Mapping function for two integers

SO,

The problem

I have two integers, which are in first case, positive, and in second case - just any integers. I need to create a map function F from them to some another integer value, which will be:

• Result should be integer value. For first case (x>0, y>0), positive integer value
• Symmetric. That means F(x, y) = F(y, x)
• Unique. That means F(x₀, y₀) = F(x₁, y₁) <=> (x₀ = x₁ ^ y₀ = y₁) V (y₀ = x₁ ^ x₀ = y₁)

My approach

At first glance, for positive integers we could use expression like F(x, y) = x² + y², but that will fail - for example, 89² + 23² = 13² + 91² As for second (common) case - that's even more complicated.

Use-case

That may be useful when dealing with some things, which supposed to be order-independent and need to be unique. For example, if we want to find cartesian product of many arrays and we want result to be unique independent of order, i.e. <x,z,y> is equal to <x,y,z>. It may be done with:

function decartProductPair($one, $two, $unique=false)
{
   $result = [];
   for($i=0; $i<count($one); $i++)
   {
      for($j=0; $j<count($two); $j++)
      {
         if($unique)
         {
            if($i!=$j)
            {
               $result[$i*$i+$j*$j]=array_merge((array)$one[$i],(array)$two[$j]);
               //           ^
               //           |
               //           +----//this is the place where F(i,j) is needed
            }
         }
         else
         {
            $result[]=array_merge((array)$one[$i], (array)$two[$j]);
         }
      }
   }
   return array_values($result);
}

Another use-case is to properly group sender and receiver in some SQL table, so that different senders/receivers will be differed while they should stay symmetric. Something like:

SELECT
  COUNT(1) AS message_count,
  sender,
  receiver
FROM
  test
GROUP BY
-- this is the place where F(sender, receiver) is needed:
  sender*sender + receiver*receiver

(By posting samples I wanted to show that issue is certainly related to programming)

The question

As mentioned, the question is - what can be used as F? I want as simple F as it's possible. Keep in mind two cases:

• Integer x>0, y>0. F(x,y) > 0
• Any integer x, y and so any integer F(x,y) as a result

May be F isn't just an expression - but some algorithm to find desired result for any x,y (so tagging with algorithm too). However, expression is better because it's more like that it will be able to use that expression in SQL or PHP or whatever. Feel free to edit tagging because I'm not sure if two tags here is enough