arXiv subjects

Xing-Yu Hu

Publications and source records attributed to Xing-Yu Hu.

6 recordsLinked to original sources

arXiv2026

Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing

For every uncountable regular cardinal $α$, $\mathbb S^{\ast}(α)$ is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is $σ$-closed and, after adjoining a formal maximum, $α$-strategically closed. For $α\geqω_2$, every nonempty countable family of designated generic branches has a stationary and costationary common trace on the generic ladder-coordinate set $L_α$, while no countable family of cofinal branches generates $L_α$. These conclusions persist under a Kurepa-style level-size bound. In the unrestricted forcing, for every infinite cardinal $μ<α$ in the ground model, some level of the generic tree contains a copy of $({}^μ2)^V$. Consequently, the endpoint-corrected restriction family indexed by $\mathcal P_{ω_2}α$ is too wide, whereas the scaled restriction system indexed by $\mathcal P_αα$ has all levels of size less than $α$ exactly when $α$ is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of $L_α$ relative to the scaled system is at least $ω_1$. The low-cofinality empty-value convention also ensures that the set of domains of $L_α$ contains no club in $\mathcal P_αα$. The unrestricted tree clause of the motivating presentation is retained, without asserting forcing equivalence.

math.LO↗

arXiv2026

Idempotent-free non-solvable evolution algebras over $\mathbb{C}$

A recent conjecture states that a finite-dimensional complex evolution algebra is solvable if and only if it has no non-zero idempotents. We exhibit a three-dimensional counterexample over $\mathbb{C}$ whose isomorphism class already appears in the classification of three-dimensional complex evolution algebras. Since the conjecture is known in dimensions one and two, this counterexample has the smallest possible dimension. The algebra is defined over every field of characteristic different from $2$ and is the exceptional member of a one-parameter family of pairwise non-isomorphic non-solvable evolution algebras whose idempotents are determined explicitly. For the exceptional parameter, the derived series stabilises at a non-zero two-dimensional subalgebra whose only idempotent is zero. Direct sums with zero algebras give complex counterexamples in every dimension at least three. We also prove the conjectured equivalence whenever the stable term of the derived series is an evolution algebra.

math.RA↗

arXiv2026

Dense-set dependence in the Katětov order for uncountable coordinate ideals

For each countable ordinal $α\geq 2$, Filipów, Kowalczuk and Kwela introduced an ideal $\mathsf{conv}_α$ on the countable compact ordinal space $ω^α+1$. Kowalczuk later proved that, for each countable limit ordinal $λ$, the ideal $\mathsf{conv}_{<λ}$ is the greatest lower bound of $\{\mathsf{conv}_β:β<λ\}$ in the Katětov order. At the first uncountable level, let $A\subseteq[2,ω_1)$ be uncountable and let $D$ be a countable dense subset of $X_A=\prod_{α\in A}(ω^α+1)$. The coordinate ideal $\mathsf{Conv}(A,D)$ on $D$ consists of those $B\subseteq D$ with $π_α[B]\in\mathsf{conv}_α$ for every $α\in A$. For a pair $D\subseteq E$ of countable dense sets, call $α$ non-small if $π_α[E\setminus D]\notin\mathsf{conv}_α$. In ZFC, if at most countably many coordinates are non-small, then $\mathsf{Conv}(A,D)\equiv_K\mathsf{Conv}(A,E)$. Under CH this countability bound is sharp: for every $A\subseteq[3,ω_1)$ with $|A|=\aleph_1$, there are countable dense sets $D\subseteq D^*\subseteq X_A$ such that $\mathsf{Conv}(A,D^*)\leq_K\mathsf{Conv}(A,D)$ but $\mathsf{Conv}(A,D)\not\leq_K\mathsf{Conv}(A,D^*)$, and in particular $\mathsf{Conv}(A,D)$ and $\mathsf{Conv}(A,D^*)$ are not Katětov equivalent. The non-reduction is obtained, under CH, by diagonalizing along $ω_1$ coordinates against the elements of $ω^ω$ that code retractions $D^*\to D$.

math.LO↗

arXiv2026

Dense-Kernel and Closed-Core Reductions in the D-space Problem for Charming Spaces

Let X be a charming space with a Lindelof Sigma kernel Y, and let B be the closure of Y in X. We show that the question whether every charming space is a D-space can be reduced first to the dense-kernel case and then to a closed core. We define Obs_cc(Y,B) as the set of boundary points x in B minus Y such that, for every open neighborhood U of x in X, the intersection of U and Y is not countably compact, and let H be the closure of Obs_cc(Y,B) in B. We prove that the intersection of H with B minus Y is exactly Obs_cc(Y,B), and our main reduction theorem shows that X is a D-space if and only if H is. We also prove the following sufficient condition. If there is a closed set S contained in B minus Y with compact covering number less than the dominating number d, such that every point of B minus Y minus S has an open neighborhood U in X for which the intersection of U and Y is countably compact, then X is a D-space.

math.GN↗

arXiv2026

Raikov Remainders of Quotients by Raikov-Complete Almost Metrizable Normal Subgroups

Let $N$ be a closed normal subgroup of a topological group $G$, and let $\widehat q:ρG\toρ(G/N)$ extend the quotient homomorphism. We prove that $r_ρ(G/N)=\widehat q\bigl(r_ρ(G)\bigr)$ holds if and only if $\widehat q$ is onto and $\widehat q^{-1}(G/N)=G$, and we show that neither condition implies the other. Both conditions hold whenever $N$ is Raikov complete and almost metrizable. Consequently, pseudocompactness of $r_ρ(G)$ passes to $r_ρ(G/N)$. In particular, the conclusion applies to closed locally compact normal subgroups.

math.GN↗

arXiv2026

Carrier ideals, tail obstructions, and remainder traces for ladder-system spaces

For a ladder-system space $X_L$ with carrier $S\subseteq E^{ω_1}_ω$, the finite-label uniformization property $M_{<ω}$ characterizes countable metacompactness, and countable metacompactness is equivalent to the $Δ$-property. Both equivalences are known for stationary carriers. For arbitrary carriers, an active-tail formulation gives a direct proof that $M_{<ω}$ is equivalent to the $Δ$-property and leads to a support-finite decomposition theorem, together with club-smallness and trace criteria that avoid explicit ladder-position thresholds. A club-gap argument, combined with Fodor's lemma, shows that finite and countable tail multiplicity determine the same carrier ideal, namely $\mathrm{NS}\restriction S$. Subsets of the isolated part that meet each ladder in only finitely many points have clopen remainder traces, and these traces form a generalized Boolean algebra. All such traces are disjoint from the carrier part of the remainder. Finally, the subcarriers whose restricted spaces are $σ$-closed discrete form an ideal $\mathcal{C}_L$ containing $\mathrm{NS}\restriction S$. If $X_L$ is a $Δ$-space, a threshold-based gluing argument shows that $\mathcal{C}_L$ is a $σ$-ideal. Whether this holds for every ladder system remains open.

math.LO↗