Completions and Complete Representations

Abstract

The title of this chapter indicates a rather technical topic, but it can also be thought of as a foundational issue in logic. The question to be considered is this: to what extent can we use an abstract mathematical language to express and reason about relations? Going back at least as far as Augustus de Morgan [Mor,60], a relation can be defined explicitly, as a set of tuples of some fixed length. This allows us to focus on the mathematical aspects of relations and ignore other more problematic features that might arise from other approaches, such as a linguistic analysis of the use of relations in natural language. In order to treat relations algebraically, we consider them abstractly, identify certain relational operations (e.g., the operation of taking the converse of a binary relation) and write down some equational axioms which are sound for the chosen kind of relations (e.g., a binary relation is equal to the converse of its converse). Ideally, our set G of equations will be equationally complete, so that any equation valid over fields of relations of a certain rank equipped with the chosen set-theoretically definable operators will be entailed by Γ.