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This work studies invariants of generative grammars, independently of the specific format in which a grammar is specified, drawing on general algebraic notions of structure and symmetry. Structural notions in generative grammar such as dominance and c-command are provably invariant in all grammars, and specific morphemes, such as case markers and voice markers, are invariant in exactly the same sense in languages which have them, so structural properties of languages are not restricted to those that are coded into tree structures or hierarchical feature structures of any kind. This work also illustrates how relations such as the anaphor-antecedent relation can be invariant in all grammars even if realized differently in different languages. Grammars with anaphora constrained by nominal case marking and others with anaphora constrained by voice marking are elaborated. Thus the existence of universal invariants does not assume that grammars of different languages are isomorphic. This work further supports a strong form of compositionality and hypothesizes that syntactically invariant morphemes denote semantically invariant objects, revealing that the relation between form and meaning is not entirely arbitrary.