementation details. A subsidiary question is when one datatype simulates another. This second question is of interest in its own right but is also important to answering the rst question since isomorphism is frequently reduced to mutual simulation. This paper formulates a number of algebraic laws for the construction of simulations and isomorphisms between datatypes and applies these laws to the construction of several iso-morphisms. Among the isomorphisms we construct are the \monad simulation theorem" and the \monad decomposition theorem". These are general theorems that are used to construct isomorphisms between monads. More speciic isomorphisms we construct are \list decomposi-tion" and \list leapfrog". \List decomposition" expresses an elementary isomorphism between list structures an instance of which is the solution to the so-called \lines-unlines problem" 13]. \List leapfrog" has a similar elementary interpretation. Surprisingly, I have been unable to nd any mention of the monad decomposition theorem in the literature (either in articles or texts on category theory or on lattice theory); even if the theorem is \well-known" its central importance does not seem to be properly recognised. The theorems we present were invented by the following process: we studied theorems in lattice theory about least xed points and closure operators with a view to whether the theorems could be adapted to \constructive" theorems about \map relators" and \monads" in our theory. Conndence that this might prove to be a fruitful process came from the idea that category theory equals constructive lattice theory. This idea is not new. In 1968, Lambek 19] began an article with the sentence 1