  The present paper studies multiple failure and signature analysis of coherent
systems using the theory of monomial ideals. While system reliability has been
studied using Hilbert series of monomial ideals, this is not enough to
understand in a deeper sense the ideal structure features that reflect the
behavior of the system under multiple simultaneous failures and signature.
Therefore, we introduce the lcm-filtration of a monomial ideal, and we study
the Hilbert series and resolutions of the corresponding ideals. Given a
monomial ideal, we explicitly compute the resolutions for all ideals in the
associated lcm-filtration, and we apply this to study coherent systems. Some
computational results are shown in examples to demonstrate the usefulness of
this approach and the computational issues that arise. We also study the
failure distribution from a statistical point of view by means of the algebraic
tools described.
