A common way to define multiplication for Dedekind cuts is to first define it for pairs of positive reals, and to then extend it to general pairs of reals case by case. Is there an alternative definition that is less ham-fisted?
edit: I suppose one way is to identify the positive and negative reals with the upward- and downward-portions of their cuts, respectively. Multiplication can then just be defined element-wise... but I think this complicates the other elementary operation, addition.
As noted in the comments, it seems unlikely to entirely avoid some sign considerations in the multiplication of Dedekind cuts. This is so because of the direction reversal when multiplying by a negative number, and so, in some sense, the two halves of a Dedekind cut need to switch place, being the source of the mild headache.
There are plenty of constructions of the real numbers (see here for a survey of 20 constructions), some of which give more direct and simple definitions of the algebraic operations. In particular, see here for a detailed construction using minimal rational filters for a construction in which the definition of addition and multiplication (as well as the order structure) are very straightforward.
