Intrinsically defined sorting functions

We exhibit intrinsic definitions of inequality tests, sorting functions and comparators as isomorphic presentations of total preorders (orders): • Inequality tests are functions of type T × T → Bool that are reflexive, transitive and total. • Sorting functions are functions of type T ∗ → T ∗ (mapping finite sequences of elements of type T ) that permute their input and do so consistently for each pair of elements. • Comparators are functions of type T × T → T × T that are permutative, transitive and idempotent. The definitions are intrinsic in the sense that they do not refer to predefined orders or to each other. Furthermore the isomorphisms can be defined parametrically polymorphically: the mapping from inequality tests to sorting functions is the class of stable comparator-based sorting algorithms; from comparators to sorting functions it is sorting networks (comparator-based sorting algorithms); the other mappings are trivial. So defining any one concrete presentation of an order automatically provides all three presentations. Whilst mathematically elementary and pleasingly simple, we believe our definitions of sorting functions and comparators as isomorphic characterizations of orders to be novel and of interest in themselves. We investigate a number of variations on the definition of sorting functions to illustrate that some care must be exercised in formulating their definition. Drawing on the resulting conceptual symmetry of inequality tests, sorting functions and comparators—no presentation has definitional preeminence over the other—we raise the question: How should orders be presented? We consider the question itself a conceptual contribution of this paper. We suggest that there are powerful reasons for choosing sorting functions instead of the more obvious inequality tests, namely the possibility of sorting in worst-case linear time), but defer showing this to two subsequent papers.