Coproducts of Distributive Lattice based Algebras

The analysis of coproducts in varieties of algebras has generally been variety-specific, relying on tools tailored to particular classes of algebras. A recurring theme, however, is the use of a categorical duality. Among the dualities and topological representations in the literature, natural dualities are particularly well behaved with respect to coproduct. Since (multisorted) natural dualities are based on hom-functors, they send coproducts into cartesian products. We carry out a systematic study of coproducts for finitely generated quasivarieties A that admit a (term) reduct in the variety D of bounded distributive lattices. In this setting we present necessary and sufficient conditions on A for the forgetful functor UA from A to D to preserve coproducts. When this is not the case, we demostrat how to obtain the lattice reduct of a coproduct of algebras in A from the lattice reducts of the algebras involved. Our results are based on an in depth analysis of the connection between Priestley duality and multisorted piggyback dualities. Given an algebra A ∈ A we present procedure to recover the Priestley dual of UA(A) from the natural dual of A. We then use this translation and the good behaviour of natural dualities with coproducts to recover distributive lattice reduct of a coproduct of algebras in A. As a byproduct, our work reveals that the type of natural duality that the class A can possess is connected with properties of coproducts in A and the way in which UA behaves with respect to them.