adj. 1. being the only value satisfying some condition. For example, the unique positive square root of 4 is 2, although -2 is another square root of 4. See also definite description. 2. unique up to (some relation). (of elements within some structure) related by a relative identity relation; identical relative to that relation; equivalent within a given structure. For example, to say the positive integers have prime factorizations that are unique up to order is to say that the index of each prime is unique, but that their order is irrelevant. Common phrases of this kind are unique up to isomorphism, unique up to permutation, unique up to a constant.