We generalize Jiroušek's (\emph {right}) \emph {composition operator} in such a way that it can be applied to distribution functions with values in a "semifield", and introduce (parenthesized) \emph {compositional expressions}, which in some sense generalize Jiroušek's "generating sequences" of compositional models. We say that two compositional expressions are \emph {equivalent} if their evaluations always produce the same results whenever they are defined. Our first result is that a set system $\cal H$ is star-like with centre $X$ \emph {if and only if} every two compositional expressions with "base scheme" $\cal H$ and "key" $X$ are equivalent. This result is stronger than Jiroušek's result which states that, if $\cal H$ is star-like with centre $X$, then every two generating sequences with base scheme $\cal H$ and key $X$ are equivalent. Then, we focus on \emph {canonical expressions}, by which we mean compositional expressions $\theta$ such that the sequence of the sets featured in $\theta$ and arranged in order of appearance enjoys the "running intersection property". Since every compositional expression, whose base scheme is a star-like set system with centre $X$ and whose key is $X$, is a canonical expression, we investigate the equivalence between two canonical expressions with the same base scheme and the same key. We state a graphical characterization of those set systems $\cal H$ such that every two canonical expressions with base scheme $\cal H$ and key $X$ are equivalent, and also provide a graphical algorithm for their recognition. Finally, we discuss the problem of detecting conditional independences that hold in a compositional model.
compositional expression, compositional model, running intersection property, perfect sequence
05C65, 05C85, 1699, 65C50, 60E99, 68T37