André Weil and the first algebra of kinship

In 1949 Claude Lévi-Strauss published Les Structures élémentaires de la parenté, which treats marriage rules as systems of exchange between groups. At the end of its first part sits a short appendix by André Weil, one of the founders of the Bourbaki group, written at Lévi-Strauss's request; the two had met in New York as French exiles during the Second World War. Its title translates as "On the algebraic study of certain types of marriage laws (Murngin system)". It is only a few pages long, and much of later kinship mathematics grows out of it.

This piece works Weil's method through by hand, using kin-type shorthand: F father, M mother, B brother, Z sister, S son, D daughter, read left to right from the person speaking (Ego). So MBD is a mother's brother's daughter and FZD a father's sister's daughter. Both are cross cousins, linked through a brother and a sister; cross and parallel cousins covers why so many terminologies single them out.

The puzzle on Lévi-Strauss's desk

The appendix names the Murngin, the label W. Lloyd Warner used for Yolŋu clans of northeast Arnhem Land, where he worked from 1927 to 1929 before publishing A Black Civilization (1937). "Murngin" is a scholars' word, not theirs. Yolŋu clans belong to one of two moieties, Dhuwa and Yirritja, and relationships are reckoned through a kinship system called gurruṯu that still governs ceremony and marriage.

Warner's account combined two features that seemed to pull in opposite directions: eight subsections, the division Lévi-Strauss read in Aranda-type systems as restricted exchange (groups swapping spouses), and marriage with the MBD, a rule that sends wives one way around a circle of groups, the mark of what Lévi-Strauss called generalized exchange. Could one consistent set of rules produce both? Lévi-Strauss wanted a way to check, and asked Weil.

The question outlived both men. Warner's material set off what became known as the Murngin controversy, and in 1967 J. A. Barnes devoted a fifty-page paper, Inquest on the Murngin, to it. Keep one distinction in view: an anthropologist's model of a system, people's own account of it, and the marriages they actually make are three different things. Weil's algebra is about the first.

Weil's move: classify marriages, not people

Weil started from marriages, not individuals or subsections. Suppose every marriage in a society falls into one of a finite number of types, and suppose (in paraphrase of his first two conditions) that:

1. each person, man or woman, is entitled to exactly one type of marriage, and
2. which type that is depends only on the person's sex and on the type of the marriage the person was born from.

A man and a woman may marry when both are destined for the same type. The second condition hands us two functions. Write f(t) for the type of marriage that a son of a type t marriage will make, and g(t) for the type that a daughter will make. Everything about the system lives in f and g.

Two consequences follow straight away. A brother and sister born of the same marriage t must never land in the same type, or the rules would let them marry, so f(t) and g(t) differ for every t. And in closed systems of the kind at issue, f and g are permutations: each type receives the sons of exactly one type and the daughters of exactly one type. Weil's third condition was the cousin rule itself: every man must be able to marry his MBD.

What a group is, in plain words

A permutation is a reshuffling of a finite set: every item goes somewhere, and no two items go to the same place. Two permutations done in turn make another permutation, any permutation can be undone, and there is a do-nothing permutation, the identity, written e. A group of permutations is a collection closed under these moves, with e included.

Keep composing f and g (and their undoings) and everything you reach forms the group generated by f and g. Each member reads as a chain of kin links: "the marriage your daughter's son will make" is g followed by f. When the order of composing never matters, so that f then g always equals g then f, the group is called abelian, or commutative.

A four-type system, worked by hand

Take a society with four sections labeled A1, A2, B1 and B2, laid out in the Kariera manner (the section system simulator runs the same rules, writing A, B, C and D for A1, B1, A2 and B2):

• A1 men marry B1 women, A2 men marry B2 women, B1 men marry A1 women, B2 men marry A2 women.
• A child's section is fixed by the father's: an A1 father has A2 children, A2 has A1, B1 has B2, and B2 has B1.

Name the four marriage types by the couple: t1 is an A1 man with a B1 woman, t2 an A2 man with a B2 woman, t3 a B1 man with an A1 woman, t4 a B2 man with an A2 woman. In a t1 marriage the father is A1, so the children are A2. A son, an A2 man, will marry a B2 woman: type t2. A daughter, an A2 woman, will marry a B2 man: type t4. Repeat for each type:

The brother and sister check passes: in every row the two right-hand entries differ.

Now compose. From t1, g then f: g(t1) = t4, and f(t4) = t3. From t1, f then g: f(t1) = t2, and g(t2) = t3. Same answer. Every row, plus each function applied twice:

The first two columns agree, so f and g commute. The last two show that f twice, or g twice, puts every type back where it started. Call h the shuffle "g, then f"; it swaps t1 with t3 and t2 with t4. The whole group has four members:

Mathematicians know it as the Klein four-group. Because e, f, g and h carry t1 to t1, t2, t4 and t3, every type can be reached from every other, so this is one society rather than two that never intermarry.

Check it genealogically. A man of section A1 has an A2 father and a B2 mother, and he marries a B1 woman. His MB is a B2 man married to an A2 woman, so their daughter is B1: a possible wife. His FZ is an A2 woman married to a B2 man, so her daughter is B1 too. Both cross cousins fall into the spouse category.

Cousin rules as equations

Weil's point was that you never need sections at all. Take any man and call the type of his parents' marriage t; he will make a marriage of type f(t).

His MBD first. His mother was a daughter of some marriage s, and her own marriage is t, so g(s) = t. Her brother is a son of s and makes a marriage of type f(s); his daughter makes g(f(s)). The man himself makes f(t) = f(g(s)). So he can marry his MBD exactly when

f(g(s)) = g(f(s))

and for every man to have that option, it must hold for every type s. In words: universal MBD marriage means that f and g commute, which makes the whole group abelian.

Now his FZD. His father was a son of some marriage u, so f(u) = t, and the man makes f(f(u)). The father's sister makes g(u), and her daughter makes g(g(u)). So FZD marriage is always available exactly when

f(f(u)) = g(g(u)) for every u.

The four-type system passes both tests: both kinds of cross cousin are spouses. Sorting systems by properties like these became a research program. Alain Gottcheiner's 2008 papers take up kinship systems whose groups are abelian and, in a sequel, nonabelian; our note on abelian kinship systems follows that thread.

Change one rule, change the algebra

Generalized exchange first. Take three patrilineal lines: L1 men marry L2 women, L2 men marry L3 women, L3 men marry L1 women. Name each marriage type by the husband's line: u1, u2, u3. A son makes the same type of marriage as his father, so f leaves every type where it is. A daughter of L1 marries an L3 man, so g(u1) = u3, and likewise g(u2) = u1 and g(u3) = u2.

The MBD test passes, since f changes nothing and so commutes with anything. The FZD test fails: f(f(u1)) = u1, but g(g(u1)) = g(u3) = u2. A man may marry his MBD and never his FZD. Wives move one way round the circle of lines, which is generalized exchange in two equations.

Now a system that quietly falls apart. Go back to the four sections and change only the descent rule, so that children take their father's own section. Then f does nothing, and a daughter of a t1 marriage is an A1 woman who marries a B1 man, type t3, so g swaps t1 with t3 and t2 with t4. Both cousin tests pass. But starting from t1, f and g only ever reach t1 and t3. A1 and B1 form one closed pair that exchanges spouses forever, A2 and B2 another, and the two pairs never become kin. Later writers call such a system reducible. The algebra spots the split before anyone counts a marriage. Weil's own pages apply the same tools to the Murngin rules as Lévi-Strauss had set them out.

After Weil

Harrison White's An Anatomy of Kinship (1963) drew directly on Weil, modeling kinship systems as clans linked by fixed rules of marriage and descent. White went on to become one of the founders of modern network analysis, and the idea of structural equivalence he sketched in this kinship book went with him; network density and centrality introduces that side of the field. Philippe Courrège (1965) recast the model with three permutations of classes, conjugal (a man's class to his wife's), maternal (a mother's class to her children's) and paternal, tied by one axiom: the father's route to his children runs through his wife's class. He applied it to the Kariera, Aranda and Murngin systems. John Paul Boyd's "The algebra of group kinship" (1969) pushed the group theory further, and Gisèle De Meur and Alain Gottcheiner (2000) joined permutations, groups and graphs, as described in our note on prescriptive systems as permutations.

All of this models rules, not lives. A permutation model assumes that everyone belongs to one class and that the rules are followed; it says nothing about how often people married the prescribed category, or about how Yolŋu understand their own system. Its strength is narrower and still useful: from the rules alone, it tells you which cousins are spouses and whether a society hangs together. Let A1 fathers have A1 children and four sections become two societies that never meet. Weil showed how to see that in a two-line calculation.

Sources and further reading

• Lévi-Strauss, Claude. 1949. Les Structures élémentaires de la parenté. Paris: Presses Universitaires de France. With André Weil's appendix. English edition: The Elementary Structures of Kinship, ed. Rodney Needham. Boston: Beacon Press, 1969.
• Warner, W. Lloyd. 1937. A Black Civilization: A Social Study of an Australian Tribe. New York: Harper.
• Barnes, J. A. 1967. Inquest on the Murngin. Royal Anthropological Institute Occasional Paper 26. London: Royal Anthropological Institute of Great Britain and Ireland.
• White, Harrison C. 1963. An Anatomy of Kinship: Mathematical Models for Structures of Cumulated Roles. Englewood Cliffs, NJ: Prentice-Hall.
• Courrège, Philippe. 1965. "Un modèle mathématique des structures élémentaires de parenté." L'Homme 5 (3-4): 248-290. https://www.persee.fr/doc/hom_0439-4216_1965_num_5_3_366751
• Boyd, John Paul. 1969. "The algebra of group kinship." Journal of Mathematical Psychology 6 (1): 139-167. https://doi.org/10.1016/0022-2496(69)90032-7
• Gottcheiner, Alain. 2008. "On some classes of kinship systems I: Abelian systems." Mathematical Anthropology and Cultural Theory 2 (4).
• Jaramillo Uribe, Juan Manuel. 2009. "Estructuralismo francés y estructuralismo metateórico." Discusiones Filosóficas 10 (15): 23-50. On Weil's conditions and Courrège's axioms (in Spanish). https://revistasojs.ucaldas.edu.co/index.php/discusionesfilosoficas/article/view/660