- Essentially unique
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 anequivalence relation .A related notion is a
universal property , where an object is not only essentially unique, but unique "up to a uniqueisomorphism " (meaning that it has trivialautomorphism group ). In general given two isomorphic examples of an essentially unique object, there is no "natural" (unique) isomorphism between them.Examples
et theory
Most basically, there is an essentially unique set of any given
cardinality , whether one labels the elements 1,2,3} or a,b,c}.In this case the non-uniqueness of the isomorphism (does one match 1 to "a" or to "c"?) is reflected in thesymmetric group .On the other hand, there is an essentially unique "ordered" set of any given finite cardinality: if one writes 1 < 2 < 3} and a, b, c}, then the only order-preserving isomorphism maps 1 to "a," 2 to "b," and 3 to "c."
Group theory
Suppose that we seek to classify all possible groups. We would find that there is an essentially unique group containing exactly 3 elements, the
cyclic group of order three. No matter how we choose to write those three elements and denote the group operation, all such groups areisomorphic , hence, "the same".On the other hand, there is not an essentially unique group with exactly 4 elements, as there are two non-isomorphic examples: the cyclic group of order 4 and the
Klein four group .Measure theory
Suppose that we seek a translation-invariant, strictly positive,
locally finite measure on thereal line . The solution to this problem is essentially unique: any such measure must be a constant multiple ofLebesgue measure . Specifying that the measure of the unit interval should be 1 then determines the solution uniquely.Topology
Suppose that we seek to classify all two-dimensional, compact,
simply connected manifold s. We would find an essentially unique solution to this problem: the2-sphere . In this case, the solution is unique up tohomeomorphism .Lie theory
A
maximal compact subgroup of asemisimple Lie group not unique, but is unique up to conjugation.ee also
*
Classification theorem
*Universal property
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