Abstracts of talks at the Conference on Order, Algebra, Logic and Real Algebraic Geometry (OAL-RAG 2024)
Mehsin Jabel Atteya (Mustansiriyah University, Iraq; mehsinatteya88@gmail.com): When Multiplicative Generalized $(\lambda, \lambda)$ Derivations Supply Commutative Ideals Over Associative Rings.
The modern definition of abstract ring appeared in 1914 while investigating reversible rings, which represent the generalization of the reduced ring. It has meaning in the ring theory. In fact, ring theory has undergone a revolution in recent years with the development of what is now known as e-reversible rings. Indeed, there are several authors interested in this algebraic area. In this manuscript, $R$ acts as a semiprime rings, $U$ is a nonzero ideal of $R$, $\lambda$ is an automorphism mapping over $R$ and $(F,f),(G,g)$ and $(H,h)$ are multiplicative generalized $(\lambda, \lambda)$ derivations of $R$. Hence, we investigate the behaviour of certain differentials identities involving three multiplicative generalized $(\lambda, \lambda)$ derivations on ideals of prime rings. We assume that $R$ is an associative ring with center $Z$. Named that $R$ is a semiprime when $R$ satisfy the expression $r_{1}Rr_{1}=0$ which yields $r_{1}=0$ and $R$ is prime if $r_{1}Rr_{2}=0$ which supply two options there either $r_{1}=0$ or $r_{2}=0$. As a factual information about the connection between the previous concepts a prime and semiprime ring mentioned as following: A prime ring forms another kind of ring, which is a semiprime, while the converse, unfortunately, is not always true. For any $x,y \in R$ the symbol $[x,y]$ represents the Lie commutator $xy-yx$ and the Jordan product $x\circ y=xy+yx$.
The commutativity of prime rings with derivation was initiated by Posner [4]. An additive map $d\colon R\to R$ is called a derivation if $d(xy)=d(x)y+xd(y)$, for all $x,y\in R.$ A derivation $d$ is said to be inner if there exists $a\in R$ such that $d(x)=[a,x]$, for all $x\in R$. Over the last several years, a number of authors studied commutativity theorems for prime rings admitting automorphisms or derivations on appropriate subsets of $R$. In [1], M. Brešar introduced it definition of generalized derivation as follows: $F\colon R \to R$ is an additive mapping which is uniquely determined by a derivation d such that $F(xy) =F(x)y+xd(y)$,for all $x,y\in R$. Obviously, every derivation is a generalized derivation. Generalized derivations have been primarily studied on operator algebras. In [2], the concept of multiplicative derivation was defined by Daif, put forward by Martindale in [3], $d\colon R \to R$ is called a multiplicative derivation if $d(xy)=d(x)y+xd(y)$, holds for all $x,y\in R$. These maps are not additive. In this article, we generalize the concept of a multiplicative generalized derivation to a multiplicative generalized $(\lambda,\lambda)$-derivation. A mapping $F \colon R \to R$ (not necessarily additive) is called a multiplicative $(\lambda, \lambda)$-derivation if there exists a map $\lambda \colon R \to R$ such that $f(xy)=f(x)\lambda(y)+\lambda(x)f(y)$, for all $x, y \in R$. A mapping $F \colon R \to R$ (not necessarily additive) is called a multiplicative generalized $(\lambda, \lambda)$-derivation if $F(xy)=F(x)\lambda (y)+\lambda (x)d(y)$, for all $x, y \in R$ where $d$ the multiplicative $(\lambda, \lambda)$-derivation of $R$.
Lemma 1: Let $R$ be a semiprime ring and $U$ a nonzero ideal of $R$. If $U\circ U\in Z$, then $R$ is a commutative ring.
Theorem 2: Let $R$ be a semiprime ring, $U$ a nonzero ideal of $R$, $\lambda$ an automorphism on $R$ and $(F,f),(G,g)$ and $(H,h)$ are a multiplicative generalized $(\lambda, \lambda)$ derivations such that $h(Z)\neq 0.$ If $F(x)G(y) +H(xy)\pm\lambda(yx)\in Z$ or $F(x)G(y)+H(xy)\pm \lambda(xy)\in Z$, for all $x,y\in U$, then $R$ has a commutative ideal.
Theorem 3: Let $R$ be a semiprime ring, $U$ a nonzero ideal of $R$, $\lambda$ an automorphism on $R$ and $(F,f),(G,g)$ and $(H,h)$ are a multiplicative generalized $(\lambda, \lambda)$ derivations such that $h(Z)\neq 0.$ If $F(x)G(y)\pm H(yx)\in Z$ or $F(x)G(y)\pm H(xy)\in Z$,for all $x, y\in U$, then $R$ has a commutative ideal.
References
[1] M. Brešar, On the distance of the compositions of two derivations to the generalized derivations, Glasgow Math. J., 33(1), (1991), 89-93.
[2] M. N. Daif, When is a multiplicative derivation additive, Int. J. Math. Math. Sci., 14(3), (1991), 615-618.
[3] W. S. Martindale III, When are multiplicative maps additive, Proc. Amer. Math. Soc., 21 (1969), 695-698.
[4] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8, (1957), 1093-1100.
Richard N. Ball* (University of Denver; rball@du.edu),
Anthony W. Hager (Wesleyan University; ahager@wesleyan.edu),
Joanne Walters-Weyland (Chapman University; joanne@waylands.com):
Uniformly continuous real functions on locales.
We extend the classical Yosida adjunction to the categories $\mathbf{uF}$ of uniform frames and $\mathbf W$ of divisible archimedean $\ell$-groups with designated weak order unit. The connecting functors are the functor $\mathcal U$, which assigns to the uniform frame $L$ its $\mathbf W$-object $\mathcal UL$ of uniform frame homomorphisms $\mathcal O\mathbb R\to L$, and the functor $\mathcal J$, which assigns to a $\mathbf W$-object its Madden frame of $\mathbf W$-kernels equipped with the uniformity generated by $\widetilde G$, the image of $G$ in its Yosida representation. This immediately yields that the full subcategory comprised of the $\omega_1$-Lindelöf real uniform frames is coreflective in $\mathbf{uF}$, and the full subcategory comprised of the $\mathbf W$-objects of the form $\mathcal UL$ is reflective in $\mathbf W$.
The significant results which emerge from this investigation include the following. Let $L$ be a real uniform frame, i.e., a frame whose uniformity $u_G$ is generated by a subobject $G\subseteq\mathcal{R}L$ such that $\mathrm{coz}\,G$ join generates $L$.
- The completion of $L$ is (isomorphic to) $(L, u_{\widehat{G^*}}) \to L$. This map is the identity on the underlying frame, but not on the uniformities.
- Let $\mathcal SL \to L$ signify the Samuel compactification of $L$, and let $X$ signify the space of points of $\mathcal SL$. Then $G$ has a natural representation as a $\mathcal W$-object in $\mathcal DX$.
- The space of Cauchy filters of $L$ is (homeomorphic to) the intersection of the domains of reality of the members of $G$ when represented as above.
- Identify a $\mathbf W$-object $G$ with its image in its reflector $G\to\mathcal{UJ}G$. Then $G$ is all of $\mathcal{UJ}G$ if and only if $G$ is complete with respect to uniform convergence, and every equiuniform good sequence in $G$ has a join in $G$.
Saugata Basu (Purdue University; sbasu@purdue.edu): Homology of symmetric semi-algebraic sets.
Studying the homology groups of semi-algebraic subsets of $\mathbb{R}^n$ and obtaining upper bounds on the Betti numbers has been a classical topic in real algebraic geometry beginning with the work of Petrovskii and Oleinik, Thom, and Milnor. In this talk I will consider semi-algebraic subsets of $\mathbb{R}^n$ which are defined by symmetric polynomials and are thus stable under the standard action of the symmetric group $\mathfrak{S}_n$ on $\mathbb{R}^n$. The homology groups (with rational coefficients) of such sets thus acquire extra structure as $\mathfrak{S}_n$-modules leading to possible refinements on the classical bounds. I will also mention some algorithmic consequences and connections with a homological stability conjecture.
Papiya Bhattacharjee (Florida Atlantic University; bpapiya@gmail.com): Prime spaces of an algebraic frame with FIP.
Studies have been done on different prime spaces of an algebraic frame $L$ with the FIP, such as $\mathrm{Spec}(L)$, $\mathrm{Min}(L)$, and $\mathrm{Min}(L)^{-1}$. Recently, the space of maximal $d$-elements ($\mathrm{Max}(dL)$) has been studied as a subspace of $\mathrm{Spec}(L)$. In this talk, a comparison of these spaces will be discussed, along with some topological properties.
Ricardo Carrera (Nova Southeastern University, Florida; ricardo@nova.edu): The essentials of essential morphisms and extensions in W and kReg.
W is the category of archimedean ℓ-groups with designated weak unit and ℓ-homomorphisms that preserve the weak unit. kReg is the category of compact regular frames and frame homomorphisms. In this talk, we provide results about essential morphisms and essential extensions in kReg and demonstrate how the similarities between the respective concepts in kReg and W leads to a theory of hull classes in kReg. Moreover, we provide the frame-theoretic counterpart of certain classical results for investigating, generating and working with hull classes in kReg. Lastly, if time permits, we discuss some of the broader applications of this work.
Themba Dube (University of South Africa),
Oghenetega Ighedo* (Chapman University; ighedo@chapman.edu):
On the torsion ideal of a homomorphism.
In their study of what they call “straight rings”, Dobbs and Picavet [1] define the torsion ideal of a ring homomorphism $\varphi:A\to B$, denoted by $T(\varphi)$, by the requirement $$b\in T(\varphi)\Leftrightarrow b\varphi(a)=0\hbox{, for some non-zerodivisor }a ∈ A.$$ Motivated by this definition, we define another torsion-like ideal of a ring homomorphism as follows. Recall that an element $a$ of a ring $A$ is said to be prime to an ideal $I$ of $A$ if, for any $x\in A$, the containment $ax\in I$ implies that $x\in I$. Equivalently, $x\notin I$ implies $ax\notin I$. Clearly, an element of $A$ is a non-zerodivisor if and only if it is prime to the zero ideal of $A$. Thus, $T(\varphi)$ consists of those elements $b$ of $B$ such that $b\varphi(a)\in\{0\}$ for some $a\in A$ that is prime to $\{0\}$.
Let Nil$(A)$ designate the nilradical of $A$. Replacing the zero ideal with the nilradical, and non-zerodivisors with elements prime to the nilradical in the definition of the torsion ideal, we define what we call the nil-torsion ideal of $\varphi$, denoted by $T^•(\varphi)$, by the requirement $$b\in T^•(\varphi)\Leftrightarrow b\varphi(a)\in \hbox{Nil}(B)\hbox{, for some } a\in A\hbox{ which is prime to Nil}(A).$$ In this talk, we compare these ideals and note that while $T(\varphi)$ is always a radical ideal, $T^•(\varphi)$ is not. We do this by means of an example.
These notions are also viewed in the context of algebraic frames with the finite intersection property. That is, in terms of two functors, RId${}:{}$CRing${}\to{}$FIPFrm, which sends a ring to the lattice of its radical ideals, and the second ZId${}:{}$CRing${}\to{}$FIPFrm but with some restrictions (with ring homomorphisms that contract $z$-ideals to $z$-ideals), which sends a ring to the lattice of its $z$-ideals.
References
[1] Dobbs, D.E., Picavet, G.: Straight rings. Comm. Algebra 37 (2009), 757–793.
Anthony Hager (Wesleyan University, Emeritus: ahager@wesleyan.edu),
Brian Wynne* (Lehman College, City University of New York: Brian.Wynne@lehman.cuny.edu):
The Freudenthal spectral theorem and sufficiently many projections in archimedean vector lattices with weak unit.
The Yosida representation for an archimedean vector lattice with distinguished weak unit reveals similarities between the ideas of the title, FST and SMP. I will discuss the following results: (i) For any archimedean $A$ (possibly without weak units), the conclusion of the FST means exactly that for each $0 < e \in A$, the Yosida space for $(e^{dd},e)$, denoted $Y_e$, has a base of clopen sets (this yields a short ``Yosida-based" proof of FST); (ii) SMP implies that each $Y_e$ has a $\pi$-base of clopen sets. While the converse of (ii) holds if $A$ has a strong unit (and in a somewhat more general situation), we present examples showing that otherwise it can fail.
Ramiro H. Lafuente-Rodriguez (University of South Dakota; Ramiro.LafuenteRodri@usd.edu): Over the Specialization Order on the Prime Spectrum of an Algebraic Frame.
Let $X$ be a topological space. The preorder $x\leq_s y$ defined as $x\in\overline{\{y\}}$ is called the specialization order on $X$. We consider the topological space Spec$(L)$, the prime spectrum of a frame $(L,\leq)$ with the Hull-kernel topology, describe the behavior of this order and its main properties when Spec$(L)$ is not $T_1$, particularly when $L$ is the frame of ideals of certain rings. We also prove that there is a Galois connection between $(\mbox{Spec}(L),\leq)$ and $(\mbox{Spec}(L),\leq_s)$. Finally, we briefly describe when Spec$(L)$ is an Alexandrov space.
Jingjing Ma (University of Houston-Clear Lake; Ma@uhcl.edu): Infinite primes for rings.
All rings are associative, with 1, and of characteristic zero. Let $R$ be a ring. A nonempty subset $S$ of $R$ is called a preprime if $S +S\subseteq S$, $SS\subseteq S$, and $−1\notin S$. A maximal preprime is called a prime. By Zorn’s Lemma, every preprime is contained in a prime. A prime $S$ is called infinite if $1\in S$, otherwise it is called finite. An infinite prime $S$ of a ring $R$ is called full if $R=S−S$.
The theory of primes was established by D. K. Harrison around 1966 (Finite and infinite primes for rings and fields, Memoirs AMS, #68, 1966), and further developed by Harrison, Dubois and their students.
Recently, it has been noticed that infinite primes and maximal partial orders have strong connections. For instance, for a number field, which is a subfield of $\mathbb C$ that is finite-dimensional over $\mathbb Q$, the infinite primes and maximal partial orders are identical. This connection enables us to use the theory of infinite primes in the research of partially ordered rings. In this talk, I will present recent developments in this direction.
James Madden (Louisiana State University; jamesjmadden@gmail.com): Posites.
A coverage on a category consists of a collection covering families that are stable under pullback. This provides the minimum structure necessary to define a notion of sheaf. In his book, Stone Spaces, Johnstone specialized this notion to meet-semilattices to create a language in which to describe presentations of frames. A meet-semilattice equipped with a coverage is called a posite; see the article ``posite'' at https://ncatlab.org/. My talk will provide a quick survey of the category of posites, and will describe connections to (a) classical topics on meet-semilattices (such injective hulls), (b) $\kappa$-frames, and (c) term rewriting systems.
Papiya Bhattacharjee (Florida Atlantic University; bpapiya@gmail.com),
Albert Madinya (Florida Atlantic University; amadinya2016@fau.edu),
Warren McGovern$^*$ (Florida Atlantic University, Wilkes Honors College; WMcGove1@fau.edu):
Algebraic Frames.
In his recent dissertation, A. Madinya, worked on finding out which results on algebraic frames with FIP would generalize to general algebraic frames. We shall discuss of his results and show that the restricted Hahn group over a poset can be used to construct examples in the theory.
Ali Mohammad Nezhad (University of North Carolina-Chapel Hill ; alimn@unc.edu): Improved effective Łojasiewicz inequality and applications.
Let $\mathrm R$ be a real closed field. Given a closed and bounded semi-algebraic set $A\subset\mathrm R^n$ and semi-algebraic continuous functions $f,g:A \rightarrow \mathrm R$, such that $f^{-1}(0)\subseteq g^{-1}(0)$, there exist an integer $N>0$ and $c\in\mathrm R$, such that the inequality (Łojasiewicz inequality) $|g(x)|^N\le c\cdot|f(x)|$ holds for all $x\in A$. In this talk, we consider the case when $A$ is defined by a quantifier-free formula with atoms of the form $P=0$, $P>0$, $P\in\mathcal{P}$, for some finite set $\mathcal{P}\subset\mathrm R[X_1,\ldots,X_n]_{\leq d}$ of polynomials of degrees $\le d$, and the graphs of $f,g$ are also defined by quantifier-free formulas with atoms of the form $Q=0$, $Q>0$, $Q\in\mathcal{Q}$, for some finite set $\mathcal{Q}\subset\mathrm R[X_1,\ldots,X_n,Y]_{\leq d}$. We prove that in this case the Łojasiewicz exponent $N$ in is bounded by $(8d)^{2(n+7)}$. Our bound depends on $d$ and $n$, but is independent of the combinatorial parameters, namely the cardinalities of $\mathcal P$ and $\mathcal Q$. We exploit this fact to improve the best known error bounds for polynomial and non-linear semi-definite systems.
Ricardo Palomino Piepenborn (University of Manchester; ricardo.palomino@manchester.ac.uk): Relative quantifier elimination for lattice-ordered modules of continuous semi-algebraic functions on a curve.
In the late 1980s, Shen and Weispfenning proved, via relative quantifier elimination in a suitable 2-sorted language, that under a mild condition on a divisible abelian lattice-ordered group $G$ of functions, the theory of $G$ is completely determined by the theory of its lattice of zero sets. In this talk I will give the relevant context and details of their result, to then explain how the ideas in their proof can be adapted to lattice-ordered modules $M$ of continuous semi-algebraic functions on a curve by enriching their 2-sorted language with a new sort for a real closed valuation ring; as a consequence of the method, decidability of $M$ is obtained whenever the base field is a recursive real closed field.
Tomás Recio$^*$ (Universidad Antonio de Nebrija, Madrid, Spain; trecio@nebrija.es),
M. Pilar Velez (Universidad Antonio de Nebrija, Madrid, Spain; pvelez@nebrija.es):
A Real Mechanical Geometer.
In our talk we introduce, first, the current performance of the programs GeoGebra and GeoGebra Discovery regarding some automatic reasoning tools features, that use computational algebraic geometry methods, mostly in the complex setting. Then we focus on the pending theoretical and algorithmic issues to extend (in the same technological and educational framework) such methods to deal with statements in the real algebraic geometry context.
Philip Scowcroft (Wesleyan University; pscowcroft@wesleyan.edu): Adjoining a strong unit to a hyperarchimedean lattice-ordered group.
This talk will describe circumstances under which a hyperarchimedean lattice-ordered group may be embedded in a hyperarchimedean lattice-ordered group with strong unit.
Marcus Tressl (University of Manchester, United Kingdom; marcus.tressl@manchester.ac.uk): Pseudo complementation on rings of continuous functions.
I'll report on joint work in progress with Guram Bezhanishvili.
Various classes of completely regular spaces $X$ are studied in the literature whose defining clause is a property of pseudo complementation of open (or cozero) sets of $X$. A key tool in this task is to mirror the pseudo complementation properties as properties of the topological space of the (z-)prime spectrum of $C(X)$. For example, $X$ is a P-space iff Spec $C(X)$ is Boolean; $X$ is an F-space iff the inverse space (in the sense of Hochster) of z-Spec $C(X)$ is normal with a patch closed set of closed points; $X$ is cozero complemented iff the space of minimal prime ideals is a compact subspace.
In this talk we will first overview a couple of classes of distributive lattices defined in terms of pseudo complementation (Heyting algebras, Stone lattices, weakly subfit lattices) and spell out Stone duality like theorems for those. Then we'll use these dualities to characterize completely regular spaces whose (z-)prime spectra occur in these dualities. A particular emphasis will be on pseudo complementation of compact open subsets of the spectrum of $C(\beta \mathbb{N})$.
Last updated May 7, 2024.