On the structure of modular lattices -- Axioms for gluing

Abstract: This paper explores alternative statements of the axioms for lattice gluing, focusing on lattices that are modular, locally finite, and have finite covers, but may have infinite height. We give a set of "maximal" axioms that maximize what can be immediately adduced about the structure of a valid gluing. We also give a set of "minimal" axioms that minimize what needs to be adduced to prove that a system of blocks is a valid gluing. This system appears to be novel in the literature. A distinctive feature of the minimal axioms is that they involve only relationships between elements of the skeleton which are within an interval $[x \wedge y, x \vee y]$ where either $x$ and $y$ cover $x \wedge y$ or they are covered by $x \vee y$. That is, they have a decidedly local scope, despite that the resulting sum lattice, being modular, has global structure, such as the diamond isomorphism theorem.