Choice-free duality for orthocomplemented lattices by means of spectral spaces

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets a Stone space depends upon Alexander’s Subbase Theorem, which asserts that X is compact if every subbasic open cover admits finite subcover. This easy consequence Ultrafilter Theorem—whose proof Zorn’s Lemma, well known to be equivalent Axiom Choice. Within this work, we give choice-free lattices by means special subclass spectral spaces; in sense our avoids use along with its associated nonconstructive choice principles. We then introduce new spaces call upper Vietoris orthospaces order characterize up homeomorphism (and isomorphism respect their orthospace reducts) proper filters used representation. It shown how constructions rise dual equivalence categories between category and orthospaces. Our duality combines Bezhanishvili Holliday’s approach for Boolean algebras Goldblatt Bimbó’s choice-dependent lattices.