# Essentially unique

Last updated

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. 

## Contents

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.

## Examples

### Set theory

At the most basic level, 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 (e.g., match 1 to $a$ or 1 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 , then the only order-preserving isomorphism is the one which maps 1 to $a$ , 2 to $b$ , and 3 to $c$ .

### Number theory

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.   

### Group theory

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. 

### Measure theory

There is an essentially unique measure that is translation-invariant, strictly positive and locally finite on the real line. In fact, any such measure must be a constant multiple of Lebesgue measure, specifying that the measure of the unit interval should be 1—before determining the solution uniquely.

### Topology

There is an essentially unique two-dimensional, compact, simply connected manifold: the 2-sphere. In this case, it 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. 

### Lie theory

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

### Category theory

An object that is the limit or colimit over a given diagram is essentially unique, as there is a unique isomorphism to any other limiting/colimiting object. 

### Coding theory

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. 

## Related Research Articles In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after early 19th century mathematician Niels Henrik Abel. In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. In mathematics, an isomorphism is a mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape". In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group. In graph theory, an isomorphism of graphsG and H is a bijection between the vertex sets of G and H

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. It is a special type of C*-algebra. In abstract algebra, a finite group is a group, of which the underlying set contains a finite number of elements.

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.

In mathematics, categorification is the process of replacing set-theoretic theorems by category-theoretic analogues. Categorification, when done successfully, replaces sets by categories, functions with functors, and equations by natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane.

Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. The classical prime number theorem serves as a prototypical example, and the emphasis is on abstract asymptotic distribution results. The theory was invented and developed by mathematicians such as John Knopfmacher and Arne Beurling in the twentieth century.

In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group. In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of Wilhelm Ackermann (1937). The Rado graph can also be constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.

In mathematics, a refinement monoid is a commutative monoid M such that for any elements a0, a1, b0, b1 of M such that a0+a1=b0+b1, there are elements c00, c01, c10, c11 of M such that a0=c00+c01, a1=c10+c11, b0=c00+c10, and b1=c01+c11.

In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a field and its absolute Galois group. It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.

In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solution xA, or the symbol ∞ if there is no such N. The p-height considers only divisibility properties by the powers of a fixed prime number p. The notion of height admits a refinement so that the p-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants.

1. "The Definitive Glossary of Higher Mathematical Jargon — Essentially unique". Math Vault. 2019-08-01. Retrieved 2019-11-22.
2. "Universal property - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-11-22.
3. Garnier, Rowan; Taylor, John (2009-11-09). Discrete Mathematics: Proofs, Structures and Applications, Third Edition. CRC Press. p. 452. ISBN   9781439812808.
4. Weisstein, Eric W. "Essentially Unique". mathworld.wolfram.com. Retrieved 2019-11-22.
5. Corry, Scott. "Classification of Groups of Order n ≤ 8" (PDF). Lawrence University. Retrieved 2019-11-21.
6. Lickorish, W. B. Raymond (2012-12-06). An Introduction to Knot Theory. Springer Science & Business Media. ISBN   9781461206910.
7. "limit in nLab". ncatlab.org. Retrieved 2019-11-22.
8. Baez, John (2015-12-01). "Golay Code". Visual Insight. American Mathematical Society . Retrieved 2017-12-02.