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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 v y) < z <-> 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 v appears to have no adjoints, while meet ^ has. The question is more subtle, however: is there adjoint for lattice meet operation in any lattice model? More interestingly, lattice model with some additional structure (additional axioms)?

3 deleted 58 characters in body; added 44 characters in body; added 19 characters in body

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 v y) < z <-> x < (y / z).


for some binary operation /? (My intuition and brute force computer search hints that it is only possible for monoidal, not lattice-like operations.)

Edit: One way to answer this is just put putting lattice axioms together with Galois condition into something like Mace4and witness it constructing a . Then finite model search reveals that lattice join v appears to have no adjoints, while meet ^ has. The question is more subtle, however: is there adjoint for lattice meet operation in any lattice modelcan one define ? More interestingly, lattice model with some additional operation / to satisfy Galois connectionstructure (additional axioms)?

2 added 274 characters in body

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 ^ v y) < z <-> x < (y <op> / z).


for some binary operation <op>/? (My intuition and brute force computer search hints that it is only possible for monoidal, not lattice-like operations.)

 Edit: One way to answer this is just put lattice axioms together with Galois condition into something like Mace4 and witness it constructing a model. The question is more subtle, however: for any lattice model can one define additional operation / to satisfy Galois connection? 
 
 
 
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