Definition:Zero


 * Set Theory and Order Theory:
 * The zero ordinal.
 * The zero cardinal.


 * Algebra:
 * The zero of the numbers:
 * The zero of the natural numbers, a concept which follows from, and can be defined from, the definition of the natural numbers as the isomorphism class of a naturally ordered semigroup. From this definition follow:
 * The zero of the integers.
 * The zero of the rational numbers.
 * The zero of the real numbers.
 * The zero of the complex numbers.


 * Abstract Algebra:
 * The zero of the naturally ordered semigroup.
 * A zero element of an algebraic structure $\left({S, \circ}\right)$: an element $z \in S$ such that $\forall s \in S: z \circ s = z = s \circ z$.
 * The zero of a ring: that element $0_R$ of a ring $\left({R, +, \times}\right)$ such that $\forall a \in R: 0_R \times a = 0_R = a \times 0_R$.


 * Analysis and Complex Analysis:
 * A zero of a function: given a function $f$ (which will usually be either real-valued or complex-valued), an element $x$ such that $f \left({x}\right) = 0$.


 * Linear Algebra:
 * A zero vector.


 * Category Theory:
 * The empty category.