Abstract
We extend the work of Galatos (2004) on nested sums, originally called gener-alised ordinal sums, of residuated lattices. We show that the nested sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2025) the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their nested sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable. Quasi relation algebras (qRAs) were first described by Galatos and Jipsen [6]. On the one hand they can be viewed as a generalisation of relation algebras, or as involutive FL-algebras with an additional negation-like unary operation. Unlike relation algebras, the variety of qRAs has a decidable equational theory. In this paper, we first extend the so-called nested sum, originally called the gener-alised ordinal sum by Galatos [5], from residuated lattices to quasi relation algebras. Since the construction for residuated lattices has appeared in tandem with " ordinal 1