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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



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