Groups and Inverse Semigroups in Lambda Calculus

2026-02-05

AI Summary

This study investigates the invertibility of λ-terms across various λ-theories, with a focus on the algebraic characterizations of finite hereditary permutations (FHPs) and hereditary permutations (HPs). By unifying FHPs and HPs within the framework of inverse semigroups and leveraging tools from λ-calculus, Böhm tree semantics, and η-expansion techniques, the work establishes that FHPs are precisely the invertible elements in every λ-theory lying between λη and H⁺. Furthermore, it demonstrates an exact correspondence between the natural order on FHPs and (infinite) η-expansions. The results not only confirm that FHPs and HPs form inverse semigroups in a broad range of λ-theories but also elucidate the semantic meaning of their order structure, thereby extending classical invertibility results to a significantly wider spectrum of λ-theories.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study invertibility of $\lambda$-terms modulo $\lambda$-theories. Here a fundamental role is played by a class of $\lambda$-terms called finite hereditary permutations (FHP) and by their infinite generalisations (HP). More precisely, FHPs are the invertible elements in the least extensional $\lambda$-theory $\lambda \eta$ and HPs are those in the greatest sensible $\lambda$-theory $H^*$. Our approach is based on inverse semigroups, algebraic structures that generalise groups and semilattices. We show that FHP modulo a $\lambda$-theory $T$ is always an inverse semigroup and that HP modulo $T$ is an inverse semigroup whenever $T$ contains the theory of B\"ohm trees. An inverse semigroup comes equipped with a natural order. We prove that the natural order corresponds to $\eta$-expansion in $\mathrm{FHP} /T$, and to infinite $\eta$-expansion in $\mathrm{HP}/T$. Building on these correspondences we obtain the two main contributions of this work: firstly, we recast in a broader framework the results cited at the beginning; secondly, we prove that the FHPs are the invertible $\lambda$-terms in all the $\lambda$-theories lying between $\lambda \eta$ and $H^+$. The latter is Morris'observational $\lambda$-theory, defined by using the $\beta$-normal forms as observables.

Problem

• lambda calculus
• invertibility
• lambda-theories
• finite hereditary permutations
• inverse semigroups

Innovation

• inverse semigroups
• lambda calculus
• finite hereditary permutations
• eta-expansion
• lambda-theories