# Unit

The smallest (positive) natural number: $1$. Multiplication of any number by $1$ produces the same number again.

An element $e$ in a set $M$ is called a left (right) unit (left (right) identity) with respect to a binary operation $*$ defined on $M$ if for any $a \in M$ the following equation holds: $$e * a = a \ \ \ (\ a * e = a\ ) \ .$$

If there exists at least one left unit and at least one right unit, then they coincide and there are no other units. If more than one binary operation is defined on $M$ (for example, addition and multiplication in a ring), then the term unit is used for only one of these operations, usually multiplication. The unit with respect to addition is called the zero element.

The unit of a lattice is its greatest element, that is, the identity with respect to the operation of intersection.

In an integral domain $D$, any invertible element, that is, an element $u$ that has an inverse $u^{-1}$ such that $u.u^{-1} = u^{-1}.u = 1$, is called a unit, or a divisor of unity. The units of an integral domain form a group under multiplication. The same terminology is sometimes preserved when passing to the field of fractions of the ring $D$ (that is, the units of $D$ itself are called units of the field of fractions). For example, the units of an algebraic number field $k$ are the units of the ring of algebraic integers of $k$, the $p$-adic units are the units of the ring of $p$-adic numbers, etc.

The unit morphism (identity morphism) of an object $X$ in a category is the (unique) morphism $1_X : X \rightarrow X$ satisfying $1_X f = f$ and $g 1_X = g$ for all $f : Y \rightarrow X$ and all $g : X \rightarrow Z$.

A standard length when measuring things or comparing objects is called a unit of length, or simply a unit. More generally, various somethings one of whose main parameters is equal to the unit length is called a "unit something" ; e.g. unit vector, unit circle, unit disc, unit cube, unit $n$-cube, unit sphere, unit ball, unit $n$-ball (unit cell), etc.