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 $𝖦$ is a category of structures, say gadgets, then the strict category of $𝑆$ -graded gadgets is $𝖦 𝗋 𝑆 𝖦$ .

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