Algebraic Property
If f : M → N is any function, then we have f idM = f = idN f (where "" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M.
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.
Read more about this topic: Identity Function
Famous quotes containing the words algebraic and/or property:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“General education is the best preventive of the evils now most dreaded. In the civilized countries of the world, the question is how to distribute most generally and equally the property of the world. As a rule, where education is most general the distribution of property is most general.... As knowledge spreads, wealth spreads. To diffuse knowledge is to diffuse wealth. To give all an equal chance to acquire knowledge is the best and surest way to give all an equal chance to acquire property.”
—Rutherford Birchard Hayes (1822–1893)