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Three basic properties that standard graded artinian $k$-algebras may or maynot enjoy are the Weak and Strong Lefschetz Properties and the Maximal RankProperty (respectively WLP, SLP, and MRP).  In this paper we will assume that the base field $k$ has characteristic zero.It is known that SLP implies MRP, which in turn implies WLP, but that bothimplications are strict. However, it surprisingly turned out that the set ofHilbert functions admitting any algebras with WLP coincides with thecorresponding set for SLP (and therefore with that for MRP). J. Migliore andthe first author, using Green's theorem and a result of Wiebe, characterizedthe Hilbert functions forcing all algebras to enjoy WLP. The purpose of thisnote is to prove the corresponding characterizations for both SLP and MRP.Unsurprisingly (or surprisingly??), the two characterizations coincide, butthey define a class of Hilbert functions strictly smaller than that determinedfor WLP.  Our methods include the Herzog-Popescu theorem on quotients of $k$-algebrasmodulo a general form, a result of Wiebe, and gins and stable ideals. At theend, we will also discuss the importance of assuming that the characteristic bezero, and we will exhibit a class of codimension 2 monomial completeintersections for which SLP (but not MRP) fails in positive characteristic.

Forcing the Strong Lefschetz and the Maximal Rank Properties