Symbols:Abstract Algebra

Commutative Operation


Often used to denote:


 * The binary operation in a general abelian group $\left({G, +}\right)$
 * The additive binary operation in a general ring $\left({R, +, \circ}\right)$.

Its $\LaTeX$ code 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 $a + a + \cdots (n \mbox{ times}) \cdots + a $.

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.

Its $\LaTeX$ code is \cdot.

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

Modulo Addition

 * $+_z$

Addition modulo $z$.

Its $\LaTeX$ code is +_z.

Modulo Multiplication

 * $\times_m$ or $\cdot_m$

Multiplication modulo $m$.

The $\LaTeX$ code for $\times_m$ is \times_m and the $\LaTeX$ code for $\cdot_m$ is \cdot_m.

General Operation

 * $\circ$

Often used to denote:


 * A general binary operation in an equally general algebraic structure $\left({S, \circ}\right)$.
 * A general ring product in an equally general ring $\left({R, +, \circ}\right)$.

Its $\LaTeX$ code is \circ.

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 $S$.

Its $\LaTeX$ code is \left|{\left({S, \circ}\right)}\right| or \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 poset $\left({S, \preceq}\right)$, $\left({T, \preccurlyeq}\right)$ 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</tt>
 * $\preccurlyeq$: \preccurlyeq</tt>
 * $\curlyeqprec$: \curlyeqprec</tt>
 * $\prec$: \prec</tt>
 * $\succeq$: \succeq</tt>
 * $\succcurlyeq$: \succcurlyeq</tt>
 * $\curlyeqsucc$: \curlyeqsucc</tt>
 * $\succ$: \succ</tt>

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.