Open problems on quasi-absorbing elements in semigroups: order generation by quasi-primitive elements, semigroup-level duality, and Rees-congruence reduction

The 2023 preprint of Hager, Hamel and Heyde introduces quasi-absorbing elements, quasi-primitive elements, and ideals/simple semigroups relative to a given ideal, motivated by the semigroup $(\mathcal{G}(\mathbb{R}^d,C),\oplus,\supseteq)$ of closed convex $C$ -invariant sets under closed Minkowski addition that underlies set optimization. Its closing section poses three open directions that together ask for an axiomatic structure theory of ordered semigroups with quasi-absorbing elements reproducing the halfspace calculus of set optimization: (1) under what conditions on an ordered semigroup does every quasi-primitive element induce a total preorder via translation; (2) under what conditions does the duality representation of each element as a supremum of its translates by quasi-primitive elements hold, beyond the separated locally convex setting where a separation argument proves it; (3) classify the special cases (e.g. bottleneck ideals) in which the ideal-relative notions reduce to the classical 0-minimal/0-simple theory via the Rees congruence, a reduction that fails in general. All three questions remain open.

Background

A semigroup $(W,+)$ is a set with an associative binary operation; an element $\alpha\in W$ is absorbing (called a zero in multiplicative language) if $\alpha+w=w+\alpha=\alpha$ for all $w\in W$ , and a subset $A\subseteq W$ is an ideal if $W+A\subseteq A$ and $A+W\subseteq A$ . An element $e$ is idempotent if $e+e=e$ ; on the set $E(W)$ of idempotents the Rees order is defined by $f\le_H e\iff f+e=e+f=f$ , and Green's relation $H$ identifies elements with the same principal left and right ideals: with $W_L(u)=(W+\{u\})\cup\{u\}$ and $W_R(u)=(\{u\}+W)\cup\{u\}$ , one has $H(u)=\{v\in W\mid W_L(v)=W_L(u),\ W_R(v)=W_R(u)\}$ . The source paper (Hager–Hamel–Heyde, arXiv:2312.09914) starts from a phenomenon typical of set optimization: after deleting the absorbing element, another element may become absorbing. The set $A(W)$ of quasi-absorbing elements consists of the idempotents $\alpha$ such that $\le_H$ totally orders the idempotents $E_{\le}(\alpha)$ below $\alpha$ and every $w\notin E_{\le}(\alpha)$ is absorbed by $\alpha$ . The elements that become absorbing at some stage of successively removing absorbing elements are instead the stepwise quasi-absorbing elements $A_s(W)$ , which coincide with the finiteness-restricted set $A_{\mathrm{fin}}(W)$ and satisfy $A_s(W)\subseteq A(W)$ in general, with the inclusion possibly strict. The set $A(W)$ is always an ideal, indeed a bottleneck ideal (an ideal comparable by inclusion with every other ideal). Removing it leaves nontrivial minimal idempotents: the quasi-primitive elements $P(A(W))$ , i.e. the $A(W)$ -primitive idempotents — those $e\in E(W)\setminus A(W)$ with no idempotent strictly below them outside $A(W)$ . The paper also relativizes the classical notions of 0-minimal ideals and 0-simple semigroups: an ideal $D$ is $A$ -minimal if $D\not\subseteq A$ and $D$ properly contains no ideal other than subsets of $A$ , and $(W,+)$ is $A$ -simple if its only ideals are $W$ and subsets of $A$ ; taking $A$ to be the absorbing singleton recovers the classical definitions. The Rees congruence generated by an ideal $A$ collapses all of $A$ to one point, and passing to the Rees quotient is the natural attempt to reduce $A$ -relative notions to the classical theory.

The motivating example is the semigroup $\mathcal{G}(\mathbb{R}^d,C)=\{A\subseteq\mathbb{R}^d\mid A=\operatorname{cl}\operatorname{conv}(A+C)\}$ of closed, convex sets invariant under addition of a fixed closed convex cone $C$ , equipped with the closed Minkowski addition $A\oplus B=\operatorname{cl}(A+B)$ and ordered by reverse inclusion $\supseteq$ ; this is the setting of set optimization, with applications to set-valued risk measures, multivariate statistics, and vector optimization. Here $\emptyset$ is absorbing, $\mathbb{R}^d$ is quasi-absorbing, and the quasi-primitive elements are exactly the halfspaces $h^+_y=\{z\mid y^{\top}z\ge 0\}$ with normal $y$ in the dual cone $C^+\setminus\{0\}$ . Two structural facts hold in this model: each quasi-primitive $h^+_y$ induces a total preorder on the whole semigroup via

A\le B\iff A\oplus h^+_y\supseteq B\oplus h^+_y,

and every element admits the duality representation

A=\sup_{h^+_y\in P(A(\mathcal{G}(\mathbb{R}^d,C)))}A\oplus h^+_y,

a 'basic duality' proved by a separation argument in locally convex spaces (the construction extends from $\mathbb{R}^d$ to any separated locally convex topological vector space with topological dual $Z^*$ ). The paper's closing section asks how far these phenomena and the $A$ -relative structure theory generalize to arbitrary (ordered) semigroups, and states that the Rees-congruence reduction fails in general while important special cases — bottleneck ideals are highlighted via a companion manuscript announced as 'under preparation' — remain to be classified.

Problem Statement

As posed in Section 6 ('Perspectives') of Hager–Hamel–Heyde (arXiv:2312.09914), the problem is to develop the structure theory of (ordered) semigroups with quasi-absorbing elements far enough to answer three linked questions, whose common objective is an axiomatic characterization of when the halfspace calculus of the set-optimization semigroup $(\mathcal{G}(\mathbb{R}^d,C),\oplus,\supseteq)$ persists in a general ordered semigroup $(W,+,\le)$ with quasi-absorbing set $A(W)$ and quasi-primitive elements $P(A(W))$ :

Order generation: under what conditions on an ordered semigroup does every quasi-primitive element $p\in P(A(W))$ make the induced relation $A\preceq_p B\iff A+p\le B+p$ (translation by $p$ , with the given semigroup order on the right) a total preorder on $W$ (equivalently, a total order on the classes $A+p=B+p$ ), as the halfspaces do in $(\mathcal{G}(\mathbb{R}^d,C),\oplus,\supseteq)$ ? Note, as the authors do, that in this situation the Rees order on the relevant idempotents coincides with the original order except at the absorbing element.
Duality representation: under which conditions on the (ordered) semigroup and its quasi-absorbing elements does the basic duality

A=\sup_{p\in P(A(W))}A+p

hold for every element $A$ , generalizing the separation-based proof available in the separated locally convex topological vector space setting?

Rees-congruence reduction: classify the special cases in which the ideal-relative notions — $A$ -minimal ideals and $A$ -simple semigroups, in particular the quasi-minimal and quasi-simple cases $A=A(W)$ , and the role of bottleneck ideals — reduce to the classical 0-minimal and 0-simple theory via the Rees congruence generated by $A$ ; the reduction is stated to fail in general, and the class of ideals for which it succeeds is undetermined.

The three questions are independent directions of the source's Section 6 and need not be answered together; none of the three presupposes another.

References

Rico Hager, Andreas H. Hamel, Frank Heyde, 'Ideals of quasi-absorbing elements in semigroups', arXiv preprint arXiv:2312.09914 [math.GR] (2023), DOI 10.48550/arXiv.2312.09914, https://arxiv.org/abs/2312.09914
Andreas H. Hamel, 'Extended real number arithmetics via Dedekind cuts', arXiv preprint arXiv:2601.02229 (2026), DOI 10.48550/arXiv.2601.02229, https://arxiv.org/abs/2601.02229