# Definition:Algebraic Structure

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## Definition

An **algebraic structure with $n$ operations** is an ordered tuple:

- $\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$

where:

- $S$ is a set
- $\circ_1, \circ_2, \ldots, \circ_n$ are $n$ binary operations which are defined on all the elements of $S \times S$.

### One Operation

An **algebraic structure with $1$ operation** is an ordered pair:

- $\struct {S, \circ}$

where:

- $S$ is a set
- $\circ$ is a binary operation defined on all the elements of $S \times S$.

### Two Operations

An **algebraic structure with $2$ operations** is an ordered triple:

- $\struct {S, \circ, *}$

where:

- $S$ is a set
- $\circ$ and $*$ are binary operations defined on all the elements of $S \times S$.

## Also known as

Some sources refer to an **algebraic structure** as an **abstract algebra**, but this term is more generally used for the branch of mathematics that studies **these structures**.

Some sources use the term **algebraic system**, which $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves for a slightly more general concept.

Some sources use the variant term **algebraic structure with $n$ compositions**.

Some sources use the notation $\gen {S, \circ_1, \circ_2, \ldots}$ for $\struct {S, \circ_1, \circ_2, \ldots}$ and so on.

## Also see

- Definition:Closed Algebraic Structure
- Definition:Magma
- Definition:Algebraic System, a slightly more general concept

- Definition:Underlying Set of Structure: the set $S$ on $\struct {S, \circ}$

- Results about
**algebraic structures**can be found**here**.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): Prologue - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 26$. Introduction - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**algebraic structure**