Enriching the Object-Oriented Paradigm via Shadows in the Context of Mathematics

public class Ring { abstract public RingElement add(RingElement,RingElement); abstract public RingElement mult(RingElement, RingElement); ... } Both methods have “ring elements” as arguments and return values. A RingElement is a class that can be (polymorphically) associated to any object (e.g. a multi-precision integer, or a vector of RingElement objects). Not all methods need to be abstract. For instance a method square can be implemented via a = a · a using the abstract multiplication method. More sophisticated non-abstract methods are exponentiation or evaluation of a polynomial. Thus, an abstract Java class is able to mimic the axiomatic definition of a mathematical entity, here a “ring”. The axioms addition and multiplication appear as abstract methods in Ring while squaring is not axiomatic and therefore may be implemented. Note that structural axioms such as associativity or distributivity cannot VOL 5, NO. 6 JOURNAL OF OBJECT TECHNOLOGY 109 ENRICHING THE OBJECT-ORIENTED PARADIGM VIA SHADOWS IN THE CONTEXT OF MATHEMATICS