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

The infinite algebras with 3-transitive groups of weak automorphisms are investigated. Among others it is shown that if an infinite algebra with 3-transitive group of weak automorphisms has a nontrivial idempotent polynomial operation then either it is locally functionally complete or it is polynomially equivalent to a vector space over the two element field or it is a simple algebra that is semi-affine with respect to an elementary 2-group. In the second and third cases the group of weak automorphisms...