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Obverse u :. v  mu lu ru  

The result of u :. v is the verb u, but with an assigned obverse v (used as the “inverse” under the conjunctions &. and ^:).
 

For example:
   y=: _4 0 4 3j4
   rp=: <@(%: , -@%:)"0        NB. Root pairs
   rp y
+--------+---+----+---------+
|0j2 0j_2|0 0|2 _2|2j1 _2j_1|
+--------+---+----+---------+

   I=: ^: _1
   rp I                        NB. No assigned obverse
rp^:_1

   rp I rp y
|domain error
|     rp I rp y

   inv=: *:@{.@,@>
   inv rp y
_4 0 4 3j4

   RP=: rp :. inv              NB. Assigned obverse in RP
   RP I RP y
_4 0 4 3j4

   rc=: <@(,: +)@(, -)@%:"0    NB. Root companions
   rc y
+---------+---+----+----------+
| 0j2 0j_2|0 0|2 _2| 2j1 _2j_1|
|0j_2  0j2|0 0|2 _2|2j_1  _2j1|
+---------+---+----+----------+

   RC=: rc :. inv
   RC I RC y
_4 0 4 3j4


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