Definition:Square/Mapping

Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $f: S \to S$ be the mapping from $S$ to $S$ defined as:

$\forall x \in S: \map f x := x \circ x$

This is usually denoted $x^2$:

$x^2 := x \circ x$

Element

A square (element of $S$) is an element $y$ of $S$ for which:

$\exists x \in S: y = x^2$

Such a $y = x^2$ is referred to as the square of $x$.

Square Function

The square mapping (or square function) is usually defined in the context of the standard number systems:

Let $\GF$ denote one of the standard number systems: $\N$, $\Z$, $\Q$, $\R$, $\C$.

Definition 1

The square (function) on $\GF$ is the mapping $f: \GF \to \GF$ defined as:

$\forall x \in \GF: \map f x = x \times x$

where $\times$ denotes multiplication.

Definition 2

The square (function) on $\GF$ is the mapping $f: \GF \to \GF$ defined as:

$\forall x \in \F: \map f x = x^2$

where $x^2$ denotes the $2$nd power of $x$.