Old and New Minimalism: a Hopf algebra comparison

• URL: http://arxiv.org/abs/2306.10270v1
• Date: Sat, 17 Jun 2023 06:31:36 GMT
• Title: Old and New Minimalism: a Hopf algebra comparison
• Authors: Matilde Marcolli, Robert C. Berwick, Noam Chomsky
• Abstract summary: We show that the newer formulation of Minimalism has a clear advantage purely in terms of the underlying mathematical structure. This mathematical structure shows that Internal and External Merge have significantly different roles in the old formulations of Minimalism.
• Score: 0.12891210250935145
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper we compare some old formulations of Minimalism, in particular Stabler's computational minimalism, and Chomsky's new formulation of Merge and Minimalism, from the point of view of their mathematical description in terms of Hopf algebras. We show that the newer formulation has a clear advantage purely in terms of the underlying mathematical structure. More precisely, in the case of Stabler's computational minimalism, External Merge can be described in terms of a partially defined operated algebra with binary operation, while Internal Merge determines a system of right-ideal coideals of the Loday-Ronco Hopf algebra and corresponding right-module coalgebra quotients. This mathematical structure shows that Internal and External Merge have significantly different roles in the old formulations of Minimalism, and they are more difficult to reconcile as facets of a single algebraic operation, as desirable linguistically. On the other hand, we show that the newer formulation of Minimalism naturally carries a Hopf algebra structure where Internal and External Merge directly arise from the same operation. We also compare, at the level of algebraic properties, the externalization model of the new Minimalism with proposals for assignments of planar embeddings based on heads of trees.

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