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// generally is used with one argument, but Map requires two
arguments, so the first argument is wrapped
This is called an operator form and is documented.
Map applies the function only to level 1 elements
Map can be applied to whatever level(s) you want using the three argument form. Check the documentation.
l = {{a, {a, a}}, {b, {b, {b}}}, {c, {c, {c, {c, c}}}}}
ClearAll@f
(* Levels 1 to 3 *)
Map[f, l, 3]
(* {f[{f[a], f[{f[a], f[a]}]}], f[{f[b], f[{f[b], f[{b}]}]}], f[{f[c], f[{f[c], f[{c, {c, c}}]}]}]} *)
(* Level 4 only *)
Map[f, l, {4}]
(* {{a, {a, a}}, {b, {b, {f[b]}}}, {c, {c, {f[c], f[{c, c}]}}}} *)
