Definition:Algebraic Structure

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