Characterization (mathematics)

In mathematics, the statement that "Property P characterizes object X" means that not only does X have property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in different words that the extension of P is a singleton set. The second says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might be involved).

Examples

• A parallelogram is a quadrilateral with opposite sides parallel. One of its characterizations is that the diagonals bisect each other. This means that the diagonals in all parallelograms bisect each other, and conversely, that any quadrilateral where the diagonals bisect each other must be a parallelogram. The latter statement is only true if inclusive definitions of quadrilaterals are used (so that, for example, rectangles count as parallelograms), which is the dominant way of defining objects in mathematics nowadays.
• "Among probability distributions on the interval from 0 to ∞ on the real line, memorylessness characterizes the exponential distributions." This statement means that the exponential distributions are the only such probability distributions that are memoryless.
• "According to Bohr–Mollerup theorem, among all functions f such that f(1) = 1 and x f(x) = f(x + 1) for x > 0, log-convexity characterizes the gamma function." This means that among all such functions, the gamma function is the only one that is log-convex. (A function f is log-convex iff log(f) is a convex function. The base of the logarithm does not matter as long as it is more than 1, but conventionally mathematicians take "log" with no subscript to mean the natural logarithm, whose base is e.)
• The circle is characterized as a manifold by being one-dimensional, compact and connected; here the characterization, as a smooth manifold, is up to diffeomorphism.