ABSTRACT
Lexsegment ideals in this generality were first introduced in polynomial rings by Hulett and Martin [1]. In extremal combinatorics one usually considers more special lexsegment ideals, which we call initial lexsegment ideals. These are ideals which in each degree are generated by initial lexsegments, that is, sets of the form: Ll(u) = {w € Md/w > u}, called simple lexsegment in the usual terminology usually are considered. The support of a monomial u G E is supp(u) = {i : e^ divides u}. The shadow of a set S of monomials in M^ is the set of the non-zero monomials Shad(S) = {sei : Vs € 5, Vz ^ supp(s)} . We define the i-th shadow recursively by Shadz(5) = Shad(Shadz~1(5)). The initial lexsegments have the nice property that their shadow is again an initial lexsegment. This important property of initial lexsegments is crucial in the proof of Kruskal-Katona theorem. This fact is not true for arbitrary lexsegments. A lexsegment L is called completely lexsegment [1], if all the iterated shadows of L are again lexsegments, that is, if for each i the set Shad'(L) is a lexsegment. De Negri and Herzog
in [2] characterized the completely lexsegments in polynomial rings and gave sufficient conditions for an ideal generated by a completely lexsegment to have a linear resolution.
