Untying of Clutter-Family Supports and Its Role in the Monotone-Structure Reliability Theory

We suggest a transformation of supports of a clutter family. It is a natural generalization of the untying transformation of a one-clutter support previously introduced by the author. We show that the new transformation does not decrease the reliability of the corresponding clutter sum. We give new bounds for the clutter-sum reliability. We demonstrate that the untying transformation of clutter-family supports and the factor transformation introduced by McDiarmid provide a combinatorial basis for the monotone-structure reliability theory.