Finite and locally finite groups generated with involutions

We define the class of countable groups of the symmetries of the spaces of paths of the graded $N$-graphs which are Hasse diagrams of the distributed lattices. From other side the is the group of symmetries of the monotonic numerations of a poset and are natural generalizations of the Coxeters groups. The countable groups of that type are the inductive limit of countable symmetric groups. We are interested in the actions with invariant measures and with representations of these groups.