In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose LCM lattice is P , and we give a complete characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atomic lattices with n ordered atoms can be realized as deformations of exponents of monomial ideals. We also give structural results for L(n). Moreover, we prove that the cellular structure of a minimal free resolution of a monomial ideal M can be extended to minimal resolutions of certain monomial ideals whose LCM lattices are greater than that of M in L(n).

In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial ideals are lattice-linear and thus their minimal resolution can be constructed as a poset resolution. We then use this result to give a description of the minimal free resolution of a larger class of rigid monomial ideals by using L(n), the lattice of all lcm-lattices of monomial ideals with n generators. By fixing a stratum in L(n) where all ideals have the same total Betti numbers we show that rigidity is a property which is upward closed in L(n). Furthermore, the minimal resolution of all rigid ideals contained in a fixed stratum is shown to be isomorphic to the constructed minimal resolution.

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We introduce the package Posets for Macaulay2. This package provides a data structure and the necessary methods for working with partially ordered sets, also called posets. In particular, the package implements methods to enumerate many commonly studied classes of posets, perform operations on posets, and calculate various invariants associated to posets. INTRODUCTION.A partial order is a binary relation over a set P that is antisymmetric, reflexive, and transitive. A set P together with a partial order is called a poset, or partially ordered set. We refer the reader to the seminal text [Stanley 2012] for definitions omitted herein.Posets are combinatorial structures that are used in modern mathematical research, particularly in algebra. We introduce the package Posets for Macaulay2 [Grayson and Stillman] via three distinct posets or related ideals which arise naturally in combinatorial algebra.We first describe two posets that are generated from algebraic objects. The intersection semilattice associated to a hyperplane arrangement can be used to compute the number of unbounded and bounded real regions cut out by a hyperplane arrangement, as well as the dimensions of the homologies of the complex complement of a hyperplane arrangement.Given a monomial ideal, the lcm-lattice of its minimal generators gives information on the structure of the free resolution of the original ideal. Specifically, two monomial ideals with isomorphic lcm-lattices have the "same" (up to relabeling) minimal free resolution, and the lcm-lattice can be used to compute, among other things, the multigraded Betti numbers

This paper gives a description of various recent results which construct monomial ideals with a given minimal free resolution. We show that these are all instances of coordinatizing a finite atomic lattice as found in [10]. Subsequently, we explain how in some of these cases ([4], [5]), where questions still remain, this point of view can be applied. We also prove an equivalence in the case of trees between the notion of maximal defined in [5] and a notion of being maximal in a Betti stratum. arXiv:1509.06298v1 [math.AC] 21 Sep 2015