In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. That is, for f being identity, the equality f(X) = X holds for all X.
- f(X) = X for all elements X in M.
In other words, the function value f(X) in M (that is, the codomain) is always the same input element X of M (now considered as the domain). The identity function on M is clearly an injective function as well as a surjective function, so it is also bijective.
The identity function f on M is often denoted by idM.
Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.
- The identity function is a linear operator, when applied to vector spaces.
- The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
- In an n-dimensional vector space the identity function is represented by the identity matrix In, regardless of the basis.
- In a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the trivial group only containing this isometry (symmetry type C1).
- In a topological space, the identity function is always continuous.
- The identity function is idempotent.
- Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978-0-8176-3248-9
- Mapa, Sadhan Kumar (7 April 2014). Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36. ISBN 978-93-80663-24-1.
- Proceedings of Symposia in Pure Mathematics. American Mathematical Society. 1974. p. 92. ISBN 978-0-8218-1425-3.
...then the diagonal set determined by M is the identity relation...
- Rosales, J. C.; García-Sánchez, P. A. (1999). Finitely Generated Commutative Monoids. Nova Publishers. p. 1. ISBN 978-1-56072-670-8.
The element 0 is usually referred to as the identity element and if it exists, it is unique
- Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
- D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.
- T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.
- James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1-85233-934-9
- Conover, Robert A. (2014-05-21). A First Course in Topology: An Introduction to Mathematical Thinking. Courier Corporation. p. 65. ISBN 978-0-486-78001-6.
- Conferences, University of Michigan Engineering Summer (1968). Foundations of Information Systems Engineering.
we see that an identity element of a semigroup is idempotent.