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|.

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</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.