Representability for distributive quasi relation algebras via generalised ordinal sums

Abstract: We extend the work of Galatos (2004) on generalised ordinal sums of residuated lattices. We show that the generalised ordinal sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a paper by Craig and Robinson (2024), the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their generalised ordinal sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.