## Garrett Birkhoff

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## Birkhoff, Garrett

# BIRKHOFF, GARRETT

(*b*. Princeton, New Jersey, 10 January 1911; *d*. Water Mill, New York, 22 November 1996)

*abstract algebra, computing*.

Birkhoff was the son of mathematician George David Birkhoff and Margaret Grafius Birkhoff. George Birkhoff, the father, was the first American mathematician to gain wide respect in Europe. Garrett Birkhoff is more remembered for promoting new conceptions than specific theorems. His most important single result was a theorem that instituted a conception, the *Birkhoff variety theorem*, originating modern universal algebra. He showed the power of deceptively simple algebraic properties and the feasibility of more complex and realistic applied mathematics, and he was among the first mathematicians to rely heavily on computers.

**Lattices and Universal Algebra.** Entering Harvard College in 1928, Birkhoff aimed at mathematical physics. Physics led him to partial differential equations, which in turn led to more abstract ideas, including Lebesgue theory and point-set topology. Curiosity led him to finite groups. After graduating in 1932, he went to Cambridge University for physics. That July, though, he visited Munich and met Constantin Carathéodory, who pointed him towards algebra and especially van der Waerden’s great new textbook *Moderne Algebra* (Berlin: Springer, 1930). Back in Cambridge he switched to algebra with group theorist Philip Hall.

Birkhoff turned the study of subgroups, subrings, and so on into two branches of mathematics. The intersection of subgroups of a single group *G* is also a subgroup of *G*. The union *H(K)* of subgroups of *G* is generally not a subgroup because an element of *H* and another of *K* may combine to give one that is in neither. Yet *H* and *K* will generate a subgroup *H/K* which is called their *join*, defined as the smallest subgroup of *G* that contains both *H* and *K*, and so is generally larger than the set theoretic union. This suggests a dual definition: the *meet H–K* is the largest subgroup of *G* contained in both *H* and *K*. In fact, the meet of subgroups is their intersection, but other structures than groups can have meets that are smaller than intersections. In England Birkhoff organized and generalized the study of such order relations into *lattice theory*. He also characterized a wide array of structures whose substructures form lattices and organized their study as *universal algebra*. Each subject had precedents, notably in the work of Richard Dedekind and Emmy Noether, but Birkhoff established them as subjects.

Birkhoff enjoyed the unity and economy of the abstract idea of an *order relation* on a set. That is any relation *x* B *y* on the elements of the set such that: 1) *x* B *x* for all elements *x* of the set; 2) for any elements *x,y,z*, if *x* B *y* and *y* B *z* then *x* B *z*; and 3) for any elements *x,y*, if *x* B *y* and *y* B *x* then *x* =*y*. The relation xBy is usually read “*x* is less than or equal to *y*” although it may have nothing to do with magnitude. He gave the example of logical propositions with *x* B *y* defined to mean *x* implies *y*, and *x* = *y* defined to mean that *x* is logically equivalent to *y*. He defined a *lattice* as an ordered set where every two elements *x,y* have a join *x/y* defined as the smallest element greater than or equal to both *x* and *y*, and a meet *x–y* defined as the largest element less than or equal to both *x* and *y*, with a few further properties. Logical propositions form a lattice where the join *x/y* of propositions is their disjunction “*x* or *y*,” and their meet *x–y* is the conjunction “*x* and *y* .” The subgroups of a group *G* form a lattice when *H* B *K* is defined to mean *H* is contained in *K*. The notion of lattice is wide enough to include many examples yet specific enough to yield many theorems. Birkhoff also found further abstract conditions characterizing various kinds of lattice. His 1940 book *Lattice Theory* is still in print, with new concepts and results tripling its original length.

Not all mathematical structures are as tidy as groups. The substructures of a given structure do not always form a lattice. So, which ones do? Birkhoff found an elegant sufficient condition.

A *Birkhoff variety* is a class containing all the structures defined by a given set of operators and equations. For example, a *commutative ring* is a set *R* with a selected zero element *0* and unit element *1* and addition, subtraction, and multiplication +,-,• satisfying equations familiar from arithmetic, such as the zero law *x+ 0 = x* and the commutative law for addition *x+ y = y + x*. These equations are understood to hold for all elements *x,y* of *R*. A *field* is a commutative ring *R* meeting a further more com-

plex condition, not an unrestricted equation but a conditional equation: If *x* ≢ 0 then *x* has an *inverse*, an element *y* in *R* with *x* •*y = 1*. A *variety* is a class of structures definable purely by operators and equations, as shown above for the class of commutative rings and not for the class of fields.

Varieties enjoy very special properties compared to other classes of structures. For example, any two rings *R,S* have a *product R×S*. An element of *R×S* is an ordered pair *x,u* with *x* and element of *R* and *u* and element of *T*. The zero element of *R×S* is the pair of zeros *0,0*, the unit is the pair of units *1,1*. The operations are defined componentwise, which for addition means *x,u+ y,v = x+y,u+v*.

The analogues hold for subtraction and multiplication, and *R×S* satisfies all the ring equations since it satisfies them all in each component. The same does not work for fields. Even if *R* and *S* are both fields, *R×S* is not. Its element *1,0* is not zero because the first component is not the zero of *R* but it has no inverse either because the second component has no inverse in *S*.

The *Birkhoff variety theorem* lists a few constructions such as products, and proves that a class of structures is a variety if, and only if, it is closed under these operations.

Given any class of structures, no matter how it was originally defined: it can be defined purely by equations if, and only if, these listed constructions apply to it and always yield results in that class. Because fields do not have products there cannot be any way to characterize fields purely by equations. These constructions imply that the substructures of any structure in a variety form a lattice.

The theorem created modern *universal algebra* defined as the study of *Birkhoff varieties*. Earlier more sweeping universal theories of algebra were not so productive as Birkhoff's.

**A Career at Harvard.** In 1933, Birkhoff returned to Harvard as a member of the Society of Fellows, and in 1936 he joined the mathematics department. He never earned a doctorate. He married Ruth Collins in 1938 and eventually had two daughters and a son (Ruth, John, and Nancy). He began teaching the new abstract algebra, which Saunders Mac Lane also taught there. Their 1941 *Survey of Modern Algebra* was the first effective English language introduction to the material of van der Waerden’s *Moderne Algebra* and was an immediate success. It was augmented by the 1967 *Algebra* with the order of the authors’ names reversed and more emphasis on category theory. These two books had a huge impact on mathematics students for fifty years and continued to shape the standard U.S. algebra curriculum in the early twenty-first century.

During World War II, Birkhoff worked on fluid dynamics, including the explosion of bazooka charges and problems of air-launched missiles entering water. Chapters of his 1950 book *Hydrodynamics:* A *Study in Logic, Fact, and Similitude* were named for various “paradoxes” where either the models idealize phenomena in unrealistic ways or basically plausible models give some bizarre results. Birkhoff emphasized group theory for handling symmetries in hydrodynamics, although one paradox was the breakdown of symmetries in some realistic hydrodynamic situations. He urged innovative numerical methods and his later more specialized hydrodynamics relied more heavily on computing.

He consulted on reactor design for the Bettis Atomic Power Laboratory from 1955 to 1961, working on numerical solutions for partial differential equations by repeatedly improving successive approximations. Starting in 1959 he consulted for General Motors on numerical description of surfaces, to guide numerically controlled machinery cutting the dies used to stamp out automobile body parts. This led him to major contributions to “spline” methods fitting segments of cubic polynomials to data points.

Birkhoff was elected to the American Academy of Arts & Sciences in 1945, the American Philosophical Society in 1960, and the National Academy of Sciences in 1968. He received honorary degrees from the National University of Mexico, the University of Lille, and Case Institute of Technology.

## BIBLIOGRAPHY

### WORKS BY BIRKHOFF

“On the Combination of Subalgebras.” *Proceedings of the*

*Cambridge Philosophical Society*29 (1933): 441–464.

*Lattice Theory*. New York: American Mathematical Society, 1940.

Third edition, greatly expanded, 1967.

With Saunders Mac Lane. *A Survey of Modern Algebra*. New

York: Macmillan, 1941.

*Hydrodynamics: A Study in Logic, Fact, and Similitude*. Princeton,

NJ: Princeton University Press, 1950.

With Gian-Carlo Rota. *Ordinary Differential Equations*. Boston:

Ginn, 1962.

Mac Lane, Saunders, and Garrett Birkhoff. *Algebra*. New York:

Macmillan, 1967.

*The Numerical Solution of Elliptic Equations*. Philadelphia:

Society for Industrial and Applied Mathematics, 1971. “Current Trends in Algebra.” *The American Mathematical*

*Monthly* 80 (1973): 760–782.

With Gerald L. Alexanderson and Carroll Wilde, “A

Conversation with Garrett Birkhoff.” *The Two-Year College Mathematics Journal*14 (1983): 126–145.

### OTHER SOURCES

Corry, Leo. *Modern Algebra and the Rise of Mathematical*

*Structures*. Basel and Boston, MA: Birkhäuser, 1996.

Describes the history and influence of Birkhoff’s lattice theory and universal algebra in several places; see the index. Mac Lane, Saunders. “Garrett Birkhoff (10 January 1911–22

November 1996).” *Proceedings of the American Philosophical Society* 142 (1998): 646–649.

Young, David. “Garrett Birkhoff and Applied Mathematics.”

*Notices of the American Mathematical Society* 44 (1997): 1446–1450.

*Colin McLarty*

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