An integer k-tuple is simply a point in . An integer tuple relation is a mapping from tuples to tuples. The tuple that is being mapped from is refered to as the input tuple and the tuple that is being mapped to is refered to as the output tuple. All the integer tuple relations we consider map from k-tuples to -tuples for some fixed k and . We refer to k and as the input and output arities respectively. A integer tuple set is a set of k-tuples, for some fixed k. We'll often abbreviate integer tuple relations and integer tuple sets as relations and sets respectively.
The sets and relations that we use may be infinite or may depend on the values of other variables, so it is generally not possible to describe them simply by enumerating their tuples or pairs of tuples. Instead, we introduce variables that correspond to each of the positions in the input and output or set tuples and construct a formula that has these and other variables as free variables. These variables are refered to as input, output or set variables as appropriate. For example, we would represent the set of all even numbers as: In this case we introduce a single set variable because we are describing a set of 1-tuples. The formula in this case is , note that i is the only free variable in this formula. If we wanted to represent the set of all positive even numbers less than n we would use: We refer to variables such as n as symbolic or global variables. They are global in the sense that can be used in more than one relation and they represent the same value in all relations. This becomes important when we apply relational operations that combine variables and constraints from different relations. As another example, to represent the relation corresponding to integer addition we would use:
The Omega library can represent relations and sets that can be described by Presburger formulas (possibly with limited uses of uninterpreted function symbols).