identity

noun

1. (countable, uncountable)Sameness, identicalness; the quality or fact of (several specified things) being the same.

   “[…] suggesting the two are different stages of the same species. The identity of the two species is further suggested by allozyme analysis […]”

2. (countable, uncountable)The difference or character that marks off an individual or collective from the rest of the same kind; selfhood; the sense of who something or someone or oneself is, or the recurring characteristics that enable the recognition of such an individual or group by others or themselves.

   “I've been through so many changes, I have no sense of identity.”

   “This nation has a strong identity.”

   “His own identity was fading out into a grey impalpable world: the solid world itself which these dead had one time reared and lived in was dissolving and dwindling.”

   “On the other hand, ME /ɛu/ preserved its identity into the early Modern English period, and the early writers on orthography and pronunciation normally distinguish between lME /iu/ and lME /ɛu/, the latter of which occurs in words with eME /ɛː/ like dew (OE dēaw), few (OE fēawe), hew (OE hēawan), sew (eME ē < ĕ, OE seowian), shrew (OE scrēawa), etc.”

3. (countable, uncountable)A name or persona—a mask or appearance one presents to the world—by which one is known.

   “This criminal has taken on several identities.”

   “In this show, the competitor's identity will remain secret until after the vote.”

4. (countable, uncountable)An equation which always holds true regardless of the choice of input variables.

   “The equation (x+y)(x−y) = x²−y² is an algebraic identity. It is true regardless of the values of x and y.”

5. (countable, uncountable)Any function which maps all elements of its domain to themselves.
6. (countable, uncountable)An element of an algebraic structure which, when applied to another element under an operation in that structure, yields this second element.

   “Zero is the identity for the addition of real numbers.”

7. (Australia, New-Zealand, countable, uncountable)A well-known or famous person.

   “The body of a well known old identity named James Conroy […] was found in the water yesterday afternoon…”