Can two different lattices of theories be related?

Yes, since lattices of theories are based at first order languages, two different lattices of theories can be related through an interpretation between their underlying languages. The notion of a first order interpretation was discussed as example 4.11 on page 74 of the book by Jon Barwise and Jerry Seligman op. Cit.. First order language interpretations, which generalize first order language morphisms, are axiomatized in the IFF Type Language Namespace.

An interpretation from one language to a second language maps the relations of the first language to the expressions (formulas) of the second language. Each first order interpretation defines a truth infomorphism between the associated truth classifications (see Barwise and Seligman op. Cit.).

In turn, this truth infomorphism defines a truth concept morphism between the associated truth concept lattices, which maps between formal truth concepts (representing object level ontologies) in a very semantic way. More.

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