From Co-Coverages to Radicals in Complete Lattices

Completeness and representation theorems in abstract algebra, lattice theory, and theoretical computer science are tied to the existence of ideal objects, and thus to transfinite methods such as Zorn’s lemma. Yet many concrete uses of those theorems appeal to finite approximations only, which carry a clear computational meaning. In this paper, we introduce co-coverages on complete lattices as a uniform way to specify ideal elements, and associate to each co-coverage a canonical closure operator with a folding property reminiscent of the covering principles at work in constructive algebra. This yields finitary, choice-free alternatives for arguments in which ideal objects are employed to reduce a computational problem to subcases. In a classical setting, our closure operators admit bases which allow to recover a host of primality principles such as the universal Krull–Lindenbaum theorem and Henkin’s lemma.