Refactoring Heterogeneous Relation Algebras around Ordered Categories and Converse

We present a reorganisation of popular theories of “reasoning with relational flavour”, including allegories, Kleene algebras, and Dedekind categories, into an relatively symmetric picture using ordered categories as common base and defining converse independently from joins and meets. As an example application, we use this to regroup results about formalisation of algebraic graph rewriting and thus exhibit opportunities for applying these approaches in new settings. Finally we discuss how this approach influences the design of compatible approaches to formalisation and mechanisation of relation-algebraic theories.