Can we characterize functions over term algebras using lambda calculus? This paper explores the lambda-representability of functions over both open and closed term algebras. By extending the standard notion of lambda-representation to open terms, the authors provide a recursion-theoretic characterization of lambda-representable functions over open term algebras. Building on this, they derive two characterizations of lambda-representable functions over closed term algebras, which represent free structures. The appendix offers a simplified formulation of recursive word functions, examining their relation to recursive functions on natural numbers and type-free pure lambda-calculus. This research deepens our understanding of the interplay between lambda calculus and term algebras.
Published in the International Journal of Foundations of Computer Science, this article directly contributes to the journal's focus on theoretical foundations of computer science. The exploration of lambda-representability and term algebras aligns with the journal's commitment to advancing the mathematical and logical underpinnings of computation.