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Monomial ideals are at the intersection between commutative algebra, combinatorics, and algebraic geometry, being in the center of many important problems in polynomial rings. Their study developed extensively especially in the last decades when it became a standard technique to get information about the algebraic and homological invariants of polynomial ideals by passing to initial monomial ideals. By using a standard procedure, one may also use specific combinatorial techniques to study invariants of the so called squarefree monomial ideals. All these new developments led to a spectacular progress in the new branch of commutative algebra, which is usually called combinatorial commutative algebra. In this book, several classes of monomial ideals are studied by using algebraic and combinatorial techniques. Special attention is given to lexsegment ideals whose properties concerning resolutions and invariants are presented in detail, and to constructible ideals, for which, deep connections with combinatorics are established. The topic reflects some of the current trends in the development of combinatorial commutative algebra.
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