Algebraic Representation of Mappings Between Submodule Lattices

Abstract

We discuss algebraic representations of mappings preserving arbitrary joins between submodule lattices. For a given join-preserving mapping \( \bar{g}:\mathfrak{L}\left( {_RM} \right)\to \mathfrak{L}\left( {_SN} \right) \) between submodule lattices, a representation is an R-balanced mapping h : B × M → N, where S B R is a bimodule such that \( \left\langle {h\left( {B\times U} \right)} \right\rangle =\bar{g}(U) \) for all \( U\in \mathfrak{L}\left( {_RM} \right) \). We begin by posing the question in a general abstract context and by defining the canonical subrepresentation, which is a representation if and only if there exists a representation. The problem is to give easy and natural conditions for the existence of a representation. We consider a very general situation for the mappings and give sufficient criteria for the existence of a representation. We also consider lattice isomorphisms.