custom notation - How to re-parse (or re-map) character '.' to user-defined function

The default behavior of the character . between two non-numeric characters is to be interpreted as the infix of the Mathematica function Dot.

Is there a command that can be placed in a package to get the front-end to parse . as an infix for another function -- say user-defined one, myDot[]?

That is,

1. 
   

   a.b should be interpreted as myDot[a, b]

   
   



2. 
   

   myDot[a.b] should map to a.b

   
   


3. 
   

   Dot[a,b] should not get mapped to a.b, but the functionality of the built-in function Dot should not be lost.

   
   


4. 
   

   3.14 should remain a single number, where here the dot indicates a decimal separator.

   
   

Answer

Updated

First, let's stop Dot from creating these box structures.

MakeBoxes[Dot[x__], form_] := 
 RowBox[{"Dot", "[", RowBox@Riffle[MakeBoxes /@ {x}, ","], "]"}]

Next, let's specify that these structures should instead be interpreted as myDot:

MakeExpression[RowBox@(row : {PatternSequence[_, "."] .., _}), form_] := 
 MakeExpression[
  RowBox@{"myDot", "[", RowBox@Riffle[row[[1 ;; -1 ;; 2]], ","], "]"}, form]

The slicing and riffling is ugly... there is probably a cleaner way to do it. ReplacePart seemed to be doing weird things when I tried it but I didn't spend any time on it.

The last step is to specify that myDot should also be rendered into the same structure:

MakeBoxes[myDot[x__], form_] := RowBox@Riffle[MakeBoxes /@ {x}, "."]

Now,

Dot[a, b, c]

Dot[a, b, c]

Dot[{a, b, c}, {x, y, z}]

a x + b y + c z

a.b.c

a.b.c

a.b.c // FullForm

myDot[a, b, c]

{a, b, c}.{x, y, z} // InputForm

myDot[{a, b, c}, {x, y, z}]

a.(b^2).c

a.b².c

% // InputForm

myDot[a, b^2, c]

In all cases you will have to use MakeExpression to override the interpretation of user input, but you may also find TagBox and InterpretationBox interesting alternatives to the MakeBoxes implementations above.