Anne C. Morel (also published as Anne C. Davis, died July 22, 1984) was an American mathematician known for her work in

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

, and

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

. She was the first female full professor of mathematics at the

University of Washington The University of Washington (UW, simply Washington, or informally U-Dub) is a public research university in Seattle, Washington. Founded in 1861, Washington is one of the oldest universities on the West Coast; it was established in Seattle a ...

Education and career

Morel graduated in 1941 from the

University of California, Los Angeles The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California St ...

. She began graduate study in mathematics in 1942 at the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

, but left her studies to serve in the

WAVES Waves most often refers to: *Waves, oscillations accompanied by a transfer of energy that travel through space or mass. * Wind waves, surface waves that occur on the free surface of bodies of water. Waves may also refer to: Music * Waves (ban ...

(the United States Naval Women's Reserve) during

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...

. She returned to her studies in Berkeley in 1946, and completed her Ph.D. in 1953. Her dissertation, ''A Study in the Arithmetic of Order Types'', was supervised by

Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

, and concerned

ordinal arithmetic In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an expl ...

. After two years as an assistant professor at Berkeley, and positions at the

University of California, Davis The University of California, Davis (UC Davis, UCD, or Davis) is a public land-grant research university near Davis, California. Named a Public Ivy, it is the northernmost of the ten campuses of the University of California system. The institut ...

and the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

(1959–1960), she joined the mathematics faculty at the

in 1960, and became a tenured associate professor there in 1961. Eventually she became the first female full professor of mathematics there, and for many years she was the university's only female professor of mathematics.

Research contributions

As part of her thesis work, in 1952, Morel found two different

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...

s whose

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

are equal. After

Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and to ...

simplified her construction, they published it jointly. In 1955, Morel published a converse to the

Knaster–Tarski theorem In the mathematics, mathematical areas of order theory, order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following: :''Let'' (''L'', ≤) ''be a complete lattice and let f : L → ...

, according to which every incomplete lattice has an increasing function with no fixed point. Her 1965 paper with Thomas Frayne and

Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, Ca ...

, "Reduced direct products", provides the main definitions of

reduced product In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let be a family of structures of the same signature σ indexed by a set ''I'', and let ' ...

s in

model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

. It was published after several important applications of those definitions had already been discovered, and has been called a "classical reference paper". Her only publication with her advisor,

, was a brief announcement of related research using reduced products in connection with the

compactness theorem In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally no ...

in mathematical logic. Among other results, it provided a proof of the compactness theorem using

ultraproduct The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors ...

s. With

Chen Chung Chang Chen Chung Chang (Chinese: 张晨钟) was a mathematician who worked in model theory. He obtained his PhD from Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text on model th ...

, she also used reduced products to show that a sufficient condition for properties to be preserved under

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...

s, derived by

Alfred Horn Alfred Horn (February 17, 1918 – April 16, 2001) was an American mathematician notable for his work in lattice theory and universal algebra. His 1951 paper "On sentences which are true of direct unions of algebras" described Horn claus ...

, was not also a necessary condition. Topics in her later research included

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

s, and

cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses the ...

universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...

. Her final publication, published posthumously, was "Cofinality of algebras" (1986).

Personal life

During her war service, Morel met and married Alan Davis, another mathematician. However, their marriage was not successful, and Davis took a position at the

University of Nevada, Reno The University of Nevada, Reno (Nevada, the University of Nevada, or UNR) is a public land-grant research university in Reno, Nevada. It is the state's flagship public university and primary land grant institution. It was founded on October 12, ...

while Morel returned to her studies at UC Berkeley. They divorced in 1955. In Berkeley, Morel began an affair with her advisor

in 1950, at approximately the same time as another student mistress of Tarski,

Wanda Szmielew Wanda Szmielew née Montlak (5 April 1918 – 27 August 1976) was a Polish mathematical logician who first proved the decidability of the first-order theory of abelian groups. Life Wanda Montlak was born on 5 April 1918 in Warsaw. She complet ...

, left Berkeley to return to Poland. Tarski was married, to Maria Witkowska (whom he had married in 1929), but when Morel divorced her husband Alan Davis in 1955, Tarski offered to divorce Maria and marry Morel instead. However, she turned him down. Instead, in 1957, she married Delos Morel, a lawyer. Although the Morels and the Tarskis remained on friendly terms until at least 1960, Morel eventually came to view Tarski's treatment of his other female students as "taking advantage of his position of power in a way she now viewed as unacceptable". The Morels had two daughters, Jeanne (born 1958) and Verena (1962–2002). Morel died on July 22, 1984. Her husband Delos became the Chief Administrative Law Judge on the Washington State Board of Industrial Insurance Appeals, and died in 2008.

Selected publications

References

{{DEFAULTSORT:Morel, Anne C. Year of birth missing 1984 deaths 20th-century American mathematicians University of California, Los Angeles alumni University of California, Berkeley alumni University of California, Berkeley faculty University of Washington faculty WAVES personnel 20th-century American women mathematicians