Latticediffers

Latticediffers are a class of algebraic constructs used to represent and manipulate the difference between two states in a lattice-ordered domain. Built on a lattice L, a latticediffer between states s and t is expressed as a pair (P, N) where P and N are elements of L that encode additions and removals needed to transform s into t. The intuition is that P collects the components that must be present in the target state but are absent in the source, while N collects components that must be removed.

Semantics and operations are defined so that the pair (P, N) acts as a delta that can be merged with other deltas. A canonical merge operation ⊕ combines two differ pairs by taking the lattice join of corresponding components: (P1, N1) ⊕ (P2, N2) = (P1 ∨ P2, N1 ∨ N2). Applying a latticediffer to a state s uses a monotone, join-preserving transformation that yields the target state t. This design ensures that the order of applying diffs does not affect the result, a property aligned with delta-based synchronization.

Properties and behavior are chosen to be consistent with lattice theory: the set of latticediffers forms a commutative monoid under ⊕, with a zero element (0, 0) representing no change. The framework supports monotone growth in many practical lattices, and the components P and N can be interpreted as additive and subtractive directions within the lattice structure.

Applications of latticediffers appear primarily in distributed systems and knowledge representation. They provide compact, composable representations of change for state synchronization, delta-based replication, and version control in environments where data states form lattice-ordered structures, such as CRDTs and lattice-structured knowledge bases. See also lattice theory, delta-state CRDTs, and symmetric difference.