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Let L be a complete orthomodular lattice. There is a one to one correspondence between complete boolean subalgebras of L contained in the center of L and endomorphisms j of L satisfying the Borceux-Van den Bossche conditions.
Let L be an arbitrary orthomodular lattice. There is a one to one correspondence between orthomodular sublattices of L satisfying an extra condition and quantic quantifiers. The category of orthomodular lattices is equivalent to the category of posets having two families of endofunctors satisfying six conditions.