Closure functions on the set of positive integers

摘要

The following question raised by P. C. Hammer in 1960 is answered: Let M be the set of all positive integers, h the closure function under multiplication defined on the power set of M and c the complementation function. Do h and c generate exactly 14 distinct functions by composition in any order? Different types of sets which yield 14 distinct sets in the set of positive intergers will also be found.