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Given a classic (not Residuated) lattice, with standard definition of partial order via lattice join and meet operations, is it possible to satisfy Galois equivalence $$ (x \vee y) < z \;\Longleftrightarrow\; x < (y / z). $$ for some binary operation $/$ ?

Edit: One way to answer this is putting lattice axioms together with Galois condition into something like Mace4. Then finite model search reveals that lattice join $\vee$ appears to have no adjoints, while meet $\wedge$ has. The question is more subtle, however: is there adjoint for lattice meet operation in any lattice model?

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  • $\begingroup$ Consider what happens when $y>z$. You cannot dualize only some parts of the definition of a Heityng algebra and blindly hope to obtain something useful. $\endgroup$
    – user46855
    Commented Feb 16, 2014 at 22:07

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