In the following, a basic calculus of operations is defined which is required to formulate a discrete problem in a functional manner: The calculation is defined exactly and can be carried out by a computer. At a first glance, such a formalism seems to be trivial due to the fact that each programming language defines basic arithmetic features. However, the handling of the underlying topological structures is not supported at all so that additional features have to be provided.
The most essential element of a description formalism is the access to a function which is defined on the cell complex. Due to data structural considerations, the domain of this function can be restricted to a certain skeleton of the underlying complex. However, this does not affect the concept of the function. In this context the definition of a quantity is employed in order to obtain the respective value for a given topological element. In the formalism such a function can be formulated as , where and denote the quantities and denotes the element on which the quantities are evaluated.
In the next step basic arithmetic operations are introduced. Based on the rules which hold for the underlying numerical data types, the operations are introduced in the following manner:
The same holds true for the notation which compromises the readability of the notation, while no additional information is added.