IMO we should interpret the intersection and the union of two (multi)sets (sets
X with X >:&# ~.X) in the same way as is done by the GCD and the LCM of two
numbers.
So for two numbers N and M with (multi)sets of primefactors A and B, we should
have (N +. M) = */(A IS B) and (N *. M) = */ (A UN B), with IS=intersection and
UN = union of the (multi)sets A and B.
Obviously we require (A IS B) =&(/:~) B IS A and also (A UN B) =&(/:~) B UN A
for all (multi)sets A and B.
E.g.
1 2 3 3 3 IS 3 3 3 3 3 4 5 6
3 3 3
1 2 3 3 3 UN 3 3 3 3 3 4 5 6
1 2 3 3 3 3 3 4 5 6
What are elegant and efficient J-definitions for IS and UN?
I have constructed
UN_reb=: ([:(~.@{.#~ {. >.//.{:) |:@,&(({.,#)/.~))
and
IS_reb=: ([:(~.@{.#~ {. <.//.{:) ([|:@, ],~ 0,.~ -.~&:({."1),
-.&:({."1))&(({. , #)/.~))
1 2 3 3 3 UN_reb 3 3 3 3 3 4 5 6
1 2 3 3 3 3 3 4 5 6
1 2 3 3 3 IS_reb 3 3 3 3 3 4 5 6
3 3 3
1 2 3 3 3 UN_reb~ 3 3 3 3 3 4 5 6
3 3 3 3 3 4 5 6 1 2
1 2 3 3 3 IS_reb~ 3 3 3 3 3 4 5 6
3 3 3
a=: 1e7?.@#1e6
ts'a UN_reb a+499999'
0.76631908 4.6996013e8
#a UN_reb a+49999
12186812
ts'a IS_reb a+499999'
0.76073743 3.3554592e8
#a IS_reb a+49999
7813188
R.E. Boss
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