  A certain type of integer grid, called here an echelon grid, is an object
found both in coherent systems whose components have a finite or countable
number of levels and in algebraic geometry. If \alpha=(\alpha_1,...,\alpha_d)
is an integer vector representing the state of a system, then the corresponding
algebraic object is a monomial x_1^{\alpha_1}... x_d^{\alpha_d} in the
indeterminates x_1,..., x_d. The idea is to relate a coherent system to
monomial ideals, so that the so-called Scarf complex of the monomial ideal
yields an inclusion-exclusion identity for the probability of failure, which
uses many fewer terms than the classical identity. Moreover in the ``general
position'' case we obtain via the Scarf complex the tube bounds given by Naiman
and Wynn [J. Inequal. Pure Appl. Math. (2001) 2 1-16].
  Examples are given for the binary case but the full utility is for general
multistate coherent systems and a comprehensive example is given.
