摘要: Lattice theoretic results regarding valuations have been formulated for the case of semimodular semilattices. Due to the modularity law of valuations, requiring both sup and inf, such results necessarily need to be formulated for semilattices embedded in a lattice. Ie they cannot be formulated as stand-alone results on semilattices. This paper makes a start at formulating similar results for semivaluations on semilattices, formulated purely in terms of the single operation of the semilattice under consideration. We generalise Birkhoff’s well known result that metric lattices are modular to the context of semilattices. In particular, we show that join semilattices equipped with a ranking function are semi-modular and obtain a suitable converse to this result, generalising similar results for lattices.AMS Subject Classification: 06A12, 20M10, 06B35, 54E35