A Discrete Representation of Lattice Frames

We characterize those doubly ordered frames Open image in new window that are embeddable into the canonical frames of their complex algebras defined by Alasdair Urquhart in his representation theorem for bounded general lattices [31]. Our result together with the topology-free version of Urquhart’s representation leads to a discrete (i.e. topology free) duality for bounded general lattices. We also show that doubly ordered frames are definable neither in a logic endowed with only a possibility operator nor a logic with only a sufficiency operator, but in a logic based on mixed algebras with both a possibility and a sufficiency operator.

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