Normal Forms and Reduction for Theories of Binary Relations

We consider equational theories of binary relations, in a language expressing composition, converse, and lattice operations. We treat the equations valid in the standard model of sets and also define a hierarchy of equational axiomatisations stratifying the standard theory. By working directly with a presentation of relation-expressions as graphs we are able to define a notion of reduction which is confluent and strongly normalising, in sharp contrast to traditional treatments based on first-order terms. As consequences we obtain unique normal forms, decidability of the decision problem for equality for each theory. In particular we show a non-deterministic polynomial-time upper bound for the complexity of the decision problems.