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 $\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.

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
• $\preccurlyeq$: \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.