Frobenius reciprocity, modular connections, lattice isomorphism theorem and abstract principal ideals

TL;DR

This paper explores the connections between Frobenius reciprocity, modular connections, and the lattice isomorphism theorem within the context of categorical algebra and ideal theory, highlighting their interrelations and foundational properties.

Contribution

It clarifies the relationship between Frobenius reciprocity and modular connections, linking them through the lattice isomorphism theorem in an abstract algebraic framework.

Findings

Frobenius reciprocity relates to modular connections in projective homological algebra.

The lattice isomorphism theorem underpins the structural connections discussed.

Abstract principal ideals are connected to Galois connections in the study.

Abstract

The purpose of this short note is to fill a gap in the literature: Frobenius reciprocity in the theory of doctrines is closely related to modular connections in projective homological algebra and the notion of a principal element in abstract commutative ideal theory. These concepts are based on particular properties of Galois connections which play an important role also in the abstract study of group-like structures from the perspective of categorical/universal algebra; such role stems from a classical and basic result in group theory: the lattice isomorphism theorem.