This work covers three important aspects of monomials ideals in the three chapters Stanley
decompositions by Jürgen Herzog Edge ideals by Adam Van Tuyl and Local cohomology by Josep
Álvarez Montaner. The chapters written by top experts include computer tutorials that
emphasize the computational aspects of the respective areas. Monomial ideals and algebras are
in a sense among the simplest structures in commutative algebra and the main objects of
combinatorial commutative algebra. Also they are of major importance for at least three
reasons. Firstly Gröbner basis theory allows us to treat certain problems on general
polynomial ideals by means of monomial ideals. Secondly the combinatorial structure of
monomial ideals connects them to other combinatorial structures and allows us to solve problems
on both sides of this correspondence using the techniques of each of the respective areas. And
thirdly the combinatorial nature of monomial ideals also makes them particularly well suited
to the development of algorithms to work with them and then generate algorithms for more
general structures.
