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.