Definition:Algebraic Structure/One Operation

Definition

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 $S \times S$.

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:Algebraic Structure with Two Operations

• 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: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures