Essentially unique

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 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

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 the symmetric 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 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

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

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.

Lie theory

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

ee also

* Classification theorem
* Universal property


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Unique games conjecture — The Unique Games Conjecture (UGC) is a conjecture in computational complexity theory made by Subhash Khot in 2002.A unique game is a special case of a two prover, one round (2P1R) game. A two player, one round game has two players (also known as… …   Wikipedia

  • Limit (category theory) — In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint… …   Wikipedia

  • Von Neumann algebra — In mathematics, a von Neumann algebra or W* algebra is a * algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann,… …   Wikipedia

  • Universal property — In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. These properties are called universal properties …   Wikipedia

  • Maximal compact subgroup — In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classification of …   Wikipedia

  • Spherical 3-manifold — In mathematics, a spherical 3 manifold M is a 3 manifold of the form M = S3 / Γ where Γ is a finite subgroup of SO(4) acting freely by rotations on the 3 sphere S3. All such manifolds are prime, orientable, and closed. Spherical 3 manifolds are… …   Wikipedia

  • 4-manifold — In mathematics, 4 manifold is a 4 dimensional topological manifold. A smooth 4 manifold is a 4 manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different.… …   Wikipedia

  • Irreducible polynomial — In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non trivial factors in a given set. See also factorization. For any field F , the ring of polynomials with coefficients in F is… …   Wikipedia

  • Exotic sphere — In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n sphere. That is, M is a sphere from the point of view of all its… …   Wikipedia

  • Alismatales — ▪ plant order Introduction   arrowhead and pondweed order of flowering plants, belonging to the monocotyledon (monocot) group, whose species have a single seed leaf. Most of the some 4,500 species are aquatic and grow submersed or partially… …   Universalium

Share the article and excerpts

Direct link
https://en-academic.com/dic.nsf/enwiki/4131086 Do a right-click on the link above
and select “Copy Link”