[[Abstract algebra MOC]]
# Graded structure

An algebraic structure $A$ is **graded** by some set $S$, typically a [[monoid]],
if it is the direct sum of subalgebras indexed by $S$ which are in some way invariant.
In addition there may be some law governing the grade of the product of elements of different grades,
A homomorphism of such structures is called **graded** or **grade-preserving** iff it maintains the grade of elements.

- [[Graded ring]]
- [[Graded vector space]]
- [[Graded module]]
- [[Graded algebra]]
  - [[Graded Lie algebra]]
- [[Graded category]], a generalization

A note on notation: If $\cat{G}$ is a category of structures, say gadgets, then the **strict category of $S$-graded gadgets** is $\cat{Gr}_{S}\cat G$.
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