This paper focuses on asymptotic properties of random monomial ideals through a statistical viewpoint. It extends the study of redundancy in monomial ideals by analyzing the poset density of the LCM-lattice. We explore how this density behaves across random algebraic models and structured networks. Experimental data reveal that the LCM-lattice exhibits sharp threshold behavior rather than changing smoothly. We observe a strong negative correlation between the number of generators and LCM-lattice density, abruptly separating three distinct regimes: a low-density Taylor-like regime, a high-density redundant regime, and a narrow transition window. We show that increasing the generator degree causes this density drop to occur at lower probability thresholds. We conclude by conjecturing that for equigenerated squarefree ideals, the LCM-lattice density undergoes a sharp phase transition, analogous to the emergence of giant components in hypergraphs. This suggests that the classical, ideal-by-ideal role of the LCM-lattice as a combinatorial invariant also admits a statistical/asymptotic counterpart: in natural random families, redundancy and resolution-complexity indicators concentrate into distinct typical regimes separated by a narrow transition window.

A monomial ideal is usually represented by its (unique) minimal set of monomial generators. We explore the alternative representation of (square-free) monomial ideals as monotone Boolean functions in order to design and implement efficient algorithms to make computations on monomial ideals. In particular, we use Binary Decision Diagrams (BDD) as an efficient data structure to represent and compute basic operations on monomial ideals (such as sums, intersections, colon ideals, degree, Alexander dual, etc). We describe the results of some computer experiments in order to assess the convenience of using the Boolean representation of monomial ideals.

We introduce the lcm-filtration and stepwise filtration, comparing their performance across various scenarios in terms of computational complexity, efficiency, and redundancy. The lcm-filtration often involves identical steps or ideals, leading to unnecessary computations. To address this, we analyse how stepwise filtration can effectively compute only the non-identical steps, offering a more efficient approach. We compare these filtrations in applications to networks, system signatures, and sensitivity analysis.

We use a monomial ideal I to model a discrete preference function on a set of n factors. We can measure the sensitivity of each point represented by a monomial m by calculating its formal partial derivatives with respect to each variable. These derivatives can be used to define the Koszul simplicial complex of the ideal I at m. We refer to points at which the homology of their Koszul complex is not null as sensitive corners. In the context of preference analysis, the ranks of the homology groups are not precise enough to distinguish between sensitive corners that have the same homology but correspond to different sensitivity behaviors. To address this issue, we propose using a filtration on the Koszul complexes of the sensitive corners based on the lcm-lattice of the ideal I. This filtration induces a persistent homology at each corner m. We then use unsupervised Machine Learning methods to classify the corners based on the distance between their persistence diagrams.

The evaluation of system reliability is an NP-hard problem even in the binary case. There exist several general methodologies to analyze and compute system reliability. The two main ones are the sum-of-disjoint-products (SDP), which expresses the logic function of the system as a union of disjoint terms, and the Improved Inclusion-Exclusion (IIE) formulas. The algebraic approach to system reliability, assigns a monomial ideal to the system and computes its reliability in terms of the Hilbert series of the ideal, providing an algebraic version of the IIE method. In this paper we make use of this monomial ideal framework and present an algebraic version of the SDP method, based on a combinatorial decomposition of the system's ideal. Such a decomposition is obtained from an involutive basis of the ideal. This algebraic version is suitable for binary and multi-state systems. We include computer experiments on the performance of this approach using the C++ computer algebra library CoCoALib and a discussion on which of the algebraic methods can be more efficient depending on the type of system under analysis.

We propose algorithms based on monomial resolution theory for simplicial homology computation. We explore some alternatives that can either be used as a preprocessing step for homology computation or as alternatives to the usual linear algebra approach. We show the results of some computer experiments to demonstrate the performance of a C++ implementation using the computer algebra library CoCoALib.

We apply the methods of algebraic reliability to the study of percolation on trees. To a complete $k$-ary tree $T_{k,n}$ of depth $n$ we assign a monomial ideal $I_{k,n}$ on $\sum_{i=1}^n k^i$ variables and $k^n$ minimal monomial generators. We give explicit recursive formulae for the Betti numbers of $I_{k,n}$ and their Hilbert series, which allow us to study explicitly percolation on $T_{k,n}$. We study bounds on this percolation and study its asymptotical behavior with the mentioned commutative algebra techniques.

The algebraic approach to the analysis of system reliability associates an algebraic object, a monomial ideal, to a coherent system (CS), and studies the reliability of the system using the Hilbert series of the monomial ideal. New capabilities of the algebraic method in system design are shown, in particular related to enumeration of working states. The algebraic method should be a useful tool in reliability, both for performing different computations on system features, and to study the structure of systems.

Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n - 1), with n the number of indeterminates of the ring, plays an important role for this algebraic description in terms of Stanley and irreducible decompositions. This feature of (n - 1)-st Koszul homology allows us to transform an algorithm that computes Koszul homology of monomial ideals to use it for the computation of irreducible and Stanley decompositions. This is an example of how algorithms and structures specifically targeted to computations on monomial ideals should take into account the combinatorial properties of them to produce efficient methods, an issue that is worth introducing into modern computer algebra systems. To illustrate this fact we present some details on the implementation of the algorithm in CoCoALib.