Ideals Generated by Reverse Lexicographic Segments

Let $k$ be a field, and let $S=k[x_1,\dots,x_n]$ be the polynomial ring in $x_1,\dots,x_n$ with coefficients in the field $k$. We study ideals of $S$ which are generated by reverse lexicographic segments of monomials of $S$. An ideal generated by a reverse lexicographic segment is called a completely reverse lexicographic segment ideal if all iterated shadows of the set of generators are reverse lexicographic segments. We characterize all completely reverse lexicographic segment ideals of $S$ and determine conditions under which they have a linear resolution.