Characterization of laws by balancing weighting against a binary operation

Let L(X) be the law of a positive random variable X, and Z be positive and independent of X. Solution pairs (L(X),L(Z)) are sought for the in-law equation (X) over cap congruent to X o Z, where is a weighted law constructed from L(X), and o is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When o is the maximum operation a complete solution in terms of a product integral is found for arbitrary weighting. Examples are given. An identity for the length biasing operator is used when o is multiplication to establish a general solution in terms of an already solved inverse equation. Some examples are given.