In mathematics, the term essentially unique is used to indicate that while some object is not the only one that satisfies certain properties, all such objects are "the same" in some sense appropriate to the circumstances. This notion of "sameness" is often formalized using an equivalence relation.
A related notion is a universal property, where an object is not only essentially unique, but unique up to a unique isomorphism (meaning that it has trivial automorphism group). In general given two isomorphic examples of an essentially unique object, there is no natural (unique) isomorphism between them.
Famous quotes containing the words essentially and/or unique:
“This is essentially a People’s contest. On the side of the Union, it is a struggle for maintaining in the world, that form, and substance of government, whose leading object is, to elevate the condition of men—to lift artificial weights from all shoulders—to clear the paths of laudable pursuit for all—to afford all, an unfettered start, and a fair chance, in the race of life.”
—Abraham Lincoln (1809–1865)
“... a unique endeavour
To bring to bloom the million-petalled flower
Of being here.”
—Philip Larkin (1922–1986)