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Is there a sensible classification of the properties of structures with a given signature $\sigma$, e.g. graphs with $\sigma = \lbrace R \rbrace$?

For example like this:

  1. properties defined by first-order sentences over $\sigma$ (with or without specific syntactical properties)

  2. properties defined by monadic second-order sentences over $\sigma$ (with or without specific syntactical properties)

  3. properties stating the existence of another structure (of the same kind or another) and a specific mapping relation to it

  4. properties stating that a given structure invariant has a specific value

  5. properties stating that a given structure invariant has a specific property

How can this list be expanded, can it be completed or can it never be exhaustive?

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Read about sketches in Johnstone's book Sketches of an Elephant (second volume). They give a hierarchy going down from your number (1) -- geometric logic, essentially algebraic logic (finite-limit theories), algebraic logic. There are many variations. There is undoubtedly no end to such classifications, both up and down from first order logic. –  SixWingedSeraph Jan 8 '10 at 15:13
    
Rather advanced, but thanks. –  Hans Stricker Jan 8 '10 at 15:19

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