Abstracts of talks at OAL-RAG 2026

Abstracts of talks at the Conference on Order, Algebra, Logic and Real Algebraic Geometry (OAL-RAG 2026)

Richard N. Ball* (University of Denver; rball@du.edu), A. W. Hager (Wesleyan University; ahager@wesleyan.edu), J. Walters-Wayland (Chapman University; joanne@waylands.com). $\lambda$-repletions in $\mathbf{W}$.

The talk builds on the analysis of the structure of a completely regular frame presented by J. Walters-Wayland. We will show that the process of hollowing out the Madden frame of a $\mathbf{W}$-object $G$, the so-called scaffolding of the frame, produces a cardinally indexed sequence of essential extensions of $G$, terminating in the essential completion of $G$. The characteristic feature of the $\lambda$th extension is that its $\lambda$-generated $\mathbf{W}$-kernels are polars, a feature which we refer to as $\lambda$-repleteness. These repletions of $G$ are connected to the quasi-$F_\lambda$-covers of the Yosida space of $G$.

The second part of the talk takes up the problem of how the elements of the $\lambda$-repletion are approximated by the elements of $G$. Here we will indulge in a brief digression on the topic of convergences in $\mathbf{W}$, for, in a development that delighted and surprised the authors (or at least the first one), it turns out that classical continuous convergence has a very pretty pointfree formulation which elegantly solves the problem.

[1] K. Yosida, On the representation of the vector lattice, Proc.\ Imp.\ Adac.\ Tokyo 18 (1942), 339--342.
[2] A. W. Hager and L. C. Robertson, Representing and ringifying a Riesz space, Sympos.\ Math.\ XXI (1977), 411--431.
[3] J. J. Madden and J. Vermeer, Epicomplete archimedean $\ell$-groups via a localic Yosida theorem, J. Pure Appl.\ Algebra 68 (1990), 243--252.

Papiya Bhattacharjee (Florida Atlantic University; bpapiya@gmail.com). Concerning Max$_j(L)$.

Given any algebraic frame $L$ satisfying the FIP, and with a unit, consider a unit system $\frak U$ of $L$. It has been known that there is an inductive nucleus $j$ associated with this unit system. Furthermore, $jL={}$Fix$(j)$ is a coherent frame, and therefore maximal elements of $jL$ exist. We call this set Max$_j(L)$. Each $m\in{}$Max$_j(L)$ is a prime element of $L$, and so Max$_j(L)$ is a topological space under the hull-kernel topology; the subspace topology inherited from Spec$(L)$. In this talk, we will discuss the space Max$_j(L)$, along with a specific example when $j$ is the $d$-nucleus.

References
[J] Johnstone, P.T., {\it Almost Maximal Ideals}, Fund. Math. {\bf 123(3)} (1984), 197--209.
[M] Martinez, J. \emph{Unit and kernel Systems in Algebraic Frames}, Alg. Univ., \textbf{55(1)} (2006), 13--43.
[W] Wallman, H. \emph{Lattices and topological spaces}, Ann. of Math. (2) \textbf{39(1)} (1938), 112--126.

Ricardo Carrera (Nova Southeastern University (Florida); ricardo@nova.edu).

Manuel Lopez del Castillo (Florida Atlantic University; mlopezdelcas2018@fau.edu). On Hyper-Complemented Rings.

A commutative ring $R$ is complemented if for every $a ∈ R$, there is an element $b ∈ R$ such that $ab = 0$ and $a + b$ is a regular element of the ring. Recall that a ring $R$ is complemented if and only if its classical ring of quotients of $q(R)$ is von Neumann regular. A ring is hyper-complemented if every reduced homomorphic image of that ring is complemented. I will present preliminary results on hyper-complemented rings and show that some classes of rings, such as von Neumann regular rings, principal ideal domains, Dedekind domains, and Noetherian rings are hyper-complemented. I will also discuss ongoing work and open questions concerning the hyper-complemented property for Bézout domains and Prüfer domains.

Christian Corbett (Florida Atlantic University).

José F. Fernando Galván (Universidad Complutense, Madrid (Spain); josefer@ucm.es). Positive semidefinite elements and sums of squares in local henselian rings.

A classical problem in Real Geometry (which goes back to Hilbert's 17th problem) concerns the representation of positive semidefinite elements (psd) of a ring $A$ as sums of squares (sos) of elements of $A$. If $A=K$ is a field and $\frac{1}{2}\in A$, this problem has always a positive solution (psd=sos) due mainly to Artin-Schreier theory of (formally) real fields, which was developed with a view towards obtaining a solution to Hilbert's 17th problem. However, the situation is in general radically different when $A$ is a (commutative unital) ring (with non-empty real spectrum), because strong dimensional restrictions appear and, apart from (formally) real fields, there are few real rings with the property psd=sos. This makes this type of rings special.

If $A$ is an excellent ring of real dimension $\geq3$, it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in $A$. In addition, if $A$ is a local henselian noetherian ring (with non-empty real spectrum) such that every positive definite element of $A$ is a sum of squares in $A$, then $A$ is real reduced, and consequently its real and Krull dimensions coincide. Thus, local excellent henselian rings with the property psd=sos have Krull dimension $\leq2$.

In this work we determine the local excellent henselian rings $A$ of embedding dimension $\leq3$ such that $\frac{1}{2}\in A$ and have the property that every positive semidefinite element of $A$ is a sum of squares of elements of $A$. We also present several families of examples (both irreducible and reducible) in higher embedding dimension that have the property psd=sos, a global-local criterion for the $2$-dimensional analytic setting and an improvement of Scheiderer's result concerning principal saturated preorderings of low order.

Zbigniew Hajto (Jagiellonian University (Poland); zbigniew.hajto@uj.edu.pl). Some applications of real differential algebra in Galois theory and affine algebraic geometry.

In this talk, we explore several applications of real differential algebra, bridging the gap between differential Galois theory and problems in affine algebraic geometry. First, we present a Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants. For integrable systems, we establish the existence and uniqueness of formally real Picard-Vessiot extensions and provide a corresponding Galois correspondence theorem. We apply this framework to characterize formally real Liouvillian extensions of real partial differential fields using split solvable linear algebraic groups. Additionally, we discuss the topological properties of real Liouville functions and their relation to the concept of tame topology in the sense of Grothendieck and Khovanskii. In the second part of the talk, we apply differential algebraic methods to the study of polynomial automorphisms and the Jacobian Conjecture. We compare two classes of automorphisms: strongly nilpotent and Pascal finite. A polynomial map $F$ is defined as Pascal finite if there exists an integer $m$ such that $\Delta_{F}^{m}(X)=0$, where $\Delta_{F}$ is a derivation on the polynomial ring $K[X]^n$ given by $\Delta_{F}(P) = P \circ F - P$. We show that while every strongly nilpotent automorphism is Pascal finite, the converse is not true, as demonstrated by the Nagata automorphism. Finally, we discuss how the derivation operator $\Delta_{F}$ serves as a link to combinatorial methods involving weighted rooted trees, providing sharp degree bounds for inverse maps.

References
[1] T. Crespo, Z. Hajto and M. van der Put, Real and p-adic Picard-Vessiot fields, Math. Ann. 365 (2016), 93--103.
[2] T. Crespo, Z. Hajto and R. Mohseni, Real Liouvillian Extensions of Partial Differential Fields, SIGMA 17 (2021), 095, 16 pages.
[3] E. Adamus, P. Bogdan, T. Crespo and Z. Hajto, An effective study of polynomial maps, Journal of Algebra and Its Applications, 16 (2017), No. 8, 1750141 [cite: 276, 278, 279].
[4] E. Adamus, A note on algorithmic approach to inverting multivariate formal power series, arXiv:2503.03520v2.

James Klinkenberg (Florida Atlantic University; jklinkenberg2020@fau.edu). Weak Disjointification and Max$_d(L)$.

An algebraic frame satisfying the finite intersection property will be denoted as an $M$-frame. An $M$-frame is said to have disjointification if for each pair of compact elements $a,b \in L$, there exist disjoint $c \wedge d = 0 \in \mathfrak{K}L$, such that $c \leq a$ and $d \leq b$, and $a \vee b = a \vee d = c \vee d$. For an $M$-frame $L$, this is if and only if $\text{Spec}(L)$ is a root system. This gives a well-defined map from $\text{Min}(L)$ to $\text{Max}_d(L)$, the space of minimal primes of $L$ to the space of maximal $d$-elements of $L$. In this talk, a weaker condition called weak disjointification is defined that leads to this same map, without requiring that $\text{Spec}(L)$ forms a root system. Assuming this condition, we will show that the space $\text{Max}_d(L)$ is Hausdorff. We will also see examples of frames having weak disjointification but not disjointification.

Wieslaw Kubis (Czech Academy of Sciences (Czech Republic) and Franz-Viktor Kuhlmann* (University of Szczecin (Szczecin, Poland); franz-viktor.kuhlmann@usz.edu.pl). Chain union closures and convexity.

Ramiro H. Lafuente-Rodriguez (University of South Dakota; Ramiro.LafuenteRodri@usd.edu).

James Madden (Louisiana State University; jamesjmadden@gmail.com).

Jim McEnerney (mcenerney.math@gmail.com). Integral closure and semialgebraic closure: are they the same?

The integral closure of an ideal is a well known operator on an ideal. Lesser known is the semi-algebraic closure of an ideal that is mostly applicable to real algebra. The purpose of this talk is to review these concepts and present evidence that they are the same for monomial ideals in the ring of polynomials over an ordered field and that the< two operators commute in the case of Noetherian rings.

Warren McGovern (Florida Atlantic University, Wilkes Honors College; WMcGove1@fau.edu).

Andrew Moshier (Chapman University; moshier@chapman.edu).

Aysha Nuhuman (Florida Atlantic University).

Ricardo Palomino Piepenborn (University of Manchester; ricardo.palomino@manchester.ac.uk).

Anand Pillay (Notre Dame University; Anand.Pillay.3@nd.edu).

John Stokes-Waters (University of Manchester; john.stokes-waters@manchester.ac.uk). A Model Companion for Abelian Lattice-Ordered Groups with a Valuation.

An abelian lattice-ordered group, or abelian ℓ-group, is an abelian group equipped with a compatible lattice ordering. In this paper, we introduce two multi-sorted extensions of abelian lattice-ordered groups inspired by the zero-set maps for continuous functions into $\mathbb R$. We demonstrate that this expansion is equivalent to equipping $G$ with a spectral subspace $X$ of ℓ-Spec$(G)$, along with the map sending $a∈G$ to $\bigvee(a∧0)∩X$. Using a classical partial quantifier elimination result originally due to Fuxing Shen and Volker Weispfenning [15], we show that one of these expansions admits a model companion, which is complete and has quantifier elimination in a small language expansion.

Marcus Tressl (University of Manchester, United Kingdom; marcus.tressl@manchester.ac.uk). Every join semilattice with smallest element is isomorphic to the semilattice of compact open sets of some space.

For a given join semilattice $S$ with smallest element, I will present (in a functorial way) a space $X$ such that $S$ is isomorphic to the join semilattice of compact and open subsets of $X$. When $S$ is a bounded distributive lattice, the construction returns the spectral space of prime ideals of $S$. When $S$ is a distributive join semilattice, the construction returns the Stone-Grätzer dual of $S$ (whose topology is the algebraic frame of ideals of $S$).

Joanne Walters-Wayland (Chapman University; jwalterswayland@gmail.com).

Brian Wynne (Lehman College, City University of New York: Brian.Wynne@lehman.cuny.edu). More existentially closed prime-model extensions of Archimedean lattice-ordered groups with strong unit.

Let $\bf{W}^+$ be the category of non-zero Archimedean $\ell$-groups with distinguished strong unit and unit-preserving $\ell$-group homomorphisms. The existentially closed (e.c.) $\bf{W}^+$-objects, which are to $\bf{W}^+$ as algebraically closed fields are to fields, may be characterized as follows: $G$ is e.c. in $\bf{W}^+$ just in case $G$ is divisible, feebly projectable, has no basic elements, and every weak unit in $G$ is a strong unit. An extension $G'$ of $G$ in $\bf{W}^+$ is called an e.c. prime-model extension of $G$ if $G'$ is e.c. in $\bf{W}^+$ and embeds over $G$ into every other e.c. extension of $G$ in $\bf{W}^+$ (this is analogous to the algebraic closure of a field). At OAL 2025, I showed that certain $\bf{W}^+$-objects, including all hyperarchimedean ones, have unique (up to isomorphism over $G$) e.c. prime-model extensions, and others have none at all. In this talk, I will give more examples of both of those situations.

Last updated April 16, 2026.