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An infinite descending chain of Boolean subfunctions consisting of threshold functions

Lehtonen, Erkko

Contributions to General Algebra 17, Proceedings of the Vienna Conference 2005 (AAA70), Verlag Johannes Heyn, Klagenfurt, (2006), 145-148

For a class C of Boolean functions, a Boolean function f is a C-subfunction of a Boolean function g, if f=g(h1,...,hn), where all the inner functions hi are members of C. Two functions are C-equivalent, if they are C-subfunctions of each other. The C-subfunction relation is a preorder on the set of all functions if and only if C is a clone. An infinite descending chain of U_\infty-subfunctions is constructed from certain threshold functions (U_\infty denotes the clone of clique functions).