  Well ordered covers of square-free monomial ideals are subsets of the minimal
generating set ordered in a certain way that give rise to a Lyubeznik
resolution for the ideal, and have guaranteed nonvanishing Betti numbers in
certain degrees. This paper is about square-free monomial ideals which have a
well ordered cover. We consider the question of subadditivity of syzygies of
square-free monomial ideals via complements in the lcm lattice of the ideal,
and examine how lattice complementation breaks well ordered covers of the ideal
into (well ordered) covers of subideals. We also introduce a family of well
ordered covers called strongly disjoint sets of simplicial bouquets
(generalizing work of Kimura on graphs), which are relatively easy to identify
in simplicial complexes. We examine the subadditivity property via numerical
characteristics of these bouquets.
