On the existence of subalgebras of direct products with prescribedd-fold projections

Abstract

Baker and Pixley have shown the equivalence of a number of conditions on a positive integerd and a varietyV be uniquely determined by its projections in thed-fold subproductsA i (1)×...×A i (d). It is shown here thatunder Baker and Pixley's conditions this uniqueness result is complemented by an existence result: SupposeA 1,...,A r ∈V, and that for everyd-tupleI={i(1),...,i(d)} a subalgebraS I ⊆A i(1)×...×A i(d) is given. Then these data are the projections of one subalgebraS⊆ A 1×...A r if and only if they are “consistent” on eachd+1-tuple {i(1),...,i(d+1)}. In the case where eachA i is the lattice {0, 1}, these results lead to the well-known description of finite distributive lattices in terms of finite partially ordered sets. Under appropriate hypotheses the above result generalizes to subalgebras of infinite direct products.