# Fred Richman ## Documents

• Walker groups, pdf file. A reformulation of Walker's theorem on the cancellation of Z says that any two homomorphisms from an abelian group B onto Z have isomorphic kernels. It does not have a constructive proof, even for B a subgroup of a free group of rank 3. In this paper we give a constructive proof of Walker's theorem for B a direct sum, over any discrete index set, of finite-rank torsion-free groups whose elements have weakly computable heights. (11 pages, 7 August 2013)

• When are the rings R(X) and R<X> clean?, (with Warren Wm. McGovern) pdf file. The answers to the title questions are: when R is clean and when R is zero dimensional, respectively. We give constructive proofs of these two theorems and a constructive proof of the known result that the two rings in question are equal exactly when R is zero dimensional. (5 pages, 27 July 2013)

• A theorem of Gilmer and the canonical universal splitting ring, pdf file. We give a constructive proof of Gilmer's theorem that if every nonzero polynomial over a field k has a root in some fixed extension field E, then each polynomial in k[X] splits in E[X]. Using a slight generalization of this theorem, we construct, in a functorial way, a commutative, discrete, von Neumann regular k-algebra A so that each polynomial in k[X] splits in A[X]. (7 pages, 4 August 2012)

• Walker's cancellation theorem, (with Robert S. Lubarsky) pdf file. Let B and C be abelian groups and Z the additive group of integers. Walker's cancellation theorem says that if the direct sum of Z with B is isomorphic to the direct sum of Z with C, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof even if B and C are subgroups of the free abelian group on two generators. (9 pages, 5 August 2012)

• A constructive theory of minimal zero-dimensional extensions, pdf file, beamer file. In this paper we prove a constructive version of Chiorescu's theorem which gives a complete set of invariants for minimal zero-dimensional extensions of a commutative ring with dimension at most one, primary zero-ideal, and Noetherian spectrum. This is done, in its full generality, without reference to prime ideals and without the hypothesis of Noetherian spectrum. (20 pages, 2 December 2010)

• Algebraic functions, calculus style, pdf file, beamer file. A look at algebraic functions according to their definition in calculus texts. (16 pages, 22 July 2010)

• Zero sets of univariate polynomials, (with Robert S. Lubarsky) pdf file. Let L be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of L. In this paper we introduce a notion of distance from a point to a subset, more general than the usual one, that allows us to measure distances to subsets like L. To verify the correctness of this notion, we show that the zero set of a polynomial cannot be empty---a weak fundamental theorem of algebra. We also show that the zero sets of two polynomials are a positive distance from each other if and only if the polynomials are comaximal. Finally, the zero set of a polynomial is used to construct a separable Riesz space, in which every element is normable, that has no Riesz homomorphism into the real numbers. (15 pages, 22 April 2009)

• Intuitionistic notions of boundedness in N, pdf file. We consider notions of boundedness of subsets of the natural numbers N that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of N to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most of the results do not require countable choice. (10 pages, 14 January 2008)

• Real numbers and other completions, pdf file. A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete archimedean Heyting field, a terminal object in the category of archimedean Heyting fields. (19 pages, 11 March 2007)

• Transient limits, (with Katarzyina Winkowska-Nowak) pdf file. Let A be a Markov matrix depending on a small parameter s, and Cn the average of the first n powers A. The stationary distributions of A are the rows of S, the limit of Cn as n goes to infinity. The limiting stationary distributions are the rows of the limit of S as s goes to zero. We investigate transient limits of the sequence Cn. These idempotent Markov matrices come up implicitly in an algorithm to compute limiting stationary distributions. They represent the intermediate-term behavior of the Markov chain at different time scales. (16 pages, 10 March 2006)

• Subrings of zero-dimensional rings, (with Jim Brewer) pdf file. When Sarah Glaz, Bill Heinzer and the junior author of this article approached Robert Gilmer with the idea of editing a book dedicated to his work, we asked him to give us a list of his work and to comment on it to the extent he felt comfortable. As usual, he was extremely thorough in his response. When the authors of this article began to consider what topic we wanted to write about, we were impressed by Robert's comment that he was particularly pleased with his series of papers with Bill on the embeddability of a ring in a zero-dimensional ring. So we decided to write about that. (16 pages, 12 December 2005)

• Near convexity, metric convexity, and convexity, Scientific WorkPlace file, pdf file. It is shown that a subset of a uniformly convex normed space is nearly convex if and only if its closure is convex. Also, a normed space satisfying a mild completeness property is strictly convex if and only if every metrically convex subset is convex. (9 pages, 22 March 2005)

• Van der Waerden's construction of a splitting field, Scientific WorkPlace file, dvi file, pdf file. In his classic book, Modern Algebra, van der Waerden gave a procedure for factoring polynomials over a finite-dimensional, separable, simple extension field. I believe that there is a nonconstructive component to his proof, and I will indicate where it comes in and why. Although I'm sure that this component could be avoided while staying within the framework that he set down, it is simpler to get around the problem by working with a splitting algebra, which is easily constructed for any polynomial and base field. The existence of a splitting field then follows from van der Waerden's argument.. (8 pages, 10 February 2005)

• Enabling conditions for interpolated rings, Scientific WorkPlace file, dvi file, pdf file. Given a ring B, a subring A, and a proposition P, we can construct a ring C between A and B so that C = B if P and C = A if not P. This construction is often used to obtain Brouwerian counterexamples. We investigate what conditions have to be put on the inclusion of A in B in order that C have some property (like being a unique factorization domain) for all P. (9 pages, 23 January 2005)

• The ascending tree condition, LaTeX file, dvi file, ps file, pdf file, html file. A strengthening of the ascending chain condition allows a choice-free constructive development of the theory of Noetherian modules. Related topics in the theory of PID's and elementary divisor rings are also explored. (10 pages, 23 December 2001)

• A division algorithm, LaTeX file, dvi file, pdf file. A divisibility test of Arend Heyting, for polynomials over a field in an intuitionistic setting, may be thought of as a kind of division algorithm. We show that such a division algorithm holds for divisibility by polynomials of content 1 over any commutative ring in which nilpotent elements are zero. In addition, for an arbitary commutative ring R, we characterize those polynomials g such that the R-module endomorphism of R[X] given by multiplication by g has a left inverse. (9 pages, 18 December 2001)

• Pre-abelian clan categories, LaTeX file, dvi file, ps file, pdf file. Categories of representations of clans without special loops, and with a linear ordering at each vertex, are studied with an eye toward identifying those that have kernels and cokernels. A complete characterization is given for simple graphs whose vertices have degree at most two. (12 pages, 18 July 2001)

• Spreads and choice in constructive mathematics, LaTeX file, dvi file, ps file, pdf file, html file. An approach to choice-free mathematics using spreads: If constructing a point satisfying property P requires choice, replace this problem by that of constructing a nonempty set of elements satisfying P. Then construct a spread, without choice, whose elements satisfy P. The theory is developed and several examples are given. (11 pages, 21 June 2001)

• Equivalence of syllogisms, LaTeX file, dvi file, pdf file. Studies of categorical syllogisms typically focus on the valid ones: which syllogisms are valid, why they are valid, how the valid ones are classified, how to derive valid ones from other valid ones. As Lear put it, "Our principal interest in invalid inferences is to discard them." Here we are interested in the invalid syllogisms too. The traditional methods for transforming one valid syllogism into another also transform any syllogism, valid or not, into an equivalent one. (22 pages, 20 December 2000)

• Weak Markov's principle, strong extensionality, and countable choice, LaTeX file, dvi file, ps file, pdf file. Ishihara showed, using countable choice, that weak Markov's principle is equivalent to all real functions on a complete metric space being strongly extensional. In this note we show that weak countable choice suffices, and that the theorem fails in sheaf models of the real numbers. (5 pages, 8 August 2000)

• Omniscience principles and functions of bounded variation, LaTeX file, dvi file, ps file, pdf file. A very weak omniscience principle is formulated, related omniscience principles are considered, and the theorem that a function of bounded variation is the difference of two increasing functions is shown to be equivalent to the omniscience principle WLPO. It is also shown that an arbitrary function (not necessarily strongly extensional) with located variation on an interval is the difference of two increasing functions. (10 pages, 31 July 2000)

• Weakly integrally closed domains: minimum polynomials of matrices, (with James Brewer) LaTeX file, dvi file, ps file, pdf file, html file. Must the coefficients of the minimum polynomial of a matrix over a domain lie in that domain? This question leads to the notion of a weakly integrally closed domain, over which the answer is "yes" for 3-by-3 matrices. It is shown that certain subalgebras of k[t] are weakly integrally closed, as are rings consisting of quadratic algebraic numbers. (15 pages, 17 November 1999)

• Constructive mathematics without choice, LaTeX file, dvi file, pdf file. What becomes of constructive mathematics without the axiom of (countable) choice? Using illustrations from a variety of areas, it is argued that it becomes better. (7 pages, 28 July 1999)

• Gleason's theorem has a constructive proof (revised), html file, dvi file, ps file, pdf file. Two recent papers have dealt with the possibility of a constructive proof of Gleason's theorem. In the first, Geoffrey Hellman claims to give an example showing that this is impossible even in three-dimensional Euclidean space. In the second, Helen Billinge suggests that some reformulation of Gleason's theorem in three-space may have a constructive proof. Douglas Bridges has noted that Hellman's example leaves open the problem of finding a constructive substitute---a theorem with a constructive proof that is easily seen to be classically equivalent to Gleason's theorem.

It turns out that Gleason's formulation admits a constructive proof as it stands. In this paper we discuss this seemingly anomalous situation. Gleason's theorem itself is somewhat peripheral to the discussion. What is interesting is the relationship of classical mathematics to constructive mathematics that is highlighted by this misunderstanding. (7 pages, 1 July 1999)

• Computing limiting stationary distributions of small noisy networks, (with Katarzyna Winkowska-Nowak) LaTeX file, dvi file, ps file, pdf file. The dynamics of opinion transformation is modeled by a neural network with a nonnegative matrix of connections. Noise is introduced at each site, and the limit of the stationary distributions of the resulting Markov chains as the noise goes to zero is taken as an indication of what configurations will be seen. An algorithm for computing this limit is given, and a number of examples are worked out. Some of the mathematical ideas developed, such as visible states, time scales, and a calculus of indexed probabilities, are of independent interest. (35 pages, 21 June 1999)

• Pointwise differentiability, LaTeX file, dvi file, ps file, pdf file. What can be done with pointwise properties as opposed to uniform properties on compact intervals? (4 pages, 7 June 1999)

• Linear independence without choice, (with Douglas Bridges and Peter Schuster) LaTeX file, dvi file, ps file, pdf file. The notions of linear and metric independence are investigated in relation to the property: if U is a set of n+1 independent vectors, and X is a set of n independent vectors, then adjoining some vector in U to X results in a set of n+1 independent vectors. It is shown that this property holds in any normed linear space. A stronger property---that finite-dimensional subspaces are proximinal---is established for strictly convex normed spaces over the real or complex numbers. It follows that metric independence and linear independence are equivalent in such spaces. Proofs are carried out in the context of intuitionistic logic without the axiom of countable choice. (9 pages, 12 December 1998)

• Trace-class operators, (with Douglas Bridges and Peter Schuster) LaTeX file, dvi file, ps file, pdf file. In this paper we define trace-class and Hilbert-Schmidt operators---the von Neumann-Schatten classes C1 and C2---without assuming the existence of an adjoint or even an absolute value. In fact, an operator is in the class Cp, for p in [1,¥), exactly when a certain supremum exists. We prove that such operators are compact, hence have adjoints. The theory is developed without appeal to separability, or to the existence of an orthonormal basis, and without using countable choice. We construct the singular values of compact operators, and characterize them, and the classes Cp, in terms of their singular values. (22 pages, 29 November 1998)

• The fundamental theorem of algebra: a constructive development without choice, LaTeX file, dvi file, ps file, pdf file. Can constructive mathematics be developed in a reasonable manner without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. Here it is shown how the fundamental theorem of algebra can be restated and proved without using countable choice, and it is argued that this is really the right way to look at it. A notion of a complete metric space, suitable for a choiceless environment, is also developed. (21 pages, 16 October 1998)

• Nontransitivity of locatedness for subspaces of a Banach space, LaTeX file, dvi file. Given subsets A contained in B of a metric space X, such that A is located in B and B is located in X, does it follow that A is located in X? It does if A and B are subspaces of a Hilbert space X, but not if X is just a Banach space. (5 pages, 27 August 1998)

• Subgroups of p5-bounded groups (with Elbert A. Walker) LaTeX file, dvi file. Each valuated module B with B(5) = 0 is a direct sum of simply presented valuated modules and copies of two valuated modules which come from (finite) hung trees. There are infinite-rank indecomposable valuated modules B with B(6) = 0. (21 pages, 8 July 1998)

• Adjoints and the image of the ball, LaTeX file, dvi file. A bounded operator between Hilbert spaces has an adjoint if and only if the image of the unit ball is located. (7 pages, 29 July 1998)

• Is 0.999... = 1?, LaTeX file, dvi file. pdf file. A skeptical look at this well-known equation. Copyright the Mathematical Association of America 1999. All rights reserved. (8 pages, 15 May 1998)

• A weak countable choice principle, (with Douglas Bridges and Peter Schuster) dvi file, ps file, pdf file. A weak choice principle is introduced that is implied both by countable choice and by the law of excluded middle. This principle suffices to prove that metric independence is the same as linear independence in an arbitrary normed space over a locally compact field, and to prove the fundamental theorem of algebra. (5 pages, 19 January 1998)

• Generalized real numbers in constructive mathematics, dvi file, LaTeX file, ps file, pdf file. Two extensions of the real number system, one given by uppercuts the other by lowercuts, are developed within a constructive framework. The first includes distances to arbitrary subsets, the second includes norms of arbitrary bounded linear operators. The intuitive meaning of comparing such quantities to ordinary real numbers is preserved. Difficulties with encompassing both kinds of numbers in a single system are considered. (15 pages, 1 November 1997)

• Adjoints, absolute value and polar decomposition, (with Douglas Bridges and Peter Schuster) dvi file, SWP file, ps file, pdf file. Bilinear forms, equivalent conditions for the existence of an adjoint, general notion of a polar decomposition. Choiceless Riesz representation theorem. (13 pages, 1 September 1997)

• Simply presented tag modules, dvi file, SWP file, ps file, pdf file. A tag module is a generalization, in any abelian category, of a simply presented torsion abelian group. The theory of such modules is developed, it is shown that countably generated tag modules are simply presented, and that Ulm's theorem holds for simply presented tag modules. Zippin's theorem is stated and proved for countably generated tag modules. (22 pages, 9 June 1997)

• Filtered modules over discrete valuation domains, (with Elbert Walker) dvi file, SWP file, ps file, pdf file. We consider a unified setting for studying local valuated groups and coset-valuated groups, emphasizing the associated filtrations rather than the values of elements. Stable exact sequences, projectives and injectives are identified in the encompassing category, and in the category corresponding to coset-valuated groups. (27 pages, 22 May 1997)

• A constructive proof of Gleason's theorem, (with Douglas Bridges) dvi file, SWP file, ps file, pdf file. Gleason's theorem states that any totally additive measure on the closed subspaces, or projections, of a Hilbert space of dimension greater than two is given by a positive operator of trace class. (25 pages, 21 May 1997)

• The regular element property, dvi file, SWP file, ps file, pdf file. The property that an ideal whose annihilator is zero contains a regular element is examined from the point of view of constructive mathematics. It is shown that this property holds for finitely presented algebras over discrete fields, and for coherent, Noetherian, strongly discrete rings that contain an infinite field. (8 pages, 25 July 1996)

• Growing forests in abelian p-groups, dvi file, ps file, pdf file. The purpose of this note is to finish a development of the theory of simply presented p-groups which exploits tree structure as much as possible. A proof that summands of simply presented p-groups are simply presented that is independent of Ulm's theorem is given. The same techniques are used to show that countable p-groups are simply presented. Indeed it is shown that a summand of an Axiom-3 p-group is simply presented, thus settling both problems, and showing that Axiom-3 p-groups are simply presented, at one go. (6 pages, 10 June 1996) Journal of Algebra, 187(1997) 289--294.

• Sets, complements and boundaries, (with Douglas Bridges and Wang Yuchuan) dvi file, ps file, pdf file. The relations among a set, its complement, and its boundary are examined constructively. A crucial tool is a theorem that allows the construction of a point where a segment comes close to the boundary of a set in a Banach space. Brouwerian examples show that many of the results are the best possible. (23 pages, 14 May 1996) Indag. Math. 7(1996) 425--445.

• Interview with a constructive mathematician, html file, LaTeX file, dvi file, ps file, pdf file. This interview is cobbled together from a series of conversations that took place on the list, l-math, during the fall of 1994. It concerns the nature of constructive mathematics, starting with the question of whether the objects studied by constructive mathematicians are the same as those studied by classical mathematicians. (26 pages, 4 March 1996) Modern Logic, 6(1996) 247--271. Click here for comments by Gabriel Stolzenberg on this paper.

• Flat dimension, constructivity, and the Hilbert syzygy theorem, dvi file, ps file, pdf file. A constructive treatment of flat dimension of modules including a proof of the Hilbert syzygy theorem. (13 pages, 13 February 1996)

• Confessions of a formalist, Platonist intuitionist, LaTeX file, dvi file, ps file, pdf file. A bit of automathography, and some musings. (1) Algebraists. (2) A Platonist in trouble. (3) Constructive mathematics and recursive function theory. (4) Different subject matter? (5) Intuitionistic logic. (6) Formalism. (7 pages, 9 April 1994)

• Intuitionism as generalization, html file, dvi file, SWP file, ps file, pdf file. Intuitionism, in its simplest form, is a generalization of classical mathematics that accomodates both classical and computational models. (5 pages, 1990)

• SuperHare, html file, dvi file. The traditional Hare proportional system has a number of flaws when used with a small electorate. For certain values of the parameters the algorithm doesn't even work. In addition, the problem of ties becomes acute, so the role of chance in determining the outcome is increased. We present here a system that deals with these flaws while still staying very much in the spirit of the original system. The main innovations are floating quotas, fractional quotas, and breaking ties by complete enumeration of possible scenarios. (9 pages, late 1980's)

• A measure of consanguinity, dvi file, ps file, pdf file. A natural numerical measure of consanguinity is developed that applies to individuals with arbitrary multiple kinship connections. For simple relationships the consanguineal distance specializes to the civil degree, less two if the relationship goes through full siblings. This measure is deduced from axioms motivated by an heuristic picture of blood mixtures. The formula suggests a quantum mechanical probability interpretation whose classical counterpart yields a generalization of the Murdock degree of consanguinity. (10 pages, 1977)