In mathematics, the phrase up to indicates that its grammatical object is some equivalence class, to be regarded as a single entity, or disregarded as a single entity. If this object is a class of transformations (such as “isomorphism” or “permutation”), it implies the equivalence of objects one of which is the image of the other under such a transformation. If X is some property or process, the phrase “up to X” means “disregarding a possible difference in X”. For instance we might follow the statement “an integer’s prime factorization is unique up to ordering”, meaning that the prime factorization is unique if we disregard the order of the factors; or we might say “the solution to an indefinite integral is , up to addition by a constant”, meaning that the added constant is not the focus here, the solution is, and that the addition of a constant is to be regarded as a background, of secondary focus. Further examples concerning up to isomorphism, up to permutations and up to rotations are described below. In informal contexts, mathematicians often use the word modulo (or simply “mod”) for similar purposes, as in “modulo isomorphism”.