Symbols:Abstract Algebra

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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.


Repeated Addition

$\cdot$

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.


Modulo Addition

$+_z$

Addition modulo $z$.


The $\LaTeX$ code for \(+_z\) is +_z .


Modulo Multiplication

$\times_m$ or $\cdot_m$

Multiplication modulo $m$.


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.