Finite implication E677 ⇒ E255

Determine whether, for finite magmas, the law x ▷ y ▷ (x ▷ ((y ▷ x) ▷ y)) (equation E677) entails the law x ▷ ((x ▷ x) ▷ x) ▷ x (equation E255); either establish the implication E677 ⊢_fin E255 for all finite magmas or construct a finite magma that satisfies E677 but not E255 to refute it.

Background

The Equational Theories Project determined the complete implication graph for 4,694 laws on magmas, including almost all finite implications. One implication remains unresolved up to duality: whether E677 implies E255 for finite magmas.

The authors were able to show that the implication E677 ⊢ E255 fails for arbitrary (possibly infinite) magmas using a greedy construction, but this method is inherently infinitary and does not settle the finite case. Earlier in the paper, they note their inability to prove or disprove the finite implication and tentatively conjecture it to be false. Resolving this question requires either a general proof for all finite magmas or an explicit finite counterexample.

References

However, there was (up to duality) precisely one finite implication which we could not settle, which we leave as an open problem:

Problem Does the law $x y (x ((y x) y))$ eq677 imply the law $x ((x x) x) x$ eq255 for finite magmas?

— The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale (2512.07087 - Bolan et al., 8 Dec 2025) in Section Implications for finite magmas (Section austin-sec)