MR3377117 Reviewed Giordano, Paolo; Nigsch, Eduard A. Unifying order structures for Colombeau algebras. Math. Nachr. 288 (2015), no. 11-12, 1286–1302. (Reviewer: Francesco Tschinke)

Colombeau Algebras are differential algebras of generalized functions (that includethe space of distributions) that are defined using a quotient set procedure involvingparticular classes of nets in a basic space E = (C∞(Ω))A, where Ω is an open subset ofRn and A is an index set. The choice of such nets depends mainly on their asymptoticbehavior over a suitable index set A. Many variants of Colombeau Algebras existingin the literature occur mainly due to different choices of the index set (and to thechoice of asymptotic behavior). A purpose of this paper is to formally unify some ofthese algebras, redefining the asymptotic behavior on an abstract (pre-ordered) set ofindices, and generalizing the corresponding “Landau big-O” notion. Such notions arereformulated in order to simplify the definition of diffeomorphism invariant algebra andto generalize some theorems which hold in the case of a special algebra