Nilpotent subsemigroups of a semigroup of order-decreasing transformations of a rooted tree

Abstract

This paper deals with a semigroup of order-decreasing transformations of a rooted tree.  Such are the transformations \(\alpha\) of some rooted tree \(G\) satisfying following condition: for any \(x\) from \(G\) \(\alpha(x)\) belongs to a simple path from \(x\) to the root vertex of \(G\). We describe all subsemigroups of the mentioned semigroup, which are maximal among nilpotent subsemigroups of nilpotency class \(k\) in our semigroup.  In the event when rooted tree is a ray we prove that all these maximal subsemigroups are pairwise nonisomorphic.