Relations into algebras of probabilistic distributions

•The paper gives a relational foundation for relations into algebras of probabilistic distributions.•Probabilistic multirelations are reformulated as certain convex relations, together with stepwise refinement.•How the convexity works in the associativity of composition of convex relations is clarified.

The paper proposes two types of convex relations into algebras of probabilistic distributions as a relational algebraic foundation of semantic domains of probabilistic systems [4], [7] and [8]. Following previous results by Tsumagari [16], we particularly focus on the associative law for the convex compositions defined via bounded combinations of probabilistic distributions, and prove that the convex compositions are associative for convex relations.