Henriksen and Isbell on f-ringsJames J. MaddenAbstract. This paper gives an account of the contributions of Melvin Henriksenand John Isbell to the abstract theory of f -rings and formally real f -rings, withparticular attention to the manner in which their work was framed by universalalgebra. I describe the origins of the Pierce-Birkhoff Conjecture and present someother unsolved problems suggested by their work.AMS Subject Classification. Primary: 06F25 Ordered rings, algebras, mod-ules, 06-03 Order, lattices, ordered algebraic structures – historical. Secondary:08B15 Lattices of varieties, 08B20 Free algebras, 01A60 Hist ory and biography,20th century.Key words. Lattice-ordered ring, f-ring, formally real f -ring, Pierce-BirkhoffConjecture, equational classesA conjecture or hypothesis may become as significant in a mathematician’s legacy as afinished piece of work, as the manner in which we refer to great mathemati cal questionsby the names of great mathematicians attests. In his wo rk on f -rings, Mel Henriksencontributed to the base of existing knowledge and also raised questions that are drivingsome of the most challenging and intriguing research I know.An f -ring—the name is short for “function ring”— is a subring of a product of totally-ordered ri ngs that is also closed under the natural lattice operations. These objects werefirst named and studied systematical ly by Birkhoff and Pierce in their paper [BP]. In [HI ],Henriksen and Isbell picked up where Birkhoff and P ierce left off, proving several deepresults about the equational theory of f -rings and adding many important results on thestructure of f -rings, as well. In the present essay, I will concentrate on the former themeand the unanswered questions it leads to. For a presentatio n of the structure theory, onemay consult [BKW], section 9.4.The notorio us problem now known as the “Pierce-Birkhoff Conjecture” was first for-mula ted by Henriksen and Isbell during their collaboration on [HI]. I heard the storydirectly from Mel, who with characteristic animati on and good humor told how his nu-merous “proofs” were shot down, one after another, by Isbell. It was like listening to afisherman talk of an encounter with a legendary fish, too big and too sly to be caught. Theconjecture is that every continuous piecewise poly nomial function on Rncan be expressedas a finite lattice-combination of polynomials, i.e., as a sup of infs of finitely many poly-nomials. Here, of course, we are concerned with piecewise polynomials that are defined bygiving a finite cover of Rnby closed semial gebraic sets and and stipulating a polynomialon each. Functions that are piecewise polynomial in a more general sense, e.g., requiringinfinitely ma ny pieces, are not generally finite l attice-combinatio ns of polynomials. Theconjecture was publicized in the early 1980s by Isbell, who beli eved that the methods ofreal-algebraic geometry then being intro duced might be capable of capturing it. UsingThom’s Lemma, Mah´e reeled in the n = 2 case soon after; see [ML1]. After this theconjecture became widely known. Since Mah´e’s work, some new techniques have beenexplored but there has been no definitive progress on the cases with n ≥ 3.1Many people ask why the names of Birkhoff and Pi erce appear in reverse alphabeticalorder i n the name of the conjecture. My guess is that it just sounds better to have the one-syllable name first. As a matter of fact, it is arguable whether Birkhoff and Pierce shouldreally be regarded as the authors of this problem. The likely inspiration i s an unsolvedproblem stat ed at the end of [BP], but a careful examination suggests that Birkhoff andPierce may actually have meant to ask something different. To be fair (and alphabetical),perhaps the name should be the “Birkhoff-Henriksen-Isbell-Pierce Conjecture,” but I don’texpect this to catch on. Whatever it’s called, it seems that everyone who has taken it uphas experienced it in much the same way as Mel did, believing at first that all the piecesof a proof are at hand only to discover that the crux of the problem has not been touched,and finally marveling at the mysterious depths.Acknowledgement. I would like to thank the referee for i nsisting on clari ty on a numberof points and for providing suggestions that were useful to this end.21. Universal algebra. I shall review some basic ideas of universal a lgebra in order t oprovide a conceptual and terminological frame of reference. Readers may skim the presentsection and refer back to it as needed, in case questions about meanings should arise. Analgebra, i n the sense of universal algebra, is a set equipped with distinguished elementsand operations. A collectio n of symbols acting as names for these elements and operationsis called the signature of the algebra. For example, a group (if written additively) is analgebra with signature (0, −, +), with the understanding that 0 names a fixed element ofthe algebra, − names a unary operat ion on it, and + names a binary operation on it. Aring with identity may be given signature (0, −, +, ·). If a signature Ω is given then analgebra with that signature is called an Ω-algebra. Though when speaking of a particularΩ-algebra I may identify the symbols in Ω with the corresponding elements or operationsof the algebra, in general one distinguishes symbols from the elements or operations theyname, so that one may speak of corresponding elements or operations in different algebras,and thus clarify not ions like homomorphism and isomorphism. For a rigorous discussionof the concept of ‘signature’ (under different names), the reader may refer to the definitionof “ language (or type) of algebras in [BS], page 2 3, or to the definition of “operator domain. . . Ω” and “Ω-algebra” in [C], page 48.Not every (0, −, +)-algebra is a group. A group satisfies t he additional requirementsthat + be associative, that 0 be a left and right identity for + and that −x be a left andright inverse for x. These requirements can be stated as equational laws, i.e., as universallyquant ified sentences in which the quanti fier-free part is an equation in the language withthe constant and function symbol s from the signature. Equational laws are also calledidentities.Fix a signature Ω, and l et X be an arbitra ry class of Ω-algebras. It is easy to see thatany equational law satisfied by every element of X is also sati sfied by the