Latticekellot

Latticekellot is a term used in theoretical mathematics and computer science to denote a class of time-annotated lattices. The name combines lattice with kellot, a word meaning clocks in some languages, signaling the addition of a temporal dimension to the standard lattice structure.

In its most common informal formulation, a latticekellot consists of a finite lattice (L, ≤) together with a time-annotation operation K that assigns to each element a finite, nonempty set of time-stamps. The time-sets are required to respect the lattice order: if x ≤ y then every time-stamp in K(x) is complemented by a time-stamp in K(y) in a way that preserves joins and meets. This yields a structure suitable for modeling time-stamped observations, events, or resource states.

Variants differ on how strictly they couple time to lattice operations; some require K to be monotone with respect to ≤, others impose compatibility with join and meet through a monotone extension of these operations to the time dimension.

Origin and use: The term emerged in late 2010s and early 2020s in theoretical discussions about extending order-theoretic constructs with temporal data. It has appeared in several papers and seminars as a flexible abstraction for time-aware data modeling, formal verification, and temporal reasoning. It is not a single, standardized notion, and precise definitions may vary by author.

Examples and applications: A simple example uses a boolean lattice with K mapping true/false to sets of timestamps indicating when a state holds. In practice, latticekellots are used to reason about the evolution of systems, track time-stamped information flows, or model versioned data.

See also: lattice, distributive lattice, temporal logic, time-stamped data, poset.