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

  • Results about algebraic structures can be found here.


Sources

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