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It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C*-algebras by Brown–Douglas–Fillmore (Lawrence G. Brown, Ronald G. Douglas, Peter Arthur Fillmore 1977).
In ring theory, combinatorics, functional analysis, and theoretical computer science, a semifield is a semiring (MSC 16Y60) (S,+,·) in which all elements have a multiplicative inverse.