# Definition:Inverse Element/Also known as

Jump to navigation
Jump to search

## Inverse Element: Also known as

An **inverse of $x$** is also known as a **two-sided inverse of $x$**, **symmetric element** or **negative of $x$**.

Some sources refer to it as a **reciprocal element**, which terminology is borrowed from the real numbers under multiplication.

The notation used to represent an **inverse of an element** is often understood to depend on the set and binary operation under consideration.

Various symbols are seen for a general **inverse**, for example $\hat x$ and $x^*$.

- If $s \in S$ has an inverse, it is denoted $s^{-1}$.

If the operation concerned is commutative, then additive notation is often used:

- If $s \in S$ has an inverse, it is denoted $-s$.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses