Properties of Multiset Orders by Minimal and Maximal Submultisets

We analyze the relations between the multisets and their submultisets involved in the multiset order M>msoand derive many properties that can be used in proofs. These properties are used to refine some proofs of known results, like the transitivity or the termination of >mso. These properties also enable a better understandingof the underlying theory and can be use din implementations of theorem provers. For two finite multisets M, N, there can be several pairs of their submultisets that satisfy M>mso N, which can be seen as solutions to the equation M>mso N. We determine the number of solutions that satisfy M>mso N and establish an order between them, not total, but admittinga minimum and a maximum. We determine the formulae for the minimal submultisets and provide several algorithmsto find the maximal submultisets. The minimal submultisetsare necessary and sufficient to determine if M>mso N. The minimal and maximal submultisets also allow for a deeperanalysis in termination problems with multiset orders, being able to determine, for instance, how fast a program can terminate.