Does every finite alternative loop have two-sided inverses?
Statement
A *loop* is a set with a binary operation and a two-sided identity element $1$ in which every left and right division is unique (a quasigroup with identity). A loop is *alternative* if it satisfies the two alternative identities $x(xy)=(xx)y$ and $x(yy)=(xy)y$. Does every finite alternative loop have *two-sided inverses* — that is, for each element $x$ an element $x^{-1}$ with $x x^{-1}=x^{-1}x=1$? An infinite alternative loop without two-sided inverses is known, so finiteness is essential. Here 'two-sided inverse' means only the existence of such an $x^{-1}$; this is strictly weaker than the inverse property, and that gap is precisely what makes the finite question sharp.
Acceptance. FULLY RESOLVES: EITHER a finite alternative loop without two-sided inverses — an explicit Cayley table with a machine checker confirming that both alternative identities x(xy)=(xx)y and x(yy)=(xy)y hold and that some element has no two-sided inverse (a finite counterexample certificate) — OR a proof that every finite alternative loop has two-sided inverses. ADVANCES: a certified exhaustive verification that every alternative loop up to order N has two-sided inverses (Mace4/GAP-LOOPS enumeration with re-runnable code), establishing the first explicit finite range in which the answer is known; the larger the N, the stronger the advance.
Background
Source: 'Problems in loop theory and quasigroup theory' (Wikipedia), open-problems section: 'Does every finite alternative loop … have 2-sided inverses?' — proposed by Warren D. Smith (after a suggestion of J. D. Phillips); the page was last edited 2025-09-05 and still marks the problem open. The infinite counterexample is N. Ormes & P. Vojtěchovský, 'Powers and alternative laws', Comment. Math. Univ. Carolin. 48 (2007) 25–40 (arXiv:1509.05698), generalizing Smith's construction. The natural structural attack fails: Artin's theorem 'alternative implies diassociative' holds for rings and algebras but NOT for loops — indeed the Ormes–Vojtěchovský counterexample shows alternative does not imply diassociative for loops (a diassociative loop has the inverse property, which would force two-sided inverses). Moufang implies alternative is strict, and no known theorem forces finite alternative loops to have two-sided inverses. GAP's LOOPS package enumerates all loops of small order and tests the alternative laws and inverses. Vetted open as of 2026-07-06: no solved-signal in 2020–2026, and the literature records no finite partial result (no 'true up to order N'), so even a certified small-order verification would be new.
References
Investigations · 0
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