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.**

**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, *u*A, 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.