Define an operator with the distributive property

Question

I would like to define the ⊕ operator with distributive, associative, and commutative properties - so that Mathematica can symbolically simplify expressions I use it in. For example:

in := Simplify[a b ⊕ a c]
out := a (b ⊕ c)

I've already given the ⊕ operator the Flat and Orderless attributes, giving the operator the other two properties I would like, but I can't figure distribution out.

Specifically, what I'm trying to do is define an operator which has the symbolic properties of Plus, but which does not evaluate against numbers, e.g

in := 2⊕2
out := 2⊕2  

is a fully simplified expression.

Answer 1

This would perhaps do what you want:

CirclePlus /: 
 Simplify[CirclePlus[pre___, a_ b_, mid___, a_ c_, post___]] := 
 Simplify@Apply[CirclePlus, 
   Simplify /@ {pre, mid, a Simplify[CirclePlus[b, c]], post}]

Since Distribute by default only works with Plus, I associate the distributive property with CirclePlus manually when Simplify is applied. I then allow Simplify to be applied recursively to the result.

Edit

Thanks to Karsten 7 for pointing out that the special case of CirclePlus with a single argument should also be handled. The Flat attribute by itself doesn't do that because it only kicks in at the pattern recognition stage and not as a simplification rule. So you could add the following:

CirclePlus[x : Except[_CirclePlus]] := x

Simplify[a b⊕a c]

(* ==> a (b⊕c) *)

I added the Except condition in case you already gave CirclePlus the Flat attribute of Plus.