Distributive lattices are studied from the viewpoint of effective algebra. In particular, we also consider special classes of distributive lattices, namely pseudocomplemented lattices and Heyting algebras. We examine the complexity of prime ideals in a computable distributive lattice, and we show that it is always possible to find a computable prime ideal in a computable distributive lattice. Furthermore, for any II01 class, we prove that there is a computable (non-distributive) lattice such that the II01 class can be coded into the (nontrivial) prime ideals of the lattice. We then consider the degree spectra and computable dimension of computable distributive lattices, pseudocomplemented lattices, and Heyting algebras. A characterization is given for the computable dimension of the free Heyting algebras on finitely or infinitely many generators.
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