Not to be confused with algebra over a field, which is a particular algebraic structure studied within algebra.
We just use "Abstract algebra" as a synonym for algebra.
A set plus any number of functions , such that each satisfies some properties of choice.
Some specific examples:
The order of a algebraic structure is just its cardinality.
The degree of some algebraic structure is some parameter that describes the structure. There is no universal definition valid for all structures, it is a per structure type thing.
This is particularly useful when talking about structures with an infinite number of elements, but it is sometimes also used for finite structures.
- the dihedral group of degree n acts on n elements, and has order 2n
- the parameter that characterizes the size of the general linear group is called the degree of that group, i.e. the dimension of the underlying matrices
mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates Interesting results in algebraic geometry accessible to 3rd year undergraduates