Representability of pairing relation algebras depends on your ontology

We consider classes of relation algebras expanded with new operations based on the formation of ordered pairs. Examples for such algebras are pairing (or projection) algebras of algebraic logic and fork algebras of computer science. It is proved by Sain and Nemeti [38] that there is no `strong' representation theorem for all abstract pairing algebras in most set theories including ZFC as well as most non-well-founded set theories. Such a `strong' representation theorem would state that every abstract pairing algebra is isomorphic to a set relation algebra having projection elements which are defined with the help of the real (set theoretic) pairing function. Here we show that, by choosing an appropriate (non-well-founded) set theory as our metatheory, pairing algebras and fork algebras admit such `strong' representation theorems.