Doroshenko V. The semigroups of monotone functions and free subsemigroups of topological semigroups

The semigroups of transformation of infinite sets and topological semigroups are investigated. It is proved that in semigroups of monotone function on sets of natural and integers numbers the minimal system of generators does not exist. The minimal systems of generators in semigroups of co-finite functions on sets of natural and integers numbers are found. The presentation of these semigroups by generators and relations are constructed. In all these semigroups the groups of automorphisms are found. It is described the conjugacy relation and cenralizators in the semigroup of co-finite monotone functions on the set of natural numbers. Sufficient conditions of almost freeness of closed subsemigroup of semigroup of all transformations of natural numbers are proved. It is proved almost freeness of semigroup of all transformations of natural numbers, semigroup of monotone functions and semigroup of injective functions. It is proved sufficient condition of almost freeness of complete metrizable topological semigroup.