Kay Schluehr wrote: > > Now lets drop the assumption that a and b commute. More general: let be > M a set of expressions and X a subset of M where each element of X > commutes with each element of M: how can a product with factors in M be > evaluated/simplified under the condition of additional information X? > > It would be interesting to examine some sorting algorithms on factor > lists with constrained item transpositions. Any suggestions? >

## Advertising

Hello Kay, take this into account: Restrictions like commutativity, associative, distributive and flexibility laws don't belong neither to operands nor to operators themselves. Instead these are properties of fields (set of numbers with respect to a certain operation). For a famous example for a somewhat "alternative" behaviour look at the Octonions (discovered in 1843 by Graves and 1845 by Cayley), which are not associative with respect to addition and/or multiplication. (http://en.wikipedia.org/wiki/Octonions) or the Quarternions, which are non-commutative (http://en.wikipedia.org/wiki/Quaternion) Obviously, it's not correct to say: addition is associative, or, that multiplication is. With the same right, you could say, multiplication is not associative. With the same reasoning, we can show that it's not easy to generalize sorting, commutation, association or distribution mechanisms. Maybe it would be a very fascinating goal to solve your algorithmic approach in such a limited environment like the Quarternions. A solution for this set of numbers, if achieved in a clean, mathematically abstract way, should hold for most other numbers/fields too, natural and real included. I guess that the approach might be this way: - define/describe the fields which shall be handled - define/describe the rules which shall be supported - find methods to reduce sequences of operations to simple binary or unary operations (tokens) - this may introduce brackets and stacking mechanisms - a weighing algorithm might be necessary to distinguish between plain numbers and place holders (variables) - application of the distributivity (as far as possible) might help to find a rather flat representation and a base for reordering according to the weights of the individual sub-expressions Nevertheless, there are lots of commercial programs which do such sort of symbolic mathematics, and which would badly fail when it would come to such awkward fields like Quarternions/Octonions. Bernhard -- http://mail.python.org/mailman/listinfo/python-list