Essentially unique

In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of essential uniqueness presupposes some form of "sameness", which 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 there can be more than one isomorphism between examples of an essentially unique object. At the most basic level, 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 (e.g., match 1 to or 1 to ) 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 is the one which maps 1 to , 2 to , and 3 to . 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. In the context of classification of groups, there is an essentially unique group containing exactly 2 elements. Similarly, there is also an essentially unique group containing exactly 3 elements: the cyclic group of order three. In fact, regardless of how one chooses to write the three elements and denote the group operation, all such groups can be shown to be isomorphic to each other, and hence are "the same". On the other hand, there does not exist an essentially unique group with exactly 4 elements, as there are in this case two non-isomorphic groups in total: the cyclic group of order 4 and the Klein four group. There is an essentially unique measure that is translation-invariant, strictly positive and locally finite on the real line.

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Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

Essentially unique

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