Abstract
Solutions of the equations X = ZX and X = XZ are found and discussed for Z, X normal terms of the lambda-calculus. Obviously fixed point combinators are of no help. Solutions will be independent from any kind of godelization or coding of data structures, they will be provided by typeless self-application. Different approaches will be shown: algebraic properties, one side invertibility and idempotency. Certain subsets of proper combinators and Church algebras between them will be proved to be domains consisting only of fixed points of combinators.
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