Abstracts of talks at the Conference on Order, Algebra, Logic and Real Algebraic Geometry (OAL-RAG 2019)
Richard N. Ball (University of Denver; rball@du.edu): Structural aspects of truncated Archimedean vector lattices: good sequences, simple elements.
Certain features of archimedean vector lattices can be conveniently articulated by means of the truncation operation. This talk will take up several such features, together with their ramifications. The talk is intended to range widely; we mention only two sample topics in this abstract.
Recall that a truncation on an archimedean vector lattice $A$ is an operation $a \mapsto\overline{a}$ on the positive cone $A^+$ satisfying three simple axioms; we refer to such vector lattices colloquially as truncs. Without loss of generality we may assume that an arbitrary trunc $A$ is a trunc in $\mathcal{D}_0 X$ for a compact Hausdorff pointed space $(X, *)$ (in which case we denote its elements $\tilde{a}$, $\tilde{b}$, etc.), or a subtrunc of the trunc $\mathcal{R}_0 L$ of pointed frame maps $\mathcal{O}_* \mathbb{R} \to L$ (in which case we denote its elements $a$, $b$, etc.).
The $n^\text{th}$ truncation of an element $a \in A^+$ is $n \overline{a/n}$, designated $a \wedge n$, and the sequence $\{a \wedge n\}$ is called the truncation sequence of $a$. Note that $a \wedge n \nearrow a$ for all $a \in A^+$.
Proposition. A sequence $\{a_n\} \subseteq A^+ \subseteq \mathcal{R}_0^+ L$ is the truncation sequence of some element $a_0 \in \mathcal{R}_0^+ L$ iff $a_n = a_{n + 1} \wedge n$ for all $n$, and $\bigvee_n a_n(-\infty, n) = \top$.
Truncation sequences are closely connected to the good sequences of Mundici, who used them to show the equivalence of unital abelian $\ell$-groups with MV-algebras. A good sequence is a sequence $\{a_n\} \subseteq A^+$ such that $a_n = \overline{a_n + a_{n + 1}}$ for all $n$, and such that $a_n \searrow 0$.
Proposition. If $\{a_n\}$ is a truncation sequence in $A^+ \subseteq \mathcal{R}_0 L$ then $\{a_{n + 1} - a_n\}$ is a good sequence. And if $\{d_n\}$ is a good sequence then $\{\sum_{1 \leq i \leq n}d_i\}$ is a truncation sequence.
A trunc $A$ is said to be replete if for every truncation sequence $\{a_n\} \subseteq A^+$ there exists $a_0 \in A^+$ such that $a_n \nearrow a_0$. By the preceding proposition, this is equivalent to the condition that every good sequence in $A^+$ has a pointwise supremum in $A$.
Theorem. A trunc $A$ is of the form $\mathcal{R}_0 L$ for some pointed frame $L$ iff it is uniformly complete and replete.
The truncation operation makes it easy to pick out those elements of a trunc which behave like step functions. They are the elements which satisfy the following lemma.
Lemma. The following are equivalent for an element $u \geq 0$ in a trunc $A$.
(1) $u$ is a unital component of $A$, i.e., $u \wedge v$ is a component of $v$ for any $v \in \overline{A}$. $\square$
(2) $u = \overline{2u}$.
(3) $\tilde{u}$ is the characteristic function $\chi_U$ of a clopen subset $U \subseteq X$ omitting the designated point $*$, i.e., $\tilde{u}(x) = 1$ for $x \in U$ and $\tilde{u}(x) = 0$ for $x \notin U$.
The set of unital components of any trunc $A$, designated $\mathcal{UC}(A)$, forms a nonempty generalized Boolean algebra, i.e., a lattice with least element $0$ which admits relative complementation: \[ \forall a,b\ \exists c\ (c \vee b = a \vee b \text{ and }c \wedge b = \bot). \] An element $a$ in a trunc $A$ is called simple if it is a linear combination of unital components. The trunc is called simple if all its elements are simple. In any trunc $A$, the set of simple elements forms a subtrunc $\sigma A$, and this is the biggest simple subtrunc of $A$.
Proposition. These three categories are equivalent to one another: the category $\mathbf{sT}$ of simple truncs, the category $\mathbf{gBa}$ of generalized Boolean algebras, and the category $\mathbf{BSp}_*$ of pointed Boolean spaces.
An element $a$ of a trunc $A$ is called bounded if it satisfies $\left|a\right| \leq n\overline{|a|}$ for some $n$. The trunc is called bounded if all its elements are bounded. In any trunc $A$, the set of bounded elements forms a subtrunc $A^*$, and this is the biggest bounded subtrunc of $A$.
Lemma. The following are equivalent for an element $a \geq 0$ in a trunc $A$.
(1) There exists $\varepsilon > 0$ such that $\tilde{a} > \varepsilon$ whenever $\tilde{a} > 0$.
(2) There exists $\varepsilon > 0$ such that $a(0, \varepsilon) = \bot$.
(3) $\overline{na} \in \mathcal{UC}(A)$ for some $n$.
We say that $a$ is bounded away from $0$ when these conditions hold. We say that the trunc $A$ is bounded away from $0$ when each $a \in A^+$ is bounded away from $0$.
Theorem. The following are equivalent for a trunc $A$.
(1) $A$ is simple.
(2) $A$ is (isomorphic to) the family $\mathcal{LC}X$ of locally constant functions which vanish at the designated point of a compact Hausdorff pointed space $X$.
(3) $A$ is bounded and bounded away from $0$.
(4) $A$ is hyperarchimedean and has enough unital components, i.e., for all $a \in A^+$ there exists $u \in \mathcal{UC}(A)$ such that $\overline{a} \leq u$.
(5) $A$ is hyperarchimedean and bounded away from $\infty$, i.e., each $a \in A^+$ has the feature that $\tilde{a}$ vanishes on a neighborhood of the designated point $*$.
Papiya Bhattacharjee (Florida Atlantic University; bpapiya@gmail.com): $\DeclareMathOperator\Max{Max}\Max(dL)$ vs. $\DeclareMathOperator\Min{Min}\Min(L)$, for $M$-Frames $L$.
The space of maximal $d$-ideals of $C(X)$ is homeomorphic to the $Z^\sharp$-ultrafilters, and this space is the minimal quasi-$F$ cover of a compact Tychonoff space $X$. In this talk the notions of maximal $d$-ideals and $Z^\sharp$-ultrafilters will be generalized for algebraic frames with the FIP. The speaker will also describe the relation between $\DeclareMathOperator\Max{Max}\Max(dL)$ and the minimal primes element spaces of an algebraic frame $L$, $\DeclareMathOperator\Min{Min}\Min(L)$ and $\DeclareMathOperator\Min{Min}\Min(L)^{-1}$.
Fred Dashiell (UCLA, and CECAT at Chapman University; dashiell@math.ucla.edu): Relative Annihilators and the Normal Completion of a Subfit Lattice.
Let $L$ be a bounded distributive lattice, with $a,b \in L$. The relative annihilator $\langle a,b\rangle$ is the ideal $\{x: x\wedge a\le b\}$. Relative annihilators have been used to characterize the normal completion (Dedekind-MacNeille completion) of a lattice. $L$ is subfit if $a < b \Rightarrow \exists\: c$ with $a\le c$ and $c\vee b = 1$. Relative annihilators are used to prove that the normal completion of a subfit lattice is a frame. This generalizes the known result of Moshier that if $L$ is a frame, then the normal completion of Coz $L$ is a frame.
Anthony Hager (Wesleyan University; ahager@wesleyan.edu) and Ricardo Carrera: A classification of hull operators in archimedean $\ell$-groups with weak unit.
$W$ is the category of the title, $B$ is the bounded monocoreflection in $W$, a hull operator (ho) is a reflection in the category with all $W$-objects, only essential embeddings as morphisms, ho$W$ is the complete lattice of all ho's (bottom the identity, top Conrad's essential completion), and is a proper class. $B$, and any ho $h$ are functions $W\to W$, and so are compositions $Bh$, $hB$, etc. "Word" is the set of all such (finite) compositions, of which there are 6. Word has the natural order of functions (and composition as multiplication), making Word lattice isomorphic to $F(2)$, the free distributive lattice on 2 generators (and making $F(2)$ into a $\ell\ell$-semigroup). An equation is an expression $E:u=v$, where $u,v$ are words; $h$ satisfies $E$ means $u(h)=v(h)$. Non-equivalent satisfiable equations number at least 5 and no more than 9 (4 being mysteries). We partition ho$W$ into 6 pieces, by "satisfaction of equations," 4 of which are known to be non-empty (2 being mysteries). Any ho $h$ defines an order-preserving quotient $Q(h)$ of Word, by the equations which $h$ satisfies. If $h$ lies in one of the 4 "known" pieces, then $Q(h)$ is a chain (lengths 2,3,3,4). Chains of length 1,5,6 do not occur. We do not know (among other things) if every $Q(h)$ is one of those 4, or if every $Q(h)$ is a chain (= one mysterious piece is empty), or if $Q(h)=F(2)$ occurs (= $h$ satisfies no equation).
Purbita Jana (Institute of Mathematical Sciences (IMSc.), Chennai, India; purbita.jana@gmail.com), Antonio Di Nola, and Revaz Grigolia: Heyting Algebra and Gödel Algebra vs. Various Topological Systems and Esakia Space: a Category Theoretic Study.
This talk will suggest a new approach to the representation of a Heyting algebra as an $I$-topological system. An $I$-topological system will be introduced following the notion of topological system introduced by S. Vickers, which is a triple $(X,\models,A)$ consisting of a non-empty set $X$, a frame $A$, and a relation $\models$ between $X$ and $A$ satisfying the logic of finite observations or geometric logic. It is well known that frame is the Lindenbaum algebra of geometric logic, whereas the Lindenbaum algebra of intuitionistic logic is Heyting algebra. Hence, to define an $I$-topological system, intuitionistic logic plays a crucial role. Moreover, we will focus on the categorical relationship between the $I$-topological system and Esakia space (and its particular case, Gödel space).
Ramiro H. Lafuente-Rodriguez (University of South Dakota; Ramiro.LafuenteRodri@usd.edu) and Warren W. McGovern: The $d$-radical of an $\ell$-group.
Let $G$ be an $\ell$-group containing weak order units. A convex $\ell$-subgroup $K$ of $G$ is a $d$-subgroup if whenever $g\in K$, then $g^{\bot\bot}\subseteq K$. In a previous work we studied the classification of when the inverse topology on the space of minimal prime subgroups of $G$ and when the space of maximal $d$-subgroups have clopen $\pi$-bases. This led naturally to the notion of the $d$-radical of an $\ell$-group, i.e., the intersection of all maximal $d$-subgroups of $G$. In this talk we describe the $d$-radical, provide some of its properties and discuss its connections with fusible $\ell$-groups. An element of $0<g\in G$ is left (right) fusible if it can be expressed as the sum of a weak order unit and a non-weak order unit (reverse the order of summands for ‘right’). An $\ell$-group is left (right) fusible of all its elements are left (right) fusible. An $\ell$-group is fusible is it is both left and right fusible.
Jingjing Ma (University of Houston-Clear Lake; Ma@uhcl.edu): Extending partial orders on commutative rings with identity.
We consider commutative rings with identity in which certain partial orders can be extended to lattice orders or total orders.
Fuchs calls a ring $R$ an $O^*$-ring if each partial order on $R$ can be extended to a total order on $R$. One open question in his book published in 1963 was to establish ring-theoretical properties for $O^*$-rings. In 1997, Steinberg answered Fuchs' question and provided a characterization for $O^*$-rings.
In this talk, I will summarize research activities in this area and present new results for commutative rings with identity element as well.
Warren Wm. McGovern (Florida Atlantic University; WMcGove1@fau.edu): The Alexandroff Duplicate of a Yosida space.
The talk will be centered around how to algebraically view the Alexandroff Duplicate of a compact Hausdorff space. For a space $X$ (Tychonoff) the Alexandroff Duplicate is the space $A(X)$ created by taking two copies of the space $X$: $$A(X)=X\cup X',$$ where for each $x\in X$ we let $x'\in X'$ denote its duplicate. Then we take the topology generated by declaring each point in $X'$ to be isolated and a basic open set around $x\in X$ to be of the form $U\cup U'\smallsetminus\{x'\}$ for some open neighborhood $U$ of $x$ in $X$. It is known that $X$ is compact Hausdorff if and only if $A(X)$ is compact Hausdorff. This construction has a history of producing very important counter-examples.
Given a W-object $(G,u)$ with Yosida space $Y$, there is an embedding $$(G,u)\longrightarrow (A(G),u)$$ such that the Yosida space of $A(G)$ is the Alexandroff duplicate of $YG$. If sufficient time is left then a discussion about a generalization of this construction will proceed.
Philip Scowcroft (Wesleyan University; pscowcroft@wesleyan.edu): Essential adjunction of a strong unit to an Archimedean lattice-ordered group.
Let $\mathcal{A}$ be the class of Archimedean $\ell$-groups. A. W. Hager and I have recently studied the following question: if $G\in\mathcal{A}$, is there $H\in\mathcal{A}$, generated as an $\ell$-group over $G$ by a strong unit of $H$, such that $H$ is an essential extension of $G$? Our results include the following: there are $G\in\mathcal{A}$, having no such $H$, which nevertheless have a $K\in\mathcal{A}$ generated over $G$ by a strong unit of $K$; even when $G$ does admit such an $H$, there may be many non-isomorphic examples.
Friedrich Wehrung (Université de Caen, Campus 2; friedrich.wehrung01@unicaen.fr): Spectra of Abelian $\ell$-groups are anti-elementary.
We consider the functor $I$ that to every Abelian $\ell$-group $G$ associates the (distributive) lattice $I(G)$ of all its principal $\ell$-ideals. We construct a non-commutative diagram of Abelian $\ell$-groups whose image under $I$ is not $I(D)$ for any commutative diagram $D$. As a consequence, the class of all $I(G)$ (which are also the dual lattices of spectra of Abelian $\ell$-groups) is not closed under $L_{\infty,\lambda}$-elementary equivalence for any infinite cardinal $\lambda$.
Brian Wynne (Lehman College, City University of New York; Brian.Wynne@lehman.cuny.edu): More on e.c. $\ell$-groups via Fraïssé's construction.
At OAL 2018 I presented a new example of an existentially closed (e.c.) Abelian lattice-ordered group ($\ell$-group). This time I will review the construction, answer several of the questions left open at the end of my previous talk, and discuss possible extensions of these results.
Last updated April 16, 2019.