Symbols:Abstract Algebra

Symbols used in Abstract Algebra

Commutative Operation

$+$

Often used to denote:

The binary operation in a general abelian group $\struct {G, +}$

The additive binary operation in a general ring $\struct {R, +, \circ}$

The additive binary operation in a general field $\struct {F, +, \times}$

both of which are commutative operations.

The $\LaTeX$ code for \(+\) is + .

Repeated Addition

$\cdot$

Often used to denote the power of the additive binary operation in a general ring $\struct {R, +, \circ}$.

In this context, $n \cdot a$ means $\underbrace {a + a + \ldots + a}_{n \text{ times}} $.

See Integral Multiple of Ring Element 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 .

Modulo Addition

$+_m$

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

where $\eqclass x m$ is the residue class of $x$ modulo $m$.

The operation of addition modulo $m$ is defined on $\Z_m$ as:

$\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$

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

Real Modulo Addition

$+_z$

Let $z \in \R$.

Let $\R_z$ be the set of residue classes modulo $z$ of $\R$.

The addition operation is defined on $\R_z$ as follows:

$\eqclass a z +_z \eqclass b z = \eqclass {a + b} z$

This operation is called addition modulo $z$.

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

Modulo Multiplication

$\times_m$ or $\cdot_m$

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

where $\eqclass x m$ is the residue class of $x$ modulo $m$.

The operation of multiplication modulo $m$ is defined on $\Z_m$ as:

$\eqclass a m \times_m \eqclass b m = \eqclass {a b} m$

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:

A general binary operation in an equally general algebraic structure $\struct {S, \circ}$

A general ring product in an equally general ring $\struct {R, +, \circ}$.

The $\LaTeX$ code for \(\circ\) is \circ .

Order of Structure

$\order {\struct {S, \circ} }$

The order of an algebraic structure $\struct {S, \circ}$ is the cardinality of its underlying set, and is denoted $\order S$.

Thus, for a finite set $S$, the order of $\struct {S, \circ}$ is the number of elements in $S$.

The $\LaTeX$ code for \(\order {\struct {S, \circ} }\) is \order {\struct {S, \circ} } .

Ordering

$\preceq, \preccurlyeq, \curlyeqprec$

Used to indicate an ordering relation on a general ordered set $\struct {S, \preceq}$, $\struct{T, \preccurlyeq}$ 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.

However, it is usually better to reserve them for the conventional orderings between numbers.