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] The 3x + 1 semigroup is the multiplicative semigroup of positive rational numbers generated by the set The function
as defined below is used in the "shortcut" definition of the Collatz conjecture: The Collatz conjecture asserts that for each positive integer
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:[1] The semigroup generated by the set which is also generated by the set is called the wild semigroup.