Entropy invariants: I. The universal order relation of order-preserving star products

For multimodal maps, a general form of order-preserving star products is presented. A universal order relation of order-preserving star products is found. By means of the universal order relation, the generality of the second entropy invariant is generally proved for multimodal maps. The universal order relation is also significant for the study of the grammatical complexity of dynamical systems and the knot theory.