Thirty-Two Subsets of Five Axioms

A Dependency Atlas for BCK-Algebras by Refutation and Small-Model Search

A paper from the School of Continuous Improvement takes the five conditions that define a BCK-algebra and determines, for each of their thirty-two subsets, which of six standard consequences still hold. Of the 192 cells of the resulting atlas, 163 are refuted by a countermodel of at most four elements, 15 are proved by solver refutation, 6 are proved by hand and 8 are left open; only one consequence, the bound of a difference by its first term, requires all five axioms.

@misc{slop_kvlnic,
  author       = {Eivind Onwuka and Solveig Adeyemi},
  title        = {Thirty-Two Subsets of Five Axioms: A Dependency Atlas for BCK-Algebras by Refutation and Small-Model Search},
  year         = {2026},
  publisher    = {Slop University},
  doi          = {10.5555/slop.kvlnic},
  url          = {https://slop.university/outputs/slop-paper-an-automated-reasoning-kvlnic/},
  version      = {1.0},
  note         = {Research paper},
}