White’s comment on Barbosa de Almeida. On the Structure of Dravidian Relationship Systems.
Douglas R. White 2010
Cultural theory and Mathematical anthropology are greatly advanced by careful ethnographic
description (e.g., Trautmann 1981), mathematical modeling (e.g., Tjon Sie Fat 1998) and
conceptual simplification, of which the Barbosa de Almeida article is an excellent example.
Trautmann (1981) did a great service in coding the variety of Dravidian kin-term systems in
India into a formal descriptive language. Tjon Sie Fat (1988) advanced our understanding of his
kin-term models into mathematical formalisms. Godelier et al.’s (1998) edited volume and its
introduction provided a synthesis of ethnographic studies and theoretical viewpoints on
understanding Dravidian kinship systems in India in contrast to Iroquois and generation-type
systems and formal Dravidianate systems in India and elsewhere. Trautmann (1981: 218, 185186, 206-7, 215-20, 228) erred, however, in disingenuously characterizing ♂eZD marriages as
outside normal Dravidian rules when such marriages are consistent with viri-sided divisions (two
sets of male lines whose sisters and daughters cross-marry). His ad hoc explanation of ♂eZD
marriage as not consistent with Dravidian terms, or with a (dubious) same-generation marriage
rule, is an unlikely explanation given its occurrence in diverse South Indian societies. Fig. 1
shows a network of couples in a typical consanguineal group descending from couple 12 and
genealogical affines shown to the left and right (patrilines of marriages 10 and 11). Each
marriage in the bottom row is characterized by one or more marriage types. Trautmann’s rules
regard marriage 5 (ZD) as non-Dravidian when it is actually Dravidian viri-sided, with marriages
10-5-6-7(3-1-12) on one side of the center (dashed line), and 6-9-11(2-4-13) on the other, with
women marrying between sides, so that each could refer to the other as “cross” or “affinal” kin.
♂ZHZHZHFZ*
(♂)ZD*
♂ZHZHDDHZFZ*
♂FZD
♂FZD
♂MBD
♂Z6HZ11HFZHZ* ♂FZHZHZHZ
♂Z5HS1D11HFZHZ
Fig. 1: Dravidian sides P-graph for a consanguineal group with two sets of non-consanguineals
(♂), Bold: Z, H = spanning 2-generations
*viri- but not uxori-sided
1
The marriages in Fig. 1 cannot be rearranged without errors, however, into opposing sets of
female-line uxori-sides with men marrying between them. The marriage type of couple 5 (♂ZD),
those of couple 10 (♂ZHZHZHFZ and ♂ZHZHFZHZ) and one of those of 9 (♂ZHZHFZHZ),
have an even number of female links (Z, D, M, W) but an odd number of male (H, F, B, S) links.
Discrepancies like this, between viri- and uxori-sidedness, can be due only to off-generation
marriages, like those of wife 10 or husband 5 (married to a ZD). Marriage 9 is viri-sided but not
uxori-sided by one cycle (9+2 -6+1 -11+4+13 -10+3 -9: 4 negatives, 5 positives) but both viri
and uxori-sided by another cycle (9+2 -5+12+1 -11+4+13 -10+3 -9: 4 negatives, 6 positives).
Marriage 9 is uniquely viri-sided like the others, but ambiguous with respect to uxori-sides
because the network cannot be resolved into uxori-sides. The network in Fig. 1 is not consistent
with Barbosa de Almeida’s or Trautmann's definition of sidedness because the product of signs
in each cycle is not positive, which requires both viri- and uxori-sidedness. The number of
female links (if coded negative) is positive for every cycle (thus viri-sided), which requires an
even number of negatives. A uxori-sided network would require an even number of male links in
every cycle.
Dravidian kinship on the ground. White (2010:16) clarifies how sidedness is defined on the
ground:
When two people in the same Dravidian language region are uncertain how they are related
… it is a computational discussion of whether they have a common close ancestor that allows
them to decide whether they are “parallel kin” or marriageable “cross” relatives. This
calculation expresses the existence of positions in the kinship network connecting same or
opposite sides of two sets of intermarrying male lines (viri-sides) so if there is an even
number of their female links – mothers of male or agnatic ascendants linking them to an
ancestor – then they are cross and marriageable (Kris Lehman, pers. comm.), as with ♂ZD,
♂FZD, ♂MBD or more remote cross-sided-kin. Otherwise they are same-sided, as with Z,
FZ, MB. In a Dravidian viri-sided system if there is a link through a father or a father’s wife
(=mother) it is the agnatically shortest path that is taken: the link to sister, for example, is FD
(i.e., parallel kin by the counting rule) and not MD. This counting rule is associative, so that
my parallel kin’s parallel kin are parallel, and so forth (//=/, /X=X/=X, and XX=/).
Algebraic Analysis. Barbosa de Almeida’s mathematical simplification of Trautmann's (1981)
model of Dravidian terminology is “inspired by Dumont's (1953, 1971[1957]) and Lévi-Strauss's
[1957[1949], 1958] theory of affinity and consanguinity, which goes back to Lewis Morgan
(1997[1871]).” It “combines ideas going back to A. Weil (1949) … as well as Lounsbury (1956,
1969[1964]), Lorrain (1975) and Tjon Sie Fat (1990, 1998).” I will extend his simplifications as
they apply to kinship behavior in actual kinship networks (Houseman and White 1998a) and
show how to improve his model for those who wish to correct for Trautmann’s error. Although
Barbosa de Almeida traces the historical precursors of his analysis from Weil’s (1949) group
theory models of kinship structures, Weil begins his analyses of mathematical group structures
very differently: the elements of Weil’s algebraic groups are marriages, as in Fig. 1, which is a P2
graph (White and Jorion 1992) in which marriages are vertices. The operators are links from one
marriage to another following gendered (♂,♀) parental (P) and progeny (p) links. In this
comment, following Weil, who began “I propose to show how a certain type of marriage laws
can be interpreted algebraically, and how algebra and the theory of groups of substitutions can
facilitate its study and classification” (Weil 1949, in Levi-Strauss 1949, English translation: p.
221). Again, following Weil, “At all events, let the number of types of marriages be n“ and let
each marriage be numbered from 1-n, as distinguished in Fig. 1: where ♂ links (and descent
lines) are shown in black and ♀ links in red.
White and Jorion (1992) generalized Weil’s approach, and Houseman and White (1998b:231232) used the P-graph approach to help characterize Dravidianate systems more generally,
including those outside Dravidian-speaking South Asia. Their key concept treats the “sidedness”
of the marriage networks in which Dravidianate terminologies are found. For m marriages in a
marriage network connected by gendered (♂,♀) links, like Fig. 1: If each cycle of a marriage
network has an even number of ♀ links it is viri-sided, and/or uxori-sided if the number of ♂
links is even. Sidedness of networks, of one or another or both of these types (in Trautmann’s
model, both), qualify as a marriage alliance structure if (1) there are a substantial number of
overlapping cycles of marriage (sharing ♂,♀ arcs not just vertices), (2) the cycles form a
bicomponent in which every marriage is connected to every other by two or more disjoint paths,
and (3) counting the number c of independent cycles (using the formula c = m – n + 1 where m
is the sum of ♂ and ♀ links), and calculating the expected number of odd and e even cycles (e =
½c is the usual equal expectation), the difference between e and c/2 is statistically significant
using the Binomial test.
Examples of Dravidianate terminologies and viri-sidedness are given in Houseman and White’s
(1998b:229, see Fig. 9-2 and 9-9) study of Lowland South America. These include the Makuna,
which are 99% viri-sided, (n = 105 and binomial p << 0. 0.0000001), who have bilateral crosscousin marriage (BCCM) but lack exogamous moieties, and the Shavante, with 95% virisidedness, (n = 149, binomial p << 0. 0.0000001), who have exogamous moieties and FZD
marriage. Uxori-sidedness in these two cases 82% and 72% respectively. Both societies have
agnatic descent, as do most of the Dravidian South Asian societies, but here only the Makuna
have BCCM, typical of South India and Trautmann’s definition of Dravidian terminologies with
exclusive same-generation marriages (mistakenly excepting ♂eZD marriage from his view as
correlated with but not part of Dravidian systems).
Pul Eliya (Houseman and White 1998a) is a rare case for the South Indian region where we have
actual marriage networks satisfying the marriage alliance structure conditions 1-3 so as to
measure both viri-sidedness (85%) and uxori-sidedness (87%). The (1998a:80) restudy of
Leach’s (1961) data, however, finds 100% viri-sidedness only among kin who have common
ancestors within four generations (ca, 70% sidedness otherwise). Of the 360 consanguineal
relatives in the network, 23 are married, all viri-sided (p < 0.0004 compared with 11.5 expected),
and 8 are close relatives (5 MBD, 3 FZD). Of distant-relative wives 6 are of a younger
3
generation than the husband and 3 of an older generation (all 9 viri-sided) while those within the
same generation, as expected, are both viri- and uxori-sided. Here, marriage to a blood relative
creates a cycle (like ♂FZD or ♂MBD in Fig. 1) with an even number of links so if female links are
even (viri-sided) then male links are even to (hence uxori-sided. Trautmann overgeneralized
from this congruence: that viri-sidedness plus same generation marriage entails uxori-sidedness.
Pul Eliya is a viri-sided network for consanguineally-related kin who intermarry (zero
exceptions) and contradicts Trautmann’s view that Dravidian kin terms are same-generation and
both uxori-and viri-sided.
In Houseman and White’s (1998b:231-232) definition:
Dravidianate terminology can be understood as an egocentric recoding of sidedness from the
point of view of a participant in marriage alliances. Looking up to the senior generation, couples
are already formed, either as one's own ancestors or collaterals, but classified as parallel or cross
for sibling links. Hence the bifurcate (same-sex sibling) merging pattern (F = FB ≠ MB, M = MZ
≠ FZ) and its attendant 0 generation parallel/cross distinctions. Looking down from one's own
generation, couples are yet to be formed, and only one member of each potential couple is
ordinarily one's descendent: ego may be a participant in making new marriages for siblings or
descendants.
In this perspective, Dravidianate terminology may be defined less as a thing in itself, than as
a systematic expression of an ongoing positive relationship between a certain behavioral
regularity, sidedness, on the one hand, a particular classificatory principle, egocentric crossness,
on the other hand…. However, such a reformulation, precisely because it is at once more realistic
and genealogically underspecified, can more easily account for both the observed variability of
Dravidianate systems (within a given community and from one society to another) and their
changes through time.
In sum, Dravidianate kinship occupies the structural space defined by intersection of
sidedness and egocentric crossness.
Comment on Barbosa de Almeida. Barbosa de Almeida’s (and Trautmann’s) formalization of
Dravidian kinship is consistent with viri-sides and constitutes a viri-Dravidianate terminology, but
attests uxori-sides that are not ethnographically justified for Dravidian. Trautmann’s
terminological rules are overspecified to include same-generation marriages, excluding
marriages such as ♂ZD. The problem with exclusive same-generation marriages is that because
uxori-sides entail viri-sides and vice versa, they jointly entail latent opposite-generation moieties
that resemble Australian section systems!
To demonstrate the compatibilities of Barbosa de Almeida’s mathematical model, I take the genders
(♂,♀), and substitute P and p for his f and f-1, and for easier mnemonics substitute o for his s to
create for his basic words w = {e, s, f, f-1} → {e, o, P, p}. That is, e for ego and sibling of same sex, o
for opposite sex sibling, P for same sex Parent, and p for same sex progeny, the P-1 inverse. Running
through a concatenation of these symbols for the links the Fig. 1 will illustrate viri-sided marriages
such as 5 (♂ZD) or 6 (♂FZ) as eop and ePop and equate them to the wife of ♂ego while preserving
Barbosa de Almeida’s axioms A, C, and D and his even/odd rule for ♀ links a viri-Dravidian
4
consanguine/ affine opposition. Marriage diagrams like Fig. 1 should fit a viri-Dravidian
terminology if they are viri-sided, and a uxori-Dravidian terminology if they are uxori-sided.
Signed graphs. For the connections between kinship terminology and kinship networks, the theorems
of signed graphs (Harary 1953) entail that if all marriages are locally sided, whether viri- or uxorisided or both, the whole marriage graph is similarly sided. In the first case the female links are (-)
and the male links (+), while in the second case the reverse is true. For signed graphs, local structure
induces global structure and vice-versa.
The fact that Dravidian terminologies do not require named moieties and that sidedness may apply de
facto only among localized groups of consanguineal kin such as intermarry frequently is consistent
with what South Asian ethnographers report regarding sidedness (personal communications, F. K. L.
Chit Hlaing aka Kris Lehman, Murray Leaf): on meeting one another, two strangers can determine
their (e.g., viri) sidedness by counting the number of female links to a common cognatic ancestor,
including but not counting the male links, observe whether their sum is even or odd, and determine
opposite-sidedness if even and same-sidedness if odd. This would explain why sided networks would
be found between those with common ancestors within a limited number of generations (memorable
for female as well as male links) but not necessarily between others in the same society or even the
same village. On a side note, if Dravidians traced uterine rather than agnatic links as the principal
descent lines, their “counting rule” for sidedness would be to count the number of male links to a
common uterine ancestor, including but not counting the female links, and determine oppositesidedness if the number of male links is even and same-sidedness if odd. (NB: previous draft had
opposite/same reversed)
Needed variants of the Dravidian terminological model. A needed modification of Barbosa de
Almeida’s mathematical Dravidian model is to provide two new alternate versions: one where the
even/odd rule is true for female links only (viri-sidedness) and the other for the male links only.
Comment on Read. I agree with Read’s comments on the structure of Dravidian kinship terms.
There need be no single prototype of “the” Dravidian terminology, and Trautmann (1981:33)
agrees. The recognition of which relatives are classificatory “blood” kin (same egocentric side,
unmarriageable) and classificatory “affinal” kin (opposite egocentric side, some marriageable) in
any of the usual variants of the Dravidian terminologies, however, are sided in precisely the
sense of “the counting rule” enunciated above, which is not recognized by Trautmann. Instead,
Trautmann (1981:47) notes that for ego, mother is parallel while for ego’s father, wife is cross,
and so forth! But by the counting rule mother=”odd” and is parallel while for father wife is by
definition cross, which makes her “even.” In the counting rule if ego’s father and mother did
have an ancestor in common, mother would be “even” to the father, so there would be
sociocentric consistency (cross) for the father and mother, while and ego’s mother is odd to the
son (unmarriageable) although his mother’s daughter (his sister) would be “even” and
marriageable, a contradiction, except that this sister would be older in relative age. A ♂ZD,
however, would be even and marriageable and it is ♂eZD that may be considered marriageable,
5
and if husband and wife in this case were close in age, the wife’s mother would have had to be
older than ego.
These types of considerations are not a part of the Western logics of kinship, but are a side
consequence of classificatory logics. Read (2007, 2009, 2010) proves that for all classificatory
systems, including Dravidian, the elder/younger sibling distinction is essential to the logical
generation of the kinship terminology. I would venture to say that in classificatory systems, men
do not marry blood kin of senior generations, including ♂MZ or ♂FMZ, who are “even sided”
by a viri-sided criterion. This is probably not the core of the puzzle, however, which would be
best treated by Read. My intuition is that the viri-sidedness of the kinship network is interpreted
through the Read’s (2010:5 Figure 1) into the sidedness of the kinship terminology.
Read (2010:1) notes that Barbosa de Almeida’s formalism has dropped, however, the principle
of relative age (relative to ego, irrespective of generation) as one of the four “principles of
opposition” that Dumont and Trautmann agreed upon, the other three being six, generation, and
crossness. Once Read unwraps the relation between network sidedness and terminological
sidedness, two observations may be relevant: (1) the terminology implies which marriages can
and cannot be made, but requires none in particular; (2) the actual marriages in cases like Pul
Eliya, among blood kin within four generations, that are 100% sided, should be consistent with
the possibilities allowed in the terminology of Pul Eliya, if not Dravidian generally.
Read’s alternate kin-term diagram. In our correspondence on this question, Read
generously provided a diagram that gives further insight. This is “another way to draw the
term map (for Tamil) that highlights a different aspect (all words in the kin term map are
terms, not genealogical relations).” The omission of O/Y distinctions doesn’t affect
argument.
6
has
kin
kin
the
Fig. 2. Tamil terminology, male speaker; cross-cousin marriage rule
need for Son of FCousin = Son and Daughter of FCousin = Daughter
Noting that the female (daughter) links in the diagram have a slightly different shape of the
arrowhead than the male links, it is evident that all solid-line genealogical routes from male self
to FCousin have two (even numbered) female links, and are therefore viri-sided as Cross
relatives. This is not the case in the genealogical space: what is suppressed by the dotted lines
(e.g., the curved dotted lines from males to their mothers) are redundancies of M = FW. This is
the same “reduction” made by P-graphs, where F and M become a single vertex. The FCousin is
therefore Cross and the OSi, YSi, with only one (odd) downward female line is Parallel. That the
MoDa link to Sister has an even number of female links is ignored and suppressed by the
terminology, just as it is suppressed in the P-graph in Figure 1. (Dravidian, then, has not only a
principle of unity of same-sex siblings, but unity of the couple). This also makes clear that in
looking back to common ancestors but erasing the Mother links it is structurally permissible to
have only two kin terms for grandparents.
Read’s Questions. Read’s queries in his discussion of Fig. 2, and my replies, are below:
7
First, sidedness. At the kin term level, sidedness (for the reference
person!) is clear for the middle 3 generations. The kin term map is
the same as for Kariera for these 3 generations. The proof that I give
showing that the section system, implicit in the terminology, is
actually absolute for the Kariera case will apply equally here, so the
reference person sidedness apparent in the diagram is absolute sidedness.
Yes: as I have noted, if all marriages are same-generation, then the system that is implied is
Kariera. This is proven in my Appendix. Note, however, that FCousin is not the only
potential Cross-marriage possibility: ♂ZD, or more likely, ♂eZD, is also a potential marriage
partner under the two (even) downward female lines rule, as with ♂FZD and ♂MBD
(FCousin). In that case, however, the same-generation rule, uxori-sidedness, and the
similarity to Kariera is nullified. ♂eZD marriages are found, however, only for a small
number of South Asian societies with Dravidian terminology.
Second, viri- versus uxori-sidedness. This does not derive from the
terminology since the terminology is symmetric with respect to sex and
so any structure implied for male terms, such as patrilines, applies
equally to female terms and matrilines. Thus viri- and uxori-sidedness
are equally likely from the viewpoint of the terminology, but not from
the data you provide. So where does the asymmetry come from?
No: It does derive from the terminology, wherein the suppression of the M = FW equivalence
is asymmetric. If the system were solely “uxori-sided Dravidian” (whose existence is not
established) then it would be F = MH that would be suppressed.
The beauty of Weil’s kinship algebras. We can now recognize the beauty of Weil’s choice, in
his kinship algebra, to make marriage the units that are related to one another in a network
kinship. This choice allows a treatment of classificatory and of descriptive principles that helps
to unify the treatment of kinship logics and terminologies with the structures of actual marriage
choices and genealogical networks, although they are completely different: kin term systems
exist as logics in their own right, independent of the genealogical grid, but also refers to those
grids, which can be defined and analyzed independently. And in classificatory systems there are
many-to-many but constrained maps between them. Many-to-one correspondences
(homomorphisms) are perhaps too strong a model for kinship when many-to-many matchings
may be needed. We are beginning to see in economic modeling, for example, theories of stability
in many-to-many matching markets (Echenique and Oveido 2006). Such markets promote
diversity and independence among players and in economics, reduce the inequalities between
employers and employees. In anthropological studies, they provide the needed modeling of
social structures that are in some senses egalitarian, as between sides or sections, if not in
relations of gender and authority.
Appendix. Conditions for sided systems to converge to section systems.
This is the more general case for Read’s (2010) demonstration in a particular case where a
Dravidian kinship terminology converges to Kariera.
8
Definitions
P-graph. Following Weil (1949), but for concrete marriages rather than marriage-types, "let the
number of types of marriages be n", and let each marriage be numbered from 1-n, as in Fig. 1 (n
= 13), where ♂ links (agnatic descent lines) are shown in black and ♀ links in red. A P-graph
has vertices that represent unmarried children or couples (with or without children) and P-links
between vertices directed from child-to-parent(s). The inverse links, parent(s) to offspring, are
given by p=P-1. Vertices represent single individuals in the case of unmarried children, whose
type of P link (♂male, ♀female) depends on their gender. All other vertices represent either a
single parent (partner unknown) or a parental couple. Their P links to parents again depend on
gender: husband to parents, wife to parents. A sub-P-graph Q consists of a set Q of vertices and
all the P links between vertices in QxQ. Although one may have more than one marriage, Pgraphs normally represent (1) real or fictive genealogy wherein no-one is their own ancestry, (2)
two-parent opposite-sex marriages, (3) only one set of parents for a child, and thus (4) no vertex
will have more than two parental links, one to a male’s parent(s) (for a couple: husband's), the
other to a female’s (for a couple: wife's) parent(s). Other variants of P-graphs may be defined.
P-graph generations. The P-graph defined by 1-4 above is a directed asymmetric graph (DAG)
with 1 ≤ gen ≤ g generations, g equal to the length of the longest directed path (e.g., g = 3 in Fig.
1), which insures that every generation will have at least one parent in the preceding generation.
Program Pajek (Batagelj and Mrvar 1992) computes g generations for P-graphs.
Sides and sidedness. These are implicit moiety structures in the marriage network, wherein a Pgraph can be uniquely partitioned into two, within each the vertices are connected by the links of
one gender and between which they are uniquely connected by the links of other gender.
V= Viri-sides. Here, a P-graph or sub-P-graph Q is split into two halves, uniquely determined,
within which vertices are connected by ♂son/parent links and between which they are uniquely
connected by ♀daughter/parent links.
U= Uxori-sides. This a P-graph or sub-P-graph Q that is split into two halves, uniquely
determined, within which vertices are connected by ♀daughter/parent links and between which
they are uniquely connected by ♂son/parent links.
S= Same-generation marriage. In general, siblings do not all have to marry spouses of the same
generation, nor does “same generation marriage” imply that generations partition by time
periods, nor that men and women share the same temporality of generations (when women marry
on average at a younger age than males, their generational time is faster than males, for
example). Nonetheless, a P-graph component (connected graph) connected only by P▪p (sibling
or parent’s child) links among marriages constitutes a pure generation iff it contains no
connecting parents P-connected to other connecting parents within it (i.e., no grandparents). If
the pure generations of the P-graph are partially ordered by the g generations in the P-graph,
then the P-graph has property S, same generation marriage. A sub-P-graph Q may also have
property S.
9
A= Implicit alternate generational moieties exist when a sub-P-graph Q is same-generation but a
larger subset R  Q has some marriages with both same-generation marriages and ones between
individuals +2 and -2 apart but none at +1 and -1 apart.
B= Bicomponent. A bicomponent of a marriage network P-graph containing marriages x and y is
a maximal sub-P-graph B containing x and y in which for any node z there are two or more
disjoint paths between very pair of nodes in B. (All links between nodes in B that occur in the
main graph are by definition also in the P-graph)
Ck= A sub-P-graph Q is one of consanguineal marriages iff every marriage within Q for which
there are two parent links is between a male and female who have a common consanguineal
ancestor less than k generations back.
Theorems
The presence of both viri-sides (condition V) and uxori-sides (condition U), for a sub-P-graph
marriage network Q of Ck consanguineal marriages, in a bicomponent B of a marriage network,
logically entails alternate generational moieties (conditions S and A) in B. This includes the
possibility of an ego at generation i, 1 ≤ i ≤ g, marrying someone at generation j, 1 ≤ j ≤ g, where
the absolute difference |i - j| is an even number, e.g., +2 or -2 generations (condition A). Further,
the presence of S, same-generation marriage, and either viri-sides V or uxori-sides U logically
entails the complementary type of sidedness.
Theorem 1. U, V, and Ck, consanguineal marriages => S and A: implicit alternategenerational moieties.
Theorem 2. U and S => V
Theorem 3. V and S => U
Proofs
Proof of 1.
Suppose U and V and consanguineal marriages. For a sub-P-graph Q in which every
marriage is between consanguines (Ck, consanguineal marriages), then marriages in which
all husband and wife marriage cycles are with ancestors that are both U and V, and each
such cycle will have even number of female links, an even number of male links, and
adding the two, an even number of total links. Thus, if the ancestral graph for each such
cycle is drawn, from parent to child in adjacent generations, then either husband and wife
will be of the same generation or one is an even number of generations above the other.
This will apply to all such marriages, and conditions S and A will be satisfied: implicit
alternate-generational moieties. Q.E.D. This proof will generalize to Australian section
systems.
Counterexample to 1, lacking consanguineal marriage, S and A are not entailed.
10
Suppose U and V and a network without Ck, consanguineal marriages, in which a man
marries a BDHBD or BDHSD, implying two nonconsanguineal marriages (they are
linked by two distinct ancestors, not a common ancestor). Then the number of males is
even (4) as is the number of females (2), qualifying for U and V but S and A are violated.
Lemma 1.
Suppose A (Implicit alternate generational moieties). This may apply so that the
generation number of each marriage is the required minimum of one below the
generation number of the parents of the husband and that of the wife, the other alternative
being an odd number below. This entails that any marriage cycle will have an even
number of links because for any distance d (odd) from one spouse to the common
ancestor, the distance is a d’ (odd) for the other spouse, for a sum of two odd numbers,
which is an even number of total links in every such cycle.
Proof of 2.
Suppose U and A. By Lemma 1, A requires that every marriage cycle will have an even
number of links ec, while U entails an even number of male links em, so the remaining
female links ef = ec - em must be even, and the network is V. Q.E.D.
Proof of 3.
Suppose V and A. Exchanging U and V above: then V and A entail U. Q.E.D.
References
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Echenique, Federico, and Jorge Oveido. 2006. A theory of stability in many-to-many matching
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