Implication between two identities in finite magmas remains undetermined

Determine whether every finite magma (a finite set equipped with a single binary operation ◇) that satisfies the identity for all elements x and y: x = y ◇ (x ◇ ((y ◇ x) ◇ y)) must necessarily also satisfy, for all elements x, the identity x = ((x ◇ x) ◇ x) ◇ x.

Background

The paper summarizes results and methodology from the Equational Theories Project (ETP), which analyzed all implications among 4,694 equational laws in magmas (algebras with a single binary operation). Automated theorem proving and model building were used extensively, and most implications were resolved.

In the finite-magma setting, one specific implication has not yet been determined: whether the identity x = y ◇ (x ◇ ((y ◇ x) ◇ y)) for all x, y implies the identity x = ((x ◇ x) ◇ x) ◇ x for all x. The author explicitly notes this as remaining unresolved in the ETP context.

References

In the restricted context of finite magmas, one implication remains undetermined: whether in all finite magma such that \forall x,\ \forall y,\ x = y \diamond (x \diamond ((y \diamond x) \diamond y)) one necessarily has \forall x, \ x = ((x\diamond x)\diamond x)\diamond x.

— Implication semilattice of 990 quasigroup equational laws  (2603.29909 - Floch, 31 Mar 2026) in Footnote in the subsection “The Equational Theories Project” (Section 2)