Hypermagmas and Colored Operads: Heads, Phases, and Theta Roles
This work provides a foundational mathematical framework for theoretical linguistics, addressing the formalization of syntactic operations and constraints, though it appears incremental in its extension of existing algebraic structures.
The paper tackles the problem of formalizing syntactic structures in linguistics by extending head functions to hypermagmas and representing phases and theta roles as colored operads, resulting in a unified framework that subsumes movement rules and compatibility conditions.
We show that head functions on syntactic objects extend the magma structure to a hypermagma, with the c-command relation compatible with the magma operation and the m-command relation with the hypermagma. We then show that the structure of head and complement and specifier, additional modifier positions, and the structure of phases in the Extended Projection can be formulated as a bud generating system of a colored operad, in a form similar to the structure of theta roles. We also show that, due to the special form of the colored operad generators, the filtering of freely generated syntactic objects by these coloring rules can be equivalently formulated as a filtering in the course of structure formation via a colored Merge, which can in turn be related to the hypermagma structure. The rules on movement by Internal Merge with respect to phases, the Extended Projection Principle, Empty Category Principle, and Phase Impenetrability Condition are all subsumed into the form of the colored operad generators. Movement compatibilities between the phase structure and the theta roles assignments can then be formulated in terms of the respective colored operads and a transduction of colored operads.