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

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