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

TL;DR

This paper introduces a novel mathematical framework using hypermagmas and colored operads to model complex syntactic structures, phases, and theta roles in linguistics, unifying various syntactic principles.

Contribution

It extends magma structures to hypermagmas with head functions, formulates syntactic structures as colored operads, and integrates movement and phase principles within this algebraic framework.

Findings

Hypermagmas model c-command and m-command relations.

Colored operads represent phases and theta roles.

Filtering syntactic objects can be reformulated via colored Merge.

Abstract

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.