In algebra, the **3 x + 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 still open Collatz conjecture or the “3

*x*+ 1 problem”. The 3

*x*+ 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 3

*x*+ 1 semigroup was introduced by H. Farkas in 2005. Various generalizations of the 3

*x*+ 1 semigroup have been constructed and their properties have been investigated.