Last Updated: March 19, 2007

Conference on Ordered Rings, Baton Rouge 2007 ("Ord07")
Honoring Melvin Henriksen on his 80th Birthday

Dates: April 25 (Wednesday) - April 28 (Saturday), 2007
Location: Louisiana State University, Baton Rouge, Louisiana, USA.

### Abstracts of talks

AUTHOR: Rick Ball, Univ. Denver
TITLE: P-frames
ABSTRACT: A space is a P-space iff every zero set is open. A frame is a P-frame iff every cozero element is complemented. (All spaces and frames are assumed completely regular.) P-frames come up so often, and in crucial ways in topology (both pointed and pointless). We give examples of several theorems on P-spaces that have frame versions with greater scope and simpler proof. In some cases, the results go well beyond what had been known in the spatial situation. We conclude the talk by outlining a proof of the existence and uniqueness of the P-space reflection of a frame and describing generalizatins to kappa-frames.

AUTHOR: Papiya Bhattacharjee from Bowling Green State University
TITLE: Minimal prime elements of an algebraic frame
ABSTRACT: I will talk about my work on the space of minimal prime elements of an algebraic frame with a unit (a compact dense element). I will discuss results concerning the hull-kernel topology and the inverse topology. The motivation for my work comes from the space of minimal prime ideals on commutative rings with identity as well as the minimal prime subgroups of an l-group.

AUTHOR: Karim Boulabiar, Univ. 7 Novembre-Carthage, Tunisia
TITLE: The Arens multiplication in a unital f-ring (20 minutes)
ABSTRACT: The Dedekind complete abelian l-group of all order bounded group homomorphisms from an abelian l-group G to the l-group R of all real numbers is denoted by G~. Let R be a unital f-ring and let R~~ indicate the Dedekind complete abelian l-group of all group homomorphisms from R~ to R. In this talk, we show that R~~ is a commutative unital f-ring with respect to the so-called Arens multiplication. This work is part of joint research with Jamel Jaber. (PDF)

AUTHOR: Gerard Buskes, Univ. Mississippi
TITLE: A detour via vector lattices to the Loomis-Sikorski Theorem in Boolean algebras
ABSTRACT: We will give a constructive proof of the Loomis-Sikorski Theorem using the theory of vector lattices.

AUTHOR: Charles N. Delzell, LSU
TITLE: The two-variable Pierce-Birkhoff conjecture for continuous, piecewise "generalized" polynomial functions on the positive orthant. (20 minutes)
ABSTRACT: The Pierce-Birkhoff conjecture states that every continuous, piecewise-polynomial function h: R^n --> R is the pointwise sup of infs of finitely many polynomial functions. Mahe proved it for n = 2. We consider "generalized" polynomials with arbitrary real exponents (and real coefficients), which in general are defined only on the positive orthant P^n of R^n. We prove that if h: P^2 --> R is a continuous and piecewise generalized polynomial function, then h is the pointwise sup of infs of finitely many generalized polynomial functions.

AUTHORS: Ralph DeMarr (presenter), Univ. of New Mexico and Donald Beken, Univ. of North Carolina at Pembroke
TITLE: Strange Inequalities in a Partially Ordered Linear Algebra (20 minutes)
ABSTRACT: We consider a partially ordered linear algebra (POLA) with identity element I. The term JEMS refers to the four types of elements in POLA: imaginary J, idempotent E, nilpotent M and involution S. We consider a POLA in which "strange" inequalities such as 2I<=E+M or I<=JS are valid. We study the properties of this POLA.

AUTHOR: Danielle Gondard-Cozette, Univ. Paris 6, France
TITLE: On Real Holormorphy Ring of Rings (joint work with Muray Marshall) (20-40 minutes)
ABSTRACT: After recalling known facts on the real holomorphy ring of a field, we present possible definitions for real holomorphy rings of rings. This may be used in obtaining a kind of complete real spectrum.

AUTHOR: Anthony W. Hager, Wesleyan University
TITLE: Uniform convergence in archimedean l-groups and f-rings (40 minutes)
ABSTRACT: In archimedean l-groups with unit and f-rings with identity, we consider uniform convergence (of sequences) of three types: Ordinary (classical), Relative (E.H.Moore, 1910; Veksler, 1968), and Indicated (Ball-Hager, 2006). Intrinsic and extrinsic constructions of the completions are discussed. These completions uG, ruG, iuG are monoreflections, are progressively larger, and are further compared. It is to be noted that iuG is the "ccc-reflection"; G = iuG means G = C(L) for a locale L. Some density theorems ( "Stone-Weierstrasse type") are presented.

Author: Melvin Henriksen, Harvey Mudd College (20 minutes)
Title: Open problems on when the ring C(X) contains "many" prime ideals P such that C(X)/P is a valuation domain
Abstract: Some unsolved problems concerning spaces X and rings C(X) such that every maximal ideal contains a minimal prime ideal P such that C(X)/P is a valuation domain.

AUTHORS: Wolf Iberkleid (presenter) and Warren Wm. McGovern, Bowling Green State Univ.
TITLE: Classes of clean rings (20 minutes)
ABSTRACT: Let $A$ be a commutative ring with identity. A ring $A$ is said to be $I$-$clean$, where $I$ is an ideal of $A$, if for every element $a$ in the ring there is an element $e = e^2$ such that $a - e$ is a unit and $ae$ belongs to $I$. A multiplicative filter of ideals of $A$ is $compact$ if every ideal in the filter contains a finitely generated ideal, also in the filter. We characterize $I$-clean rings for the ideals $0$, $n(A)$, $J(A)$, and $A$, in terms of the frame of compact multiplicative filters of ideals of $A$, the hull-kernel topology of $A$, and in terms of more classical ring properties.

AUTHORS: Don Johnson (joint work with A. W. Hager)
TITLE: Adjoining an identity element to a reduced archimedean f-ring II: Algebras
ABSTRACT: In "Part I" (presented at Ord05 (Oxford, MS)) we discussed, for reduced archimedean f-rings, the canonical extension of such a ring, A, to one with identity, uA, and the class U of u-extendable maps (i.e., homomorphisms which lift over the functions u to identity preserving homomorphisms). We showed that U is a category and u becomes a functor from U which is a monoreflection; the maps in U were characterized. This paper addresses the interaction between our functor u; and v, the vector lattice monoreflection in archimedean l-groups (due to Conrad and Bleier). In short, v restricts to a monoreflection of reduced archimedean f-rings into reduced archimedean f-algebras if and only if v preserves membership in U, and vu is a monoreflection into reduced archimedean f-algebras with identity. (Longer version in PDF.)

Author: Manfred Knebusch, Univ. Regensburg, Germany
Title: Positivity and convexity in rings of fractions
Abstract: Given a commutative ring A equipped with a preordering A^+ (in a very general sense), we look for a fractional ring extension (= \ring of quotients" in the sense of Lambek) as big as possible such that A^+ extends to a preordering R^+ of R (i.e. with the intersection of A and R^+ equal to A^+) in a natural way. We then ask for subextensions A within B within R such that A is convex in B with respect to B+ := B meet R+. Perhaps surprisingly this study leads to hard problems. (PDF)

AUTHOR: Suzanne Larson, Loyola Marymount University
TITLE: Images and Open Subspaces of SV Spaces (40 minutes)
ABSTRACT: A topological space is finitely an F-space if its Stone-Cech compactification is a union of finitely many closed F-spaces and a space is SV if C(X) has the property that C(X)/P is a valuation domain for each prime ring ideal P of C(X). We will investigate whether images of these spaces under open continuous functions and whether cozero and open subspaces of these spaces are themselves finitely an F-space or are SV. This will lead to a condition under which the Stone extension of a continuous open function is itself open.

AUTHOR: Francois Lucas, Univ. Angers, France
TITLE: Spectra of ordered groups and rings
ABSTRACT:This talk is based on joint work with M.Dickmann and D.Gluschankof studying the structure of spectra of a lattice-ordered groups (and real spectra of commutative rings). The principal focus is on the construction of a family of rings, proving that each jump dense complete root system is isomorphic to the real spectrum of a commutative unitary ring.

AUTHOR: Jorge Martinez, Univ. of Florida
TITLE: Archimedean frames revisited (40 minutes)
ABSTRACT: An algebraic frame A is said to be archimedean if for each compact c in A, the infimum of all the elements m maximal beneath c is 0. We propose a concept -- called "join fitness" -- which is free of the references to algebraic frames and to "Choice", yet equivalent to "archimedean" in the presence of both.

AUTHOR: Jim McEnerney, Lawrence Livermore National Laboratory
TITLE: Applications of the semi-linear spectrum over an ordered field (20 minutes)
ABSTRACT: This talk presents results on three topics: an infinitesimal criteria to determine when an ordered field is real closed, a generalized optimization theory over an ordered field, and the striking resemblance to the real numbers exhibited by the closed bounded semi-linear spectrum.

AUTHOR: Dr. Michael Olubukola OLUWATUKESI, Univ. Of Ado-Ekiti, Nigeria
TITLE: Algebraic Properties of the Uniform Closure of Spaces of Continuous Function (20 minutes)
ABTRACT: For a completely regular space X, C(X) denotes the algebra of all real-valued and continuous functions over X. This paper deals with the problem of knowing when the uniform closure of certain subsets of C(X) has certain algebraic properties. In this context we give an internal condition, "property A," to characterize the linear subspaces whose uniform closure is an inverse-closed subring of C(X).

AUTHOR: Homeira Pajoohesh, Georgia Southern University
TITLE: (Positive) derivations on (l-) rings of matrices (20 minutes)
ABTRACT: We consider a commutative l-ring R with unit. We show that the only positive derivation on the l-ring M_n(R) is zero if and only if the only positive derivation on R is zero. Also, if the only positive derivation on R is zero, we prove that for certain semiprime sub l-rings S of M_n(R) the only positive derivation on S is zero. Then, we study positive derivations on sub l-rings of
M_n(R) where R is an Archimedean l-ring.

AUTHOR: Alex Prestel, Univ. Konstanz, Germany
TITLE: Positive Elimination in Valued Fields
ABTRACT: Let K be an algebraically closed field with a valuation ring O or a real closed field with a convex valuation ring O. We show that the projection of a basic subset of K^n ×O^m to K^n is again basic. A basic set is given by a finite union of finite intersections of sets defined by = (or, in the case of a real closed field, the order relation) and divisibility in the valuation ring O.

AUTHOR: Robert Redfield, Hamilton College
TITLE: Super valuation groups
ABTRACT: A left valuation domain is a ring without divisors of zero such that, for any nonzero elements x and y, either x divides y or y divides x on the left. An SV-ring (super valuation ring) is a ring R such that for all prime ideals P, the quotients R/P are left valuation domains. Henriksen et al. have investigated f-rings that are SV-rings. Since, in this situation, the order and the addition seem to me to play a more prominent role than the multiplication, I decided to investigate the group-theoretic analogue of SV-rings. I will report on the results of this investigation.

AUTHORS: D. Schaub, Univ. Angers, France and M. Spivakovsky, Univ. Toulouse, France
TITLE: The Pierce--Birkhoff conjecture and approximate roots of a valuation: Parts I (40 minutes) and II (40 minutes)
ABSTRACT: A (continuous) real-valued function f: R^n --> R is said to be piecewise polynomial if there exist a finte decomposition R^n into closed semi-algebraic sets on each of which f is polynomial. The Pierce-Bikhoff conjecture says that every piecewise polynomial on R^n can be expressed as a sup of infs of finitely many polynomials. In this and the companion talk by Spivakovsky, we will discuss some recent results on the Pierce-Birkhoff Conjecture (PBC) obtained in collaboration with F. Lucas and J. Madden. D. Schaub's lecture will be devoted to introducing the theory of approximate roots. Fix a regular local ring A with regular system of parameters u and a valuation v, centered at A. To this data we canonically associate a well ordered set Q of elements of A, described, in principle, by explicit formulae, such that the valuation v is completely determined by the Q and and the values of the elements of Q. This means that every v-ideal of A is generated by products of elements of Q. In particular, the images of the elements of Q in the graded algebra associated to v, genereate that algebra. M. Spivakovsky will recall a reformulation of the Pierce-Birkhoff conjecture in terms of the real spectrum X of the ring of real polynomials. We state a new conjecture, called the connectedness conjecture, and prove that it implies the Pierce-Birkhoff conjecture. The connectedness conjecture asserts that given any two points a, b in X, there is a connected subset of X having certain properties. We will describe this set iin terms of approximate roots and explain our approach to proving connectedness.

AUTHOR: Niels Schwartz, Univ. Passau
TITLE: Convex extensions of partially ordered polynomial rings (40 minutes)
ABSTRACT: The study of convex subrings of partially ordered rings (= po-rings) is an important and difficult problem in real algebra (with many applications in real geometry). One approach is to consider a given po-ring and ask whether, and how, it can be a convex subring of some larger po-ring. The larger po-ring is then called a convex extension. It is always possible to construct a proper convex extension, using polynomial rings. The situation is different if one asks for convex extensions that are rings of quotients of the given po-ring. Substantial answers are available for po-rings with bounded inversion. Very little is known if bounded inversion does not hold, e.g., for partially ordered algebraic function rings over real closed fields. A first step in this direction is to study polynomial rings and their convex extensions. First results and some observations will be presented in the lecture.

AUTHOR: AshishKumar Srivastava, Ohio University, Athens
TITLE: Sums of units in right self-injective rings (20 minutes)
ABTRACT: A classical result of Zelinski states that every linear transformation on a vector space V, except when V is one-dimensional over Z_2, is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if and only if no factor ring of R is isomorphic to Z_2. We also give a complete charaterization of unit sum numbers of right self-injective rings and answer some questions of Henriksen on rings generated by their units.

AUTHOR: Stuart Steinberg, University of Toledo, Ohio
TITLE: Henriksen-Isbell-Weinberg's solution to Hilbert's 17th Problem
ABTRACT: Artin's solution to Hilbert's 17th problem along with McKenna's converse states: Every positive semidefinite rational function over the totally orderd field F is a sum of squares if and only if the field has a unique total order and is dense in its real closure. A Henriksen and Isbell, Weinberg proof can be given using the identification of the formally real f-rings given by Henriksen and Isbell as presented by Weinberg.

AUTHOR: Marcus Tressl, Univ. Regensburg, Germany
TITLE: Super real closed rings (40 minutes)
ABTRACT: A super real closed ring is a commutative unital ring together with functions $F_A:A^n\to A$ for all $n\in {\mathbb N}$ and every continuous function $F:{\mathbb R}^n\to {\mathbb R}$, such that the composition rules of the $F$'s are valid for the $F_A$'s, i.e. $(F\circ (...,G,...))_A=F_A\circ (...,G_A,...)$. For example, every ring $C(X)$ is a super real closed ring, where $F_{C(X)}$ is composition with $F$; also, super real fields in the sense of Dales-Woodin at prime $z$-ideals are naturally equipped with a super real closed ring structure. In contrast to rings of continuous functions, the category of super real closed rings is very flexible w.r.t. (real) algebraic operations: e.g. taking residues, various convexity constructions and all kinds of quotient constructions. The sheaf of abstract semi-algebraic functions defined on a space of so called super radical primes is a sheaf of super real closed rings. In the talk I will overview the properties of the category of super real closed rings and address the problem of amalgamating super real closed fields.

AUTHOR: Dejan Veluscek, University of Ljubljana
TITLE: Central extensions of *-ordered skew fields (joint work with Igor Klep) (20 minutes)
ABTRACT: A *-ordering of a skew field D induces an ordering of the field Kof its central symmetric elements. Let F be an ordered field extension of K. We will show that the central extension of D by F exists and admits a *-ordering extending the given *-ordering of D and ordering of F. Moreover, we show that every *-ordered skew field can be extended to a *-ordered skew field containing the real numbers in its center.

AUTHOR: Joanne Walters-Wayland
TITLE: Framing "Mel"
ABTRACT: As a tribute to Mel Henriksen and, in thanks for, his ability to inspire and motivate, I would like to give a few snapshots of work he has done over the years, taken through a localic lens. Starting with work presented to the AMS in 1954 on finitely generated ideals, the "raison d'etre" of F-frames, P-frames etc, including his work on prime ideals (1965), quasi-F-covers (1987) and pretty bases (1991), and concluding with some results about cozero complemented spaces (2003).

AUTHORS: P. Wojciechowski, Univ. Texas at El Paso
TITLE: Application of lattice ordered rings in enumeration of multiplicative bases of matrices
ABTRACT: In a finite-dimensional algebra over a field F, a basis B is called a multiplicative basis provided that B together with 0 is a semigroup under multiplication. We will describe all multiplicative bases of F_n, the algebra of n by n matrices over a subfield of the real numbers. Every such basis is associated with a nonsingular zero-one matrix via a lattice order on F_n. This correspondence yields an enumeration method for nonequivalent multiplicative bases of F_n. (PDF)

AUTHORS: Eric Zenk, Univ. Denver (presenter) and Jorge Martinez, Univ. of Florida
TITLE: $z$-Dimension of $C(X)$ revisited (40 minutes)
ABTRACT: An ideal $I$ in $C(X)$ is a $z$-ideal if whenever $f\in I$ and $z(f)\subseteq z(g)$ it follows that $g\in I$. The $z$-dimension of $C(X)$ is the least upper bound of lengths of chains of prime $z$-ideals. The talk will discuss the following result due to Martinez and Zenk. Theorem: Suppose $X$ is regular and Lindelof. The $z$-dimension $C(X)$ is zero if and only if $X$ is a P-space. For $n\geq 0$, $C(X)$ has $z$-dimension at most $n+1$ if and only if, for each cozero set $U$ in $X$, the $z$-dimension of $C(\overline{U}\setminus U)$ is at most $n$. The proof of this result also holds for regular Lindelof locales, which extends its validity to CCC archimedean $f$-rings, by a result of Madden. The talk may also discuss the topological problems which must be solved to obtain a direct (non-inductive) characterization of when $C(X)$ has finite $z$-dimension.