Abstract
Cover systems abstract from the properties of open covers in topology, and have been used to construct lattices of propositions for various modal and non-modal substructural logics. Here we explore cover systems on the set of principal filters of a lattice and their role in lattice representations. A particular system with finite covers gives a lattice of propositions isomorphic to the ideal completion of the original lattice. Relaxing the finiteness condition yields another cover system whose lattice of propositions gives a presentation of the MacNeille completion. This is analysed further for ortholattices. For Heyting algebras a stricter cover relation is shown to have the properties of a Grothendieck topology while its lattice of propositions is still the MacNeille completion.
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