We can now define how the function maps OIL axioms and class definitions into (sets of) axioms. The definition is given in Figures 2 and 3, where is a class name (or a concept name), is a slot name (or role name), (possibly subscripted) is a class expression, (possibly subscripted) is a class or concrete type expression, is a class expression or a class description (super-classes plus slot constraints), (possibly subscripted) is a slot constraint, is a constraint (on a slot), is an OIL individual, is the primitive concept used to represent the OIL individual , is a concrete data value (an integer or a string), is a non-negative integer and is a unary predicate (i.e., and is a concrete data value).
In addition, the set of disjointness axioms is defined as: