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:
“The classicist, and the naturalist who has much in common with him, refuse to see in the highest works of art anything but the exercise of judgement, sensibility, and skill. The romanticist cannot be satisfied with such a normal standard; for him art is essentially irrational—an experience beyond normality, sometimes destructive of normality, and at the very least evocative of that state of wonder which is the state of mind induced by the immediately inexplicable.”
—Sir Herbert Read (1893–1968)
“Most women of [the WW II] generation have but one image of good motherhood—the one their mothers embodied. . . . Anything done “for the sake of the children” justified, even ennobled the mother’s role. Motherhood was tantamount to martyrdom during that unique era when children were gods. Those who appeared to put their own needs first were castigated and shunned—the ultimate damnation for a gender trained to be wholly dependent on the acceptance and praise of others.”
—Melinda M. Marshall (20th century)