Theta Theory: operads and coloring
Matilde Marcolli, Richard K. Larson
TL;DR
The paper develops a formal algebraic framework for theta theory within Minimalism by embedding theta role assignment into colored operads and modeling structure generation via the Merge operad. It introduces bud generating systems and two theta-colored layers (bare and complete) to realize theta grids and the theta criterion, using coproducts on workspaces to enable recursive checking and movement tracing. A key result is that the External/Internal Merge dichotomy and movement restrictions follow from the coloring construction, with movement constrained to non-theta positions and traces preserved for interpretation. The approach also demonstrates the equivalence of viewing filters as post-hoc constraints on fully formed structures or as dynamic, colored structure-building steps, offering a modular, computation-friendly account that decouples syntax formation from semantic filtering.
Abstract
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.