On the structure of modular lattices -- Unique gluing and dissection

TL;DR

This paper proves that in the class of modular, locally-finite lattices with finite covers, the process of gluing finite lattices to form larger ones is bijective, establishing a unique dissection and reconstruction method.

Contribution

It provides the first proof that gluing and dissection processes are inverses in this lattice class, confirming bijectivity.

Findings

Gluing of finite lattices is bijective in the specified class.

Unique dissection of lattices into finite components is possible.

Gluing and dissection are inverse processes in this context.

Abstract

This work proves that the process of gluing finite lattices to form a larger lattice is bijective, that is each lattice is the glued sum of a unique system of finite lattices, provided the class of lattices is constrained to modular, locally-finite lattices with finite covers. The results of this work are not surprising given the prior literature, but this seems to be the first proof that the processes of gluing and dissection can be made inverses, and hence that gluing is bijective.