Algebra

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 $S$ plus any number of functions $f_i:S\times S\to S$, such that each $f_i$ satisfies some properties of choice.

Key examples:

Some specific examples:

The order of a algebraic structure is just its cardinality.

Sometimes, especially in the case of structures with an infinite number of elements, it is often more convenient to talk in terms of some parameter that characterizes the structure, and that parameter is usually called the degree.

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.

Examples:

Examples:

mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates Interesting results in algebraic geometry accessible to 3rd year undergraduates