nonwelldefined

Nonwelldefined is a term used to describe objects, definitions, or operations in mathematics and related fields whose output depends on how the input is represented, rather than on the input itself. An expression is well-defined if, whenever two representations denote the same input, the expression yields the same output. When this fails, the construction is non-well-defined.

Common sources of non-well-definedness include multiple representations of the same object and not constraining a definition to a canonical form. For example, consider a function defined on fractions by f(p/q) = p for integers p and q with q ≠ 0. The rational number p/q can also be written as 2/3 or 4/6; applying f yields different outputs (2 and 4), so the function is not well-defined on the rationals unless restricted to reduced fractions.

Another familiar case arises with quotient constructions. If one defines a map on pairs (a,b) with b ≠ 0 by f(a,b) = a, then on the same rational number represented by different pairs (ka,kb) the values would differ unless a canonical representative is chosen. This shows that a function on an equivalence class must be defined with respect to a specific representative or a canonical form to be well-defined.

In practice, ensuring well-definedness often involves reducing representations to a standard form, specifying equivalence classes explicitly, or redefining the object so that it is taken as a single, canonical value. Non-well-definedness is distinct from merely undefined values: it signals ambiguity due to representation rather than absence of a value.