In this licentiate thesis, we consider questions related to the so called weak or strong Lefschetz properties. These are properties of a graded artinian algebra which asks for the existence of a linear form in the algebra such that the multiplication by that linear form, or multiplications by all powers of it, gives a map which always has full rank on the algebra.

In the general introduction, we give background material for understanding the Lefschetz properties, their definitions, and mention some other standard tools used when studying graded artinian algebras. We then introduce a selection of other common methods for proving that an algebra does or does not have the weak or strong Lefschetz property. This includes monomial orders, Macaulay's inverse system, preservation results and more.

Paper I is joint with Samuel Lundqvist and Lisa Nicklasson. It concerns binomial complete intersections of a specific form we call normal form. For a collection of binomials written on normal form, we associate a family of directed labelled graphs that let us determine several properties of such a family of binomials. We give a monomial basis for the associated algebra, its Macaulay dual generator, and a formula for the resultant. 

Paper II gives an answer to the following question. Given a fixed number of variables and fixed number of minimal generators that are possible for a quadratic artinian ideal, does there exist a quadratic artinian monomial ideal with those specifications having the strong Lefschetz property? The main result of this second paper is showing that this always has a positive answer when working over a field of characteristic zero, or large enough characteristic, by giving a concrete construction of such an ideal. Along the way, several interesting facts about the Hilbert series of these ideals are also established.