Hypermagmas and Colored Operads: Heads, Phases, and Theta Roles

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.

Theta Theory: operads and coloring

We give an explicit construction of the generating set of a colored operad that implements theta theory in the mathematical model of Minimalism in generative linguistics, in the form of a coloring algorithm for syntactic objects. We show that the coproduct operation on workspaces allows for a recursive implementation of the theta criterion. We also show that this filtering by coloring rules on structures freely formed by Merge is equivalent to a process of structure formation by a colored version of Merge: the form of the generators of the colored operad then implies the dichotomy is semantics between External and Internal Merge, where Internal Merge only moves to non-theta positions.