Ideals generated by reverse lexicographic segments

Let k be a field, and let S = k[x_1, . . . , x_n] be the polynomial ring in x_1, . . . , 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.