Combinatorially rich sets in partial semigroups

There are several notions of size for semigroups that have natural analogues for partial semigroups. Among these are thick, syndetic, central, piecewise syndetic, IP, J, and the more recently introduced notion of combinatorially rich, abbreviated CR. We investigate the notion of CR set for adequate partial semigroups, its relation to other notions, especially J sets, and some surprising differences among them.