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 $\left({R, +, \circ}\right)$.

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.

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

Modulo Addition

 * $+_z$

Addition modulo $z$.

Modulo Multiplication

 * $\times_m$ or $\cdot_m$

Multiplication modulo $m$.

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 $\left({R, +, \circ}\right)$.

Variants

 * $\triangle, \triangledown, \intercal, *$

Used by certain authors to denote a general binary operation.

$\triangle$ and $\triangledown$ can be found in, while $\intercal$ is found in and given the name truc  (pronounced trook, French for trick or technique).



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 $S$.

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

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.