arXiv subjects
Publications and source records attributed to Andriy Oliynyk.
It is examined finite state automorphisms of regular rooted trees constructed to represent groups $GL(n,\mathbb{Z})$. The number of states of automorphisms that correspond to elementary matrices is computed. Using the representation of $GL(2,\mathbb{Z})$ over an alphabet of size $4$ a finite state representation of the free group of rank $2$ over binary alphabet is constructed.
For any nontrivial abelian group $\mathbb{X}$ we construct a reversible (bireversible in case the order of $\mathbb{X}$ is odd) automaton such that its set of states and alphabet are identified with $\mathbb{X}$, transition and output functions are defined via the left and the right regular actions correspondingly and its group splits into the restricted wreath product $\mathbb{X} \wr \mathbb{Z}$, i.e. is a lamplighter group.
For every prime $p$ it is shown that a wide class of HNN extensions of free abelian groups admit faithful representation by finite $p$-automata.
We classify minimal transitive subsemigroups of the finitary inverse symmetric semigroup modulo the classification of minimal transitive subgroups of finite symmetric groups; and semitransitive subsemigroups of the finite inverse symmetric semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups.