Definition:Algebraic Structure/Two Operations
Definition
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 defined as
Some sources define an algebraic structure such that the underlying set is specifically non-empty.
Others allow the concept of an empty algebraic structure, while acknowledging that it is usual for it to be non-empty.
Some sources categorize modules and vector spaces as algebraic structures, but in the strict definition as given on $\mathsf{Pr} \infty \mathsf{fWiki}$, technically they are not.
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: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures