The aim of this paper is the introduction and the initial study of a lattice-ordered semigroup S which satisfies \(ab=(a\vee b)(a\wedge b)\) for all a,b\(\in S\). This axiom is of course a well-known theorem of the classical theory of divisibility over commutative and cancellative semigroups. Furthermore, the paper establishes some connections between the subjacent lattice and semigroup structures of such a lattice-ordered semigroup which the authors have called arithmetical lattice-semigroup (shortly: a.l.-semigroup). The reason why the authors chose this name was mainly suggested by both the fact that N is such an a.l.-semigroup and the decomposition theorem given in this paper.