Polarization and depolarization of monomial ideals with application to multi-state system reliability

Abstract

Polarization is a powerful technique in algebra which provides combinatorial tools to study algebraic invariants of monomial ideals. We study the reverse of this process, depolarization which leads to a family of ideals which share many common features with the original ideal. Given a squarefree monomial ideal, we describe a combinatorial method to obtain all its depolarizations, and we highlight their similar properties such as graded Betti numbers. We show that even though they have many similar properties, their differences in dimension make them distinguishable in applications in system reliability theory. In particular, we apply polarization and depolarization tools to study the reliability of multi-state coherent systems via binary systems and vice versa. We use depolarization as a tool to reduce the dimension and the number of variables in coherent systems. arXiv:1807.05743v2 [math.AC]

ALGEBRAIC RELIABILITY OF MULTI-STATE k-OUT-OF-n SYSTEMS

“…In order to illustrate the algebraic method for system reliability analysis, we will use a simple example in which we will use all the concepts involved. A general detailed description and plenty of more elaborate examples can be found in [35,[42][43][44][45][46] where the interested reader can find full proofs of the relevant results for this approach.…”

A C++ class for algebraic reliability computations

“…The approach we follow was developed in a series of papers, e.g. [25,26,27]. The main idea is to associate an algebraic object to a coherent system and obtain information about the structure and reliability of the system by investigating the properties of the algebraic object.…”

“…Our future work includes the design of specialized algebraic algorithms for particular kinds of systems. The structure of particular systems induces a particular structure in the associated ideals which can be studied using algebraic and combinatorial tools allowing the design of more efficient algorithms, as described for instance in [26,27]. Another direction of improvement is to adapt the algorithm for systems with non-independent components.…”

Support posets of some monomial ideals

“…, x n and the minimal monomial generators of I. It was introduced in [12] to study the set of all depolarizations of a given squarefree monomial ideal. Since many relevant features of a given monomial ideal are shared by the ideals in the same polarity class, the study of polarization and depolarization has become relevant in the last years cf.…”