Essentially unique

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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.

Examples[edit]

Set theory[edit]

Most basically, there is an essentially unique set of any given cardinality, whether one labels the elements or . In this case the non-uniqueness of the isomorphism (does one match 1 to a or to c?) is reflected in the symmetric group.

On the other hand, there is an essentially unique ordered set of any given finite cardinality: if one writes and , then the only order-preserving isomorphism maps 1 to a, 2 to b, and 3 to c.

Number theory[edit]

The fundamental theorem of arithmetic establishes that the factorization of any positive integer into prime numbers is essentially unique, i.e., unique up to the ordering of the prime factors.[1]

Group theory[edit]

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 are isomorphic, 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[edit]

Suppose that we seek a translation-invariant, strictly positive, locally finite measure on the real line. The solution to this problem is essentially unique: any such measure must be a constant multiple of Lebesgue measure. Specifying that the measure of the unit interval should be 1 then determines the solution uniquely.

Topology[edit]

Suppose that we seek to classify all two-dimensional, compact, simply connected manifolds. We would find an essentially unique solution to this problem: the 2-sphere. In this case, the solution is unique up to homeomorphism.

In the area of topology known as knot theory, there is an analogue of the fundamental theorem of arithmetic: the decomposition of a knot into a sum of prime knots is essentially unique.[2]

Lie theory[edit]

A maximal compact subgroup of a semisimple Lie group may not be unique, but is unique up to conjugation.

Coding theory[edit]

Given the task of using 24-bit words to store 12 bits of information in such a way that 7-bit errors can be detected and 3-bit errors can be corrected, the solution is essentially unique: the extended binary Golay code.[3]

See also[edit]

References[edit]

  1. ^ Garnier, Rowan; Taylor, John (2009-11-09). Discrete Mathematics: Proofs, Structures and Applications, Third Edition. CRC Press. p. 452. ISBN 9781439812808.
  2. ^ Lickorish, W. B. Raymond (2012-12-06). An Introduction to Knot Theory. Springer Science & Business Media. ISBN 9781461206910.
  3. ^ Baez, John (2015-12-01). "Golay Code". Visual Insight. American Mathematical Society. Retrieved 2017-12-02.