Every incidence structure ${\mathcal J}$ (understood as a triple of sets $(G, M, I)$, ${I}\subseteq G \times M$) admits for every positive integer $p$ an incidence structure ${\mathcal J}^p=(G^p, M^p, \mathrel {{\mathrm I}^p})$ where $G^p$ ($M^p$) consists of all independent $p$-element subsets in $G$ ($M$) and $\mathrel {{\mathrm I}^p}$ is determined by some bijections. In the paper such incidence structures ${\mathcal J}$ are investigated the ${\mathcal J}^p$’s of which have their incidence graphs of the simple join form. Some concrete illustrations are included with small sets $G$ and $M$.