If A is a graded module and there exists a degree-preserving linear map ϕ:A⊗A->A, then (A, ϕ) is called a graded algebra. Cohomology is a graded algebra. In addition, the grading set is monoid having a compatibility relation such that if A is in the a grading of the algebra M, and B is in the b grading of the algebra M, then A B is in the a b grading of the algebra (where A and B are multiplied in M, and a and b are multiplied in the index monoid). For example, cohomology of a space is a graded algebra over the integers (i.e., a graded ring), since if A is an n-dimensional cohomology class and B is an m-dimensional cohomology class, then the cup product A B is an m + n dimensional cohomology class.
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