A semigroup over a generalized tree, denoted by the term ℳL-semigroup, is a compact semigroup S such that Green's relation H is a congruence on S and S/H is an abelian generalized tree with idempotent endpoints and E(S/H) a Lawson semilattice. Each such semigroup is characterized as being constructible from cylindrical subsemigroups of S and the generalized tree S/H in a manner similar to the construction of semigroups over trees and of the hormos. Indeed, semigroups over trees are shown to be particular examples of the construction given herein.
