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The functional is a homomorphism from the monoid of simplicial complexes to the rationals: the formula Dim^+(G + H) = Dim^+(G) + Dim^+(H) holds for the join + similarly as for the augmented inductive dimension dim^+(G) = dim(G)+1 where dim is the inductive dimension dim(G) = 1+ sum_x dim(S(x))/|G| with unit sphere S(x) (a recent theorem of Betre and Salinger). 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