Semilattice modes. II: The amalgamation property (Q1902533)

Although varieties of semilattice modes behave in many ways like those of modules, there is at least one important difference: varieties of semilattice modes do not necessarily have the amalgamation property. However, if the semiring \(R({\mathcal V})\) shown to be associated with each variety \({\mathcal V}\) of semilattice modes in the first of these two papers (see the preceding review) is a bounded distributive lattice, then \({\mathcal V}\) has the congruence extension property. Conversely, if \({\mathcal V}\) is locally finite and has the amalgamation property, then \(R({\mathcal V})\) is a bounded distributive lattice. An example (based on the unit interval in the real numbers) is given to show that if \({\mathcal V}\) is not locally finite, then it can have the amalgamation property even if \(R({\mathcal V})\) is not a bounded distributive lattice.