Componentwise Combination Operator

Hi,

I am looking to find a componentwise combination operator to resolve a problem. I would really appreciate help (maybe this sounds obvious to you, but combinatorics are not my speciality).

I want to operationalize a problem in which I would repeat a componentwise operation for all possible pairs from a given set, but where the ordering of pairs does not matter. For instance, let A be the set \{1,2,3\}. I would need an operator that means something like A <<insert operator here>> A = \{(1,2),(1,3),(2,3)\}.

This is close to be a Cartesian product, but the Cartesian product A X A gives \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}, which is not what I am looking for. For combinations (unordered pairs), (1,2) = (2,1), so they must not be repeated, and the pairs must comprise different elements (i.e. not including (i,i)s, but only (i,j)s where j \neq i).

Which type of operator would fill this purpose?

Ideally, I would like to find an operator with some known properties (so that it is useful to examine a full equation in its general form and find properties), generalizable to an n-dimensional space (i.e. not only for pairs but for any n-tuple), and most importantly componentwise.

By componentwise, I mean that the operator would have the following sense:

<<Repeat the following operation for all combinations, taking into account the characteristics of each component of those combinations>> <<The operation in question>>.

(Note: I am not looking to implement this with a programming language, which would be easy from the line above, but to find a known mathematical operator to develop the problem).

Thanks a lot for any answer, hint, reference, or idea on this.