Kourovka Notebook Problem 12.54

Definitions and mathematical statement

An $\ell$-group is a group $G$, with identity $e$, equipped with a lattice order such that $x\leq y$ implies $axb\leq ayb$ for all $a,b,x,y\in G$. An $\ell$-subgroup is a subgroup closed under the lattice operations. Such a subgroup $H$ is convex if $h_1\leq x\leq h_2$, with $h_1,h_2\in H$ and $x\in G$, implies $x\in H$. Let \[ C(G)=\{H\leq G:H\text{ is a convex }\ell\text{-subgroup}\}, \] ordered by inclusion. This is a lattice: meets are intersections, and joins are the smallest convex $\ell$-subgroups containing the relevant unions.

For $g\in G\setminus\{e\}$, a value of $g$ is a convex $\ell$-subgroup $V$ maximal among those not containing $g$. Its cover is \[ V^+=\bigcap\{H\in C(G):V\subsetneq H\}. \] For a value $V$ of $g$, this is the smallest convex $\ell$-subgroup containing $V\cup\{g\}$, and it strictly contains $V$. Use the standard convention that $G$ is normal valued if every value $V$ of every nonidentity element is normal in its cover $V^+$.

Does there exist a normal-valued $\ell$-group $G$ such that, for every abelian $\ell$-group $A$, \[ C(A)\not\cong C(G), \] where $\cong$ denotes an isomorphism of lattices? No finiteness or cardinality restriction on $G$ or $A$ is imposed by the supplied statement.

The supplied source additionally reports that such an example exists when the requirement that $G$ be normal valued is omitted. This contextual assertion is retained as supplied and is not independently verified here.

Short English statement

Is there a normal-valued lattice-ordered group whose lattice of convex lattice-ordered subgroups cannot be realized by any abelian lattice-ordered group? The supplied registry labels this question open and reports an affirmative answer when the normal-valued condition is dropped.

Sources

• Local UnsolvedMath registry, permanent ID $700873$, source code catalogue entry, titled Kourovka Notebook Problem 12.54. The supplied record attributes the problem to S. McCleary, with proposed year 1992, and labels it open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 12.54, 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 and the contextual assertion concerning groups without the normal-valued restriction are supplied attribution, not independently verified assessments.