Maximal (semi-)lattices of fractions and injective hulls

We show that the maximal (semi-)lattice of fractions of a (semi-)lattice may be obtained as a canonical subsemilattice of the bicommutator of an injective hull of the (semi-)lattice under consideration, whithin a suitably defined category of semilattices. The construction is roughly similar to that of a maximal semigroup of quotients, but there are significant differences due to the commutativity and idempotency of (semi-)lattice operations. In particular, the construction is effective in the sense that it avoids the use of Zorn's lemma to obtain the required injective hulls.