# 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 `+`

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### 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`

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### 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`

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### 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`

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### 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`

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The $\LaTeX$ code for \(\cdot_m\) is `\cdot_m`

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### 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`

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### 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} }`

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### 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.