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Henriksen and Isbell on f rings James J Madden Abstract This paper gives an account of the contributions of Melvin Henriksen and John Isbell to the abstract theory of f rings and formally real f rings with particular attention to the manner in which their work was framed by universal algebra I describe the origins of the Pierce Birkhoff Conjecture and present some other unsolved problems suggested by their work AMS Subject Classification Primary 06F25 Ordered rings algebras modules 06 03 Order lattices ordered algebraic structures historical Secondary 08B15 Lattices of varieties 08B20 Free algebras 01A60 History and biography 20th century Key words Lattice ordered ring f ring formally real f ring Pierce Birkhoff Conjecture equational classes A conjecture or hypothesis may become as significant in a mathematician s legacy as a finished piece of work as the manner in which we refer to great mathematical questions by the names of great mathematicians attests In his work on f rings Mel Henriksen contributed to the base of existing knowledge and also raised questions that are driving some 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 totallyordered rings that is also closed under the natural lattice operations These objects were first named and studied systematically by Birkhoff and Pierce in their paper BP In HI Henriksen and Isbell picked up where Birkhoff and Pierce left off proving several deep results about the equational theory of f rings and adding many important results on the structure of f rings as well In the present essay I will concentrate on the former theme and the unanswered questions it leads to For a presentation of the structure theory one may consult BKW section 9 4 The notorious problem now known as the Pierce Birkhoff Conjecture was first formulated by Henriksen and Isbell during their collaboration on HI I heard the story directly from Mel who with characteristic animation and good humor told how his numerous proofs were shot down one after another by Isbell It was like listening to a fisherman talk of an encounter with a legendary fish too big and too sly to be caught The conjecture is that every continuous piecewise polynomial function on Rn can be expressed as a finite lattice combination of polynomials i e as a sup of infs of finitely many polynomials Here of course we are concerned with piecewise polynomials that are defined by giving a finite cover of Rn by closed semialgebraic sets and and stipulating a polynomial on each Functions that are piecewise polynomial in a more general sense e g requiring infinitely many pieces are not generally finite lattice combinations of polynomials The conjecture was publicized in the early 1980s by Isbell who believed that the methods of real algebraic geometry then being introduced might be capable of capturing it Using Thom s Lemma Mahe reeled in the n 2 case soon after see ML1 After this the conjecture became widely known Since Mahe s work some new techniques have been explored but there has been no definitive progress on the cases with n 3 1 Many people ask why the names of Birkhoff and Pierce appear in reverse alphabetical order in the name of the conjecture My guess is that it just sounds better to have the onesyllable name first As a matter of fact it is arguable whether Birkhoff and Pierce should really be regarded as the authors of this problem The likely inspiration is an unsolved problem stated at the end of BP but a careful examination suggests that Birkhoff and Pierce 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 t expect this to catch on Whatever it s called it seems that everyone who has taken it up has experienced it in much the same way as Mel did believing at first that all the pieces of 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 insisting on clarity on a number of points and for providing suggestions that were useful to this end 2 1 Universal algebra I shall review some basic ideas of universal algebra in order to provide a conceptual and terminological frame of reference Readers may skim the present section and refer back to it as needed in case questions about meanings should arise An algebra in the sense of universal algebra is a set equipped with distinguished elements and operations A collection of symbols acting as names for these elements and operations is called the signature of the algebra For example a group if written additively is an algebra with signature 0 with the understanding that 0 names a fixed element of the algebra names a unary operation on it and names a binary operation on it A ring with identity may be given signature 0 If a signature is given then an algebra with that signature is called an algebra Though when speaking of a particular algebra I may identify the symbols in with the corresponding elements or operations of the algebra in general one distinguishes symbols from the elements or operations they name so that one may speak of corresponding elements or operations in different algebras and thus clarify notions like homomorphism and isomorphism For a rigorous discussion of the concept of signature under different names the reader may refer to the definition of language or type of algebras in BS page 23 or to the definition of operator domain and algebra in C page 48 Not every 0 algebra is a group A group satisfies the additional requirements that be associative that 0 be a left and right identity for and that x be a left and right inverse for x These requirements can be stated as equational laws i e as universally quantified sentences in which the quantifier free part is an equation in the language with the constant and function symbols from the signature Equational laws are also called identities Fix a signature and let X be an arbitrary class of algebras It is easy to see that any equational law satisfied by every element of X is also satisfied by the subalgebras products and homomorphic images that may be formed from the algebras in X The smallest class containing X and closed under these formations can be built in three steps first take all products of elements of X then adjoin all sub algebras of these and finally adjoin all

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