TL;DR
This paper introduces the concept of differential lattices, explores their isomorphisms, classifications, and derivations, and investigates the lattice structure of derivations on various types of lattices.
Contribution
It defines differential lattices with derivation-like maps, classifies them for basic lattices, and studies the structure of derivations on different lattice classes.
Findings
Derivations on finite distributive lattices form a lattice.
Derivations on complete infinitely distributive lattices form a complete lattice.
Conjecture: the poset of derivations on a general lattice is a lattice that determines the lattice.
Abstract
This paper studies the differential lattice, defined to be a lattice equipped with a map that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications of differential lattices are obtained for some basic lattices. Several families of derivations on a lattice are explicitly constructed, giving realizations of the lattice as lattices of derivations. Derivations on a finite distributive lattice are shown to have a natural structure of lattice. Moreover, derivations on a complete infinitely distributive lattice form a complete lattice. For a general lattice, it is conjectured that its poset of derivations is a lattice that uniquely determines the given lattice.
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