Inmathematics, the termessentially 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 anequivalence relation.

A related notion is auniversal property, where an object is not only essentially unique, but uniqueup to a uniqueisomorphism^[1] (meaning that it has trivialautomorphism 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 givencardinality, whether one labels the elements${\displaystyle \{1,2,3\}}$ or${\displaystyle \{a,b,c\}}$.In this case, the non-uniqueness of the isomorphism (e.g., match 1 to${\displaystyle a}$ or 1 to${\displaystyle c}$) is reflected in thesymmetric group.

On the other hand, there is an essentially uniquetotally ordered set of any given finite cardinality that is uniqueup to unique isomorphism: if one writes and, then the onlyorder-preserving isomorphism is the one which maps 1 to${\displaystyle a}$, 2 to${\displaystyle b}$, and 3 to${\displaystyle c}$.

Thefundamental theorem of arithmetic establishes that thefactorization of any positiveinteger intoprime numbers is essentially unique, i.e., unique up to the ordering of the primefactors.^[2][3]

In the context of classification ofgroups, there is an essentially unique group containing exactly 2 elements.^[3] Similarly, there is also an essentially unique group containing exactly 3 elements: thecyclic 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 beisomorphic 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 theKlein four-group.^[4]

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

There is an essentially unique two-dimensional,compact,simply connectedmanifold: the2-sphere. In this case, it is unique up tohomeomorphism.

In the area of topology known asknot theory, there is an analogue of the fundamental theorem of arithmetic: the decomposition of a knot into a sum ofprime knots is essentially unique.^[5]

Amaximal compact subgroup of asemisimple Lie group may not be unique, but is unique up toconjugation.

An object that is thelimit or colimit over a given diagram is essentially unique, as there is aunique isomorphism to any other limiting/colimiting object.^[6]

Given the task of using 24-bit words to store 12 bits of information in such a way that 4-bit errors can be detected and 3-bit errors can be corrected, the solution is essentially unique: theextended binary Golay code.^[7]

• Classification theorem
• Modulo, a mathematical term pertaining to the equivalence of objects
• Universal property
• Up to

• Mathematical terminology