Kourovka Notebook Problem 17.27

Definitions and mathematical statement

Under the usual convention that an ordered group has a total order invariant under both left and right multiplication, let $(G_i,\leq_i)$, for $i\in\{1,2\}$, be ordered groups. Thus, for all $a,b,x,y\in G_i$, \[ a\leq_i b\implies xay\leq_i xby. \] Let $H_i\leq G_i$ carry the restricted order, and let $\varphi:H_1\to H_2$ be a group isomorphism that is also an order isomorphism: \[ h\leq_1 k\quad\Longleftrightarrow\quad\varphi(h)\leq_2\varphi(k)\qquad(h,k\in H_1). \] Form the free product with amalgamation \[ P=G_1*_{\varphi}G_2 =(G_1*G_2)\big/\big\langle\!\big\langle h\varphi(h)^{-1}:h\in H_1\big\rangle\!\big\rangle, \] where the displayed elements are understood through the canonical inclusions into the free product and the double brackets denote normal closure.

A lattice order on $P$ is a partial order $\preccurlyeq$ such that every pair of elements has a greatest lower bound and a least upper bound, and \[ a\preccurlyeq b\implies xay\preccurlyeq xby\qquad(a,b,x,y\in P). \] The group $P$ is lattice orderable if it admits such an order. It is orderable, in the convention used here, if it admits a total order invariant under both left and right multiplication.

Question (Bludov--Glass). Do there exist $(G_1,\leq_1)$, $(G_2,\leq_2)$, $H_1$, $H_2$, and $\varphi$ as above such that $P$ admits a lattice order but admits no total order invariant under both left and right multiplication?

Unresolved qualifications. The supplied statement does not explicitly specify whether "ordered" means invariance under both left and right multiplication. The formulation above uses that standard convention. It also interprets orderability and lattice orderability as properties of the abstract amalgamated product; the supplied statement does not explicitly say whether an order on $P$ must extend the prescribed orders on its factors. These conventions require mathematical review.

Short English statement

Can two totally ordered groups, amalgamated along subgroups identified by an order-preserving group isomorphism, yield a group admitting a translation-invariant lattice order but no translation-invariant total order? This formulation uses orders invariant under both left and right multiplication; the intended convention and any requirement to extend the factor orders need confirmation.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701329$, exact source problem code catalogue entry, title Kourovka Notebook Problem 17.27.
• V. V. Bludov and A. M. W. Glass, problem proposed in 2010, The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), Problem 17.27. The supplied record also references the 21st issue edition (2026), arXiv:1401.0300v48; the collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied background and status review report an affirmative answer labelled "Unverified AI claim," attributed to Achyuth Jayadevan, Preprint (2026): https://zenodo.org/records/22937468. The status review says that this claim is marked with $\#$ in the October 2026 version of the 21st issue. This is a supplied report of an unverified claim, not an established resolution.
• Supplied status-review date: 2026-10-16. The registry retains the label open, with evidence classified as unrefereed_claim. This assessment is supplied dated curation and has not been independently verified.