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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Notes
- 1.
In (Goldblatt 1975) the relation \(\perp _L\) was defined on the set of all proper filters of L by putting \(x\perp _L y\) iff \(\exists a\in L\) such that \(a'\in x\), \(a\in y\). That is equivalent to (18) when restricted to principal filters, and is also irreflexive, and hence satisifies (17) vacuously, when restricted to proper filters.
- 2.
Those settings typically use down-sets rather than up-sets. Here we are following the order-convention most commonly used in the relational semantics of intuitionistic, relevant and other substructural logics.
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Goldblatt, R. (2017). Representing and Completing Lattices by Propositions of Cover Systems. In: Yang, SM., Lee, K., Ono, H. (eds) Philosophical Logic: Current Trends in Asia. Logic in Asia: Studia Logica Library. Springer, Singapore. https://doi.org/10.1007/978-981-10-6355-8_1
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