# Definition:Algebraic Structure

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## Definition

An **algebraic structure** is an ordered tuple:

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

where $S$ is a set which has one or more binary operations $\circ_1, \circ_2, \ldots, \circ_n$ defined on all the elements of $S \times S$.

An **algebraic structure** with one (binary) operation is thus an ordered pair which can be denoted $\struct {S, \circ}$ or $\struct {T, *}$, and so on.

## Also known as

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

## Also denoted as

Some sources use the notation $\gen {S, \circ}$ for $\struct {S, \circ}$.

## 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): $\S 6$ - 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): Entry:**algebraic structure**