  What is ergonomic syntax for relations? In this first paper in a series of
two, to answer the question we define regular calculi: a suitably structured
functor from a category representing the syntax of regular logic to the
category of posets, that takes each object to the poset of relations on that
type. We introduce two major classes of examples, regular calculi corresponding
to regular theories, and regular calculi corresponding to regular categories.
For working in regular calculi, we present a graphical framework which takes as
primitive the various moves of regular logic. Our main theorem for regular
calculi is a syntax-semantics $2$-dimensional adjunction to regular categories.
