Over

is a primitive dyadic operator which takes two function operands and produces a derived function which pre-processes the argument(s) using the monadic right operand, before applying the left operand on/between the result(s).

Explanation
When the resulting function is used monadically, it has the same behaviour as if the Beside or Atop operator had been used: When the resulting function is used dyadically, both arguments are pre-processed:

Close composition
In SHARP APL and J, Over is implemented as a close composition, meaning that (using SHARP syntax)  has the overall function rank of. J uses  for the close form and   for the rankless form that appears in modern APLs.

History
Ken Iverson defined Over in 1978 as part of Operators and Functions, with the glyph. He called it Composition, as there was no Atop operator. It was added to SHARP APL as a close composition with glyph  and name "on", with a limited implementation in 1981 followed by a full implementation in 1983 with the introduction of function rank.

The glyph  appeared in NARS2000, with name "Composition", by 2011. The name "Over" was used in Dyalog APL 18.0, suggested by Adám Brudzewsky. Over also appears in dzaima/APL (since 2018, before Dyalog 18.0), April, and KAP.

Lessons

 * APL Cultivation

Documentation

 * Dyalog
 * J Dictionary, NuVoc
 * BQN