  Geometric motivations warranted the study of hypergraphs on ordered vertices
that have no pair of hyperedges that induce an alternation of some given
length. Such hypergraphs are called ABA-free, ABAB-free and so on. Since then
various coloring and other combinatorial results were proved about these
families of hypergraphs. We prove a characterization in terms of their
incidence matrices which avoids using the ordering of the vertices. Using this
characterization, we prove new results about the dual hypergraphs of ABAB-free
hypergraphs. In particular, we show that dual-ABAB-free hypergraphs are not
always proper $2$-colorable even if we restrict ourselves to hyperedges that
are larger than some parameter $m$.
