Alexander Duality and Moments in Reliability Modelling

Abstract.There are strong connections between coherent systems in reliability for systems which have components with a finite number of states and certain algebraic structures. A special case is binary systems where there are two states: fail and not fail. The connection relies on an order property in the system and a way of coding states ${{\alpha=(\alpha_1, \ldots, \alpha_d)}}$ with monomials ${{x^\alpha=(x_1^{{\alpha_1}}, \ldots, x_d^{{\alpha_d}})}}$. The algebraic entities are the Scarf complex and the associated Alexander duality. The failure ‘‘event’’ can be studied using these ideas and identities and bounds derived when the algebra is combined with probability distributions on the states. The xα coding aids the use of moments μα=E(Xα) with respect to the underlying distribution.