The 3x+1 semigroup

The 3x+1 semigroup is the multiplicative semigroup S of positive rational numbers generated by together with {2}. This semigroup encodes backwards iteration under the 3x+1 map, and the 3x+1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals in lowest terms with , and so contains all positive integers.