Construction of semigroups with some exotic properties

This article is devoted to a construction of finitely presented semigroups with some exotic properties. We consider elements of the finitely presented semigroup as a line of finite automata, which are locally interacting. So we create a virtual object (computational process) within a mathematical one. The construction is based on the submersion of a virtual computer (which is similar to Minsky Machine) into a semigroup. The main result of this paper shows the existence of a finitely presented semigroup with non-integer Gelfand-Kirillov dimension.