Construction of semigroups with some exotic properties

An example of finitely presented associative algebra of intermediate growth, as well as the results which deal with the embedding of recursive presented objects into finitely presented groups was presented. It was shown that a finitely presented semigroup exists with noninteger Gelfand-Kirillov dimension. It was also demonstrated that for any rational α>5, there exists a finitely presented semigroup with Gelfand-Kirillov dimension α. The study show that by means of a manager symbols, the number α can be approximated by the sum of the finite number of power of the heads.