formatting - How does MakeBoxes handle an n-ary operator?

I want to format results from Reduce using the $\cup$ symbol instead of the $\lor$ symbol. For example, Reduce[x^2 > 4, x, Reals] produces $x<-2\lor x>2$ while I would like it to produce $x<-2\cup x>2$.

My current solution is to add the head xO to use in place of the Or head and associate the appropriate formatting with the xO head using MakeBoxes.

xO /: MakeBoxes[xO[x_, y_], form_] :=                                                             
    RowBox[{MakeBoxes[x, form], "\[Union]", MakeBoxes[y, form]}];

When I want to invoke the special formatting, I simply substitute xO for Or.

Reduce[x^2 > 4, x, Reals] /. Or -> xO

This produces the desired result of $(x<-2)\cup (x>2)$. I am feeling like a master bit-twiddler at this point.

Of course, my next test case didn't work.

Reduce[x^3 + 2 x^2 - x - 2 == 0] /. Or -> xO
(* xO(x==-2,x==-1,x==1) *)

My MakeBoxes[xO[x_, y_]] definition above does not match the case when xO has three arguments. I added the following to limp along a little further.

xO /: MakeBoxes[xO[x_, y_, z_], form_] :=                                                         
    RowBox[{MakeBoxes[x, form], "\[Union]", MakeBoxes[y, form], "\[Union]", MakeBoxes[z, form]}];

This works for three arguments, but is clearly not a robust solution. What is the general method for an n-ary function?

I tried several variations of the following:

xO /: MakeBoxes[xO[x_, y__], form_] :=                                                            
    RowBox[{MakeBoxes[x, form], "\[Union]", MakeBoxes[y, form]}];     

which produced the error:

MakeBoxes::argt: "MakeBoxes called with 3 arguments; 1 or 2 arguments are expected"

when applied to the Reduce[x^3 + 2 x^2 - x - 2 == 0] /. Or -> xO test case.

I am probably missing something obvious. Any ideas?

Answer

You were almost there. You just need to use the multiple-argument pattern, and generalize your code accordingly to create the internals of RowBox programmatically:

xO /: MakeBoxes[xO[x___], form_] :=
  RowBox[
    Riffle[
      Map[MakeBoxes[#, form] &, {x}],
      "\[Union]"

    ]
  ]

Note however that the above implementation leaks evaluation. It may or may not be a problem, but for example here:

x = 1;
xO[x == 1, x > 1, x < 1]

one may argue that the desired result should not be sensitive to the possible global values that x may have, so the result:

(* True \[Union] False \[Union] False *)

may be unsatisfacory. Thus, here is a more careful version:

ClearAll[xO];
SetAttributes[xO, HoldAllComplete]; 
xO /: MakeBoxes[xO[x___], form_] :=
  RowBox@Riffle[
     List @@ Replace[
       HoldComplete[x],
       elem_ :> With[{eval = MakeBoxes[elem, form]}, eval /; True],
       {1}
     ],

     "\[Union]"
  ]

which now gives

xO[x == 1, x > 1, x < 1]

(* x == 1 \[Union] x > 1 \[Union] x < 1 *)