A rooted directed tree $${\mathscr {T}}=(V, E)$$ with root $${\textsf{root}}$$ can be extended to a graph $$\mathscr {T}_\infty =(V_\infty , E_\infty )$$ by adding vertex $$\infty $$ V and declaring each in as parent of . One may associate the {T}}_{\infty }$$ family semigroup structures $$\sqcup _{\mathfrak b}$$ extreme ends being induced join operation meet $$\sqcap from lattice theory (corre...