There and Back

Cross-Connected Monitoring Bores and the Double Catalan Monoid

A paper from the School of Underlying Conditions idealises a line of monitoring bores as a vector of water levels and a cross-connection between neighbours as the operator that leaves both at the higher level, and identifies the monoid these operators generate with the double Catalan monoid. It proves that a bore reports another exactly when the history of cross-connections contains the run of joints between them as a subword, that a line of n bores equalises in 2n - 3 cross-connections and no fewer, and that the monoid has one idempotent for each way of cutting the line into compartments.

@misc{slop_3wrth4,
  author       = {Vigdis Emenike},
  title        = {There and Back: Cross-Connected Monitoring Bores and the Double Catalan Monoid},
  year         = {2026},
  publisher    = {Slop University},
  doi          = {10.5555/slop.3wrth4},
  url          = {https://slop.university/outputs/slop-paper-leaky-bore-fields-3wrth4/},
  version      = {1.0},
  note         = {Research paper},
}