### Local Greatest Equivalence Classes of ω-trees

#### Abstract

In \cite{ntcz1} we defined the concept of $\omega$-labeled tree as a binary, ordered and labeled tree with several features concerning the labels and order between the direct descendants of a node. In \cite{ntcz2} we introduced an equivalence relation $\simeq$ on the set $OBT(\omega)$ of $\omega$-trees and a partial order on the factor set $OBT(\omega)/_{\simeq}$. In this paper we decompose the factor set $OBT(\omega)/_{\simeq}$ into disjoint "local" subsets $K$, we show that if the relation defined by the mapping $\omega$ is a noetherian one then every local subset $K$ has a greatest element, we define an increasing operator on the set $OBT(\omega)/_{\simeq}$, which allows to obtain the greatest element of a local subset. In order to relieve the local features of a subset $K$ we give an example which shows that the greatest element of $K$ is not necessarily a maximal element of the factor set.