There is nothing to say that ≼ can not be used to denote a total ordering. A partial ordering becomes total if it can compare any two x and y.
You need more than a way to compare boxes. You need a way to compare any two arrays, and that's what f=: 0 1 -: /:@,&< does. In other words, you don't open boxes and then compare the contents (≼&>); you box the two arrays and then compare (grade) them (/:@,&<). In an array language with a different notation of arrays, total ordering would be defined differently, and the differences are subtle. See *TAO Axioms <http://www.jsoftware.com/papers/TAOaxioms.htm>* . On Sat, Mar 2, 2019 at 5:10 AM R.E. Boss <r.e.b...@outlook.com> wrote: > I remember the “curly less than” (or equal) as a notation for partial > ordering, like that of the complex numbers or vectors. > But such an ordering cannot be used for grading, as grading needs a > complete ordering. > And since grading uses the lexicographical and type ordering (as you > showed in your last example), we could easily define comparison on > characters and boxes by > > < ←→ ≼ for characters > < ←→ ≼&> for boxes > > It wouldn’t even hurt existing programs. > > > R.E. Boss > > p.s. I would have preferred all primitives to have infinity rank anyhow, > but that is another discussion > > > > -----Oorspronkelijk bericht----- > > Van: Programming <programming-boun...@forums.jsoftware.com> > > Namens Roger Hui > > Verzonden: zaterdag 2 maart 2019 04:47 > > Aan: programm...@jsoftware.com > > Onderwerp: Re: [Jprogramming] Comparing comparisons > > > > For the ordering used in sorting, what you want is the symbol denoted by > ≺ > > "curly less than" U+227A or ≼ "curly less than equal" U+227C in > conventional > > mathematical notation, with infinite dyadic ranks. It's not enough to > make < > > or <: work on characters because those functions have 0 dyadic rank. > > > > ≼ ←→ 0 1 -: /:@, > > > > f=: 0 1 -: /:@,&< > > 'foo' f 'upon' > > 1 > > 'syzygy' f 'chthonic' > > 0 > > 'pi' f 3.14159 > > 0 > > > > > > > > On Fri, Mar 1, 2019 at 5:10 PM Jimmy Gauvin <jimmy.gau...@gmail.com > <mailto:jimmy.gau...@gmail.com>> > > wrote: > > > > > Hi, > > > > > > 'c' < 'b' doesn't work in APL either but 'c'='d' works in J and in APL. > > > > > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
