Nonembeddings of Combinatory Algebras

TL;DR

This paper proves that certain embeddings between models of combinatory algebras cannot be reversed, including effective and relativized versions, addressing open questions in the field.

Contribution

It establishes nonembedding results for key models of combinatory algebras, clarifying the structure and limitations of these models.

Findings

Embeddings between Kleene's second model, van Oosten's model, and Scott's graph model are non-reversible.

Nonembedding results extend to effective versions of these models.

Relativized embeddings are also shown to be non-reversible.

Abstract

In the theory of combinatorial algebras, there is a sequence of embeddings between Kleene's second model, van Oosten's model, and Scott's graph model. We prove that none of these embeddings can be reversed. We also prove nonembedding results for the effective versions of these models, and in addition we discuss relativized embeddings. This answers several questions from the literature.