In algebra, the 3x + 1 semigroup is a special subsemigroup of the multiplicative semigroup of all positive rational numbers. The elements of a generating set of this semigroup are related to the sequence of numbers involved in the yet to be proved conjecture known as the Collatz conjecture or the "3x + 1 problem". The 3x + 1 semigroup has been used to prove a weaker form of the Collatz conjecture. In fact, it was in such context the concept of the 3x + 1 semigroup was introduced by H. Farkas in 2005. Various generalizations of the 3x + 1 semigroup have been constructed and their properties have been investigated.
The 3x + 1 semigroup is the multiplicative semigroup of positive rational numbers generated by the set
The function T : Z → Z, where Z is the set of all integers, as defined below is used in the "shortcut" definition of the Collatz conjecture:
The relation between the 3x + 1 semigroup and the Collatz conjecture is that the 3x + 1 semigroup is also generated by the set
The weak Collatz conjecture asserts the following: "The 3x + 1 semigroup contains every positive integer." This was formulated by Farkas and it has been proved to be true as a consequence of the following property of the 3x + 1 semigroup. "The 3x + 1 semigroup S equals the set of all positive rationals a/b in lowest terms having the property that b ≠ 0 (mod 3). In particular, S contains every positive integer."
The semigroup generated by the set