Essential and retractable Galois connections

TL;DR

This paper introduces new classes of Galois connections for bounded lattices that preserve important module-theoretic properties and establish bijections between closed elements, unifying various module concepts.

Contribution

It defines cyclically essential, retractable, and UC Galois connections and explores their properties and applications in module theory, unifying several module classes.

Findings

Essential retractable Galois connections preserve uniform dimension.

Essential retractable UC Galois connections induce bijections between closed elements.

Applications to submodule lattice Galois connections unify module concepts.

Abstract

For bounded lattices, we introduce certain Galois connections, called (cyclically) essential, retractable and UC Galois connections, which behave well with respect to concepts of module-theoretic nature involving essentiality. We show that essential retractable Galois connections preserve uniform dimension, whereas essential retractable UC Galois connections induce a bijective correspondence between sets of closed elements. Our results are applied to suitable Galois connections between submodule lattices. Cyclically essential Galois connections unify semi-projective and semi-injective modules, while retractable Galois connections unify retractable and coretractable modules.