In
algebra, the 3x + 1 semigroup is a special
subsemigroup of the multiplicative
semigroup of all positive
rational numbers.^{
[1]} The elements of a generating set of this semigroup are related to the sequence of numbers involved in the still open
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.^{
[2]} Various generalizations of the 3x + 1 semigroup have been constructed and their properties have been investigated.^{
[3]}

Definition

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:

$T(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n{\text{ is even}}\\[4px]{\frac {3n+1}{2}}&{\text{if }}n{\text{ is odd}}\end{cases}}$

The Collatz conjecture asserts that for each positive integer n, there is some iterate of T with itself which maps n to 1, that is, there is some integer k such that T^{(k)}(n) = 1. For example if n = 7 then the values of T^{(k)}(n) for k = 1, 2, 3,... are 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1 and T^{(11)}(7) = 1.

The relation between the 3x + 1 semigroup and the Collatz conjecture is that the 3x + 1 semigroup is also generated by the set

$\left\{{\dfrac {n}{T(n)}}:n>0\right\}.$

The weak Collatz conjecture

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:^{
[1]}

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.

^H. Farkas (2005). "Variants of the 3 N + 1 problem and multiplicative semigroups", Geometry, Spectral Theory, Groups and Dynamics: Proceedings in Memor y of Robert Brooks. Springer.