# Symbols:Abstract Algebra

## Commutative Operation

$+$

Often used to denote:

The $\LaTeX$ code for $+$ is + .

See Set Operations and Relations and Arithmetic and Algebra for alternative definitions of this symbol.

$\cdot$

Often used to denote the power of the additive binary operation in a general ring $\left({R, +, \circ}\right)$.

In this context, $n \cdot a$ means $\underbrace{a + a + \ldots + a}_{n \text{ times}}$.

See Powers of Ring Elements‎ for an example of how this can be used.

Also often used for the binary operation in a general group which is not necessarily abelian.

The $\LaTeX$ code for $\cdot$ is \cdot .

See Vector Algebra, Arithmetic and Algebra and Logical Operators: Deprecated Symbols for alternative definitions of this symbol.

$+_z$

The $\LaTeX$ code for $+_z$ is +_z .

## Modulo Multiplication

$\times_m$ or $\cdot_m$

The $\LaTeX$ code for $\times_m$ is \times_m .

The $\LaTeX$ code for $\cdot_m$ is \cdot_m .

## General Operation

$\circ$

Often used to denote:

The $\LaTeX$ code for $\circ$ is \circ .

### Variants

$\bigtriangleup, \bigtriangledown, \intercal$

Used by certain authors to denote a general binary operation.

$\bigtriangleup$ and $\bigtriangledown$ can be found in 1965: Seth Warner: Modern Algebra, while $\intercal$ is found in 1975: T.S. Blyth: Set Theory and Abstract Algebra and given the name truc (pronounced trook, French for trick or technique).

Other symbols used for a general binary operation on $\mathsf{Pr} \infty \mathsf{fWiki}$ include $*$, $\oplus$ and $\odot$.

The $\LaTeX$ code for $\bigtriangleup$ is \bigtriangleup .
The $\LaTeX$ code for $\bigtriangledown$ is \bigtriangledown .
The $\LaTeX$ code for $\intercal$ is \intercal .
The $\LaTeX$ code for $*$ is *  or \ast.
The $\LaTeX$ code for $\oplus$ is \oplus .
The $\LaTeX$ code for $\odot$ is \odot .

## Order

$\left|{\left({S, \circ}\right)}\right|$

The order of the algebraic structure $\left({S, \circ}\right)$.

It is defined as the cardinality $\left|{S}\right|$ of its underlying set $S$.

The $\LaTeX$ code for $\left\vert{\left({S, \circ}\right)}\right\vert$ is \left\vert{\left({S, \circ}\right)}\right\vert .

See Arithmetic and Algebra, Complex Analysis and Set Operations and Relations for alternative definitions of this symbol.

## Orderings

$\preceq, \preccurlyeq, \curlyeqprec$

Used to indicate an ordering relation on a general ordered set $\left({S, \preceq}\right)$, $\left({T, \preccurlyeq}\right)$ etc.

Their inverses are $\succeq$, $\succcurlyeq$ and $\curlyeqsucc$.

We also have:

• $\prec$, which means "$\preceq$ or $\preccurlyeq$, etc. and $\ne$"
• $\succ$, which means "$\succeq$ or $\succcurlyeq$, etc. and $\ne$".

Their $\LaTeX$ codes are as follows:

• $\preceq$: \preceq
• $\preccurlyeq$: \preccurlyeq
• $\curlyeqprec$: \curlyeqprec
• $\prec$: \prec
• $\succeq$: \succeq
• $\succcurlyeq$: \succcurlyeq
• $\curlyeqsucc$: \curlyeqsucc
• $\succ$: \succ

The symbols $\le, <, \ge, >$ and their variants can also be used in the context of a general ordering if desired, but it is usually better to reserve them for the conventional orderings between numbers.