When I first came across algebras as a student I availed myself of the mnemonic that these are just vector spaces with a true multiplication thrown in; recently, in a text on differential geometry, I encountered a characterization of algebras as rings with scalar multiplication added to the mix: clearly a case of six of one and half a dozen of the other.
Perhaps most of us think of algebras that way — certainly in the easy comfort and loose informality of our inner dialogues. Aber, to quote Einstein, ist dass wirklich so? Or, to quote Gershwin, it ain’t necessarily so: as is so often the case we’ve failed to factor in the logicians’ way of doing things.
On p. 8 of George Grätzer’s Universal Algebra we read that “[a] universal algebra, or, briefly, algebra … is a pair , where A is a nonvoid set and F is a family of finitary operators on A. F is not necessarily finite [!] and may be void [?!]…” This is a level of abstraction that would make Bourbaki blush. But we must give the logicians their due. Working in such great generality one gets very strong theorems, if one gets anywhere at all, and this is abundantly borne out by, for example, the treatment given on p. 57ff of my (and probably everyone’s) favorite theme in basic abstract algebra, the homomorphism and isomorphism theorems, evoking thoughts (memories, spectra, simulacra) of Emmy Nöther, Emil Artin, and B. L. van der Waerden (the latter two of whom figured prominently in my pre-qualifying examination days some thirty years ago: how times have changed…).
Anyhow, returning to p. 8ff of Grätzer, we encounter there an assortment of familiar examples to bring us in from out of the cold of high abstraction, ranging from posets and lattices to groups, rings and even division rings. Says Grätzer on p. vii of the Introduction to the First Edition: “Thus universal algebra is the study of finitary operations of a set, and the purpose of research is to find and develop the properties which such diverse algebras as rings, fields, Boolean algebras, lattices and groups have in common.” Voilà: the raison d’être of the book under review.
To be sure, coming in at 580 pages, Grätzer’s Universal Algebra is a real tour de force. Originally published in 1968 (written in 1964–65, when Grätzer was only 27 years old), the book’s Second Edition appeared thirteen years later. Again, Grätzer: “It is my opinion that the definitions and results presented in the book form a foundation of universal algebra as much today as they did a decade ago” (p. v). A Monthly review at that time touted Universal Algebra as “the standard reference” in the field.
On the other hand, as is so often the case with works, or, rather, mathematical subjects, that carry such heavy overtones of formal logic, the question of popularity in mainstream mathematical circles is a different matter. Grätzer points out that Alfred North Whitehead was the first to write seriously on the subject of universal algebra (pace Leibniz?) and goes on to note (p. vii) that “he had no results.” He goes on to observe that the subject didn’t get off the ground properly until Garrett Birkhoff got hold of it in the 1930s: consider in this connection the role played by lattice theory in the pages of Universal Algebra. But in due course the subject grew into a proper branch of mathematics, with such notables as Alfred Tarski, H. Jerome Keisler, Irving Kaplansky, J. Donald Monk, and B. Jónsson playing parts in the tale.
Grätzer peppers the book with exercises of varying degrees of difficulty as well as problems meant as springboards for research, and he appends a playful epilogue in which the older Grätzer encounters himself at 27. It is a revealing bit of prose with appropriate elements of wistfulness, criticism and humor mixed in. It fits well with the preceding eight chapters (or actually nine: chapter 0 counts, of course) and seven appendices of serious scholarship. What the Monthly claimed about the book some forty years ago still seems to hold true today: this is “the standard reference in a field notorious for its lack of standardization.”
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.