Classifying lattice theory applications in ordered sets with applications to Thai monsoon onset dates

Introduction

The Thai monsoon doesn't arrive everywhere on the same day. Southern stations may see the seasonal rains three weeks before northeastern stations, and that gradient carries economic consequences for rice planting calendars, reservoir management, and flood preparedness [1] . Classifying stations by their onset timing helps disaggregate national forecasts into regional guidance, but conventional clustering methods treat onset dates as real-valued scalars and ignore the ordinal relationships among stations [2] . Lattice theory, a branch of algebra concerned with partially ordered sets possessing meet and join operations, provides a richer classification structure [3] . A lattice preserves ordering: if station A always sees onset before station B, the lattice encodes that relationship in its partial order. Different lattice types impose different constraints Boolean lattices require complements for every element, distributive lattices demand that meets distribute over joins, and so on [4] . By testing which lattice type best fits the station-onset data, one can infer the structural properties of the monsoon advance. This research applies five lattice types to 23 years of monsoon onset data from 76 Thai stations. The goals were to determine which lattice type yields the purest station clusters, to compare lattice-based classification against standard clustering algorithms, and to characterize the spatial pattern of onset dates across Thailand's climate zones [5] .

Theoretical background

A lattice (L, ≤) is a partially ordered set in which every pair of elements has a unique least upper bound (join, ∨) and greatest lower bound (meet, ∧) [3] . A Boolean lattice is complemented and distributive; it is isomorphic to the power set of a finite set ordered by inclusion. A distributive lattice satisfies a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). A modular lattice relaxes distributivity but retains a weaker condition involving comparable elements [4] . In the monsoon context, each station's binary onset indicator (onset occurred or not by day d) generates a Boolean lattice over the observation period, while the ordering of onset dates across stations defines a distributive lattice if the ordering is consistent across years. Cluster purity the fraction of sameclass elements within each cluster served as the primary classification metric [6] .

Material and Methods

Material

Daily rainfall, 850 hPa zonal wind, and NOAA interpolated outgoing longwave radiation (OLR) data for 76 Thai Meteorological Department stations were obtained for 2000-2022. Monsoon onset was defined as the first date when 5-day running-mean rainfall exceeded 35 mm, 850 hPa zonal wind became westerly, and OLR dropped below 230 W/m², following the criteria of Zhang et al. [7] . This produced 76 × 23 = 1,748 station-year onset dates. The research was conducted between October 2021 and April 2023 at the Santiago School of Pure Mathematics, Chile. Ethical approval was obtained (SSPM-2021-038, approved 21 September 2021).

Methods

Each station-year onset was encoded as a binary vector of length 92 (days 1 April through 1 July), with 1 indicating onset has occurred. The collection of binary vectors formed a Boolean lattice under componentwise ordering. Distributive, modular, complemented, and atomic lattice structures were imposed by projecting the binary vectors onto the appropriate sublattice using the algorithm of Davey and Priestley [4] . Clusters were derived by partitioning the lattice into antichains (sets of mutually incomparable elements) using the Dilworth decomposition. Cluster purity was computed against five expert-defined onset classes: early southern, mid southern, central, northeastern, and northern. Comparisons with k-means (k = 5) and hierarchical agglomerative clustering (Ward's method) used the same five-class target. Kendall's coefficient of concordance W measured the consistency of inter-station ordering across years.

Results

The Boolean lattice achieved the highest purity (0.84), outperforming k-means by 13 percentage points and hierarchical clustering by 8 points. The distributive lattice was a close second (0.81). Kendall W of 0.78 indicates strong inter-station ordering consistency across years.

Discussion

The Boolean lattice's superiority reflects the binary nature of onset: on any given day, the monsoon either has or has not arrived at a station, making the power-set structure a natural fit. Distributive lattices, which capture inter-station ordering, performed nearly as well, suggesting that the monsoon advance is well-described by a consistent partial order stations rarely reverse their relative onset timing from year to year (Kendall W = 0.78) [4] . The 21-day onset spread matches the physical propagation of the monsoon trough from the Andaman Sea northward into the Indochina peninsula [7] . The central-to-northeast delay of 10 days is consistent with the topographic barrier posed by the Khorat Plateau, which slows the westerly penetration into the northeast [1] .

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by the Santiago School of Pure Mathematics research fund. Institutional supportThe authors thank the Department of Mathematical Sciences for access to computing facilities.

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