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In mathematics, the jargon term "up to xxxx" is used to
describe a situation in which members of an equivalence class can
be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from
the same equivalence class, i.e. one which is considered equivalent to it. In group theory, for example, this may be a group
action.
Example: in the eight queens puzzle, if the eight queens
are considered to be distinct, there are 3 709 440 distinct solutions. Normally however, the queens are considered to
be identical, and one says "there are 92 (= 3709440/8!) unique solutions up to permutations of the queens", signifying that two different arrangements of the queens are considered equivalent
if the queens have been permuted, but the same squares on the chessboard are
occupied by them.
If, in addition to treating the queens as identical, rotations and reflections of the board were allowed, we would have only 12 distinct solutions up
to symmetry, signifying that two arrangements that are symmetrical to each
other are considered equivalent.
Another typical example is the statement in group theory that "there are
two different groups of order 4 up to isomorphism". This means that there are two equivalence classes of groups of order 4,
if we consider groups to be equivalent if they are isomorphic.
In informal contexts, mathematicians often use the word modulo for
the same purpose, as in "modulo isomorphism, there are two groups of order 4", or "there are 92 solutions modulo the names of the
queens". This is different from, but probably derived from, the use of the word "modulo" in number theory.
See also:
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