Kourovka Notebook Problem 12.52

Definitions and mathematical statement

An $\ell$-group is a group $G$ equipped with a partial order $\leq$ for which every pair has a join $x\vee y$ and a meet $x\wedge y$, and which is invariant under left and right multiplication: for all $a,b,x,y\in G$, \[ x\leq y\implies axb\leq ayb. \] An $\ell$-homomorphism is a group homomorphism that preserves both joins and meets.

For an integer $\eta$ with $1<\eta<\infty$, let $\mathcal F_{\eta}$ be the free $\ell$-group on generators $x_1,\ldots,x_{\eta}$. Thus, for every $\ell$-group $H$ and every choice $h_1,\ldots,h_{\eta}\in H$, there is a unique $\ell$-homomorphism $f:\mathcal F_{\eta}\to H$ satisfying $f(x_i)=h_i$ for all $i$.

The question of S. McCleary asks whether, for every such finite rank $\eta$, the group $\mathcal F_{\eta}$ is Hopfian in the category of $\ell$-groups; explicitly, whether every surjective $\ell$-homomorphism \[ \varphi:\mathcal F_{\eta}\longrightarrow\mathcal F_{\eta} \] is injective. Equivalently, must every such $\varphi$ satisfy $\ker\varphi=\{1\}$?

Short English statement

For every finite rank greater than one, must every surjective map from the free lattice-ordered group to itself that preserves the group and lattice operations be one-to-one? The supplied registry labels this question open; that status is not independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $700871$, exact source code catalogue entry, title Kourovka Notebook Problem 12.52; supplied complete source record.
• S. McCleary, proposer, 1992. The supplied attribution identifies Problem 12.52 in The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992).
• The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently verified.