Oliynyk A. Finite state representations of groups and semigroups

For different residually finite groups and semigroups we construct faithful actions on rooted trees defined by finite automata. Faithful actions on rooted trees defined by finite automata for free products of finite groups and amalgamated free products of infinite cyclic groups are presented. Additionally, it is shown that one can define these actions for free products using automata over some alphabets with only 2 inner states. Faithful representations by finite state automorphisms of a p-Sylow subgroup of the automorphism group of p-regular rooted tree for those amalgamated free product of infinite cyclic groups which are residually finite p-groups are obtained. It is described a continuum family of automata with 2 states over an infinite alphabet such that each of them generates the free semigroup of rank 2. It is constructed faithful representations by finite state infinite unitriangular matrices over finite fields of free products of simple cyclic groups. It is introduced the finite state wreath power of a transformation semigroups and some its basic properties are obtained. In particular, faithful representations of free products of two monogenic semigroups with index 1 in such finite state wreath powers are constructed. A criterion under which an inverse semigroup admits a level transitive action by partial finite automaton permutations over a given alphabet is presented.