| Abstract: | Two genes in a pedigree are identical by descent if they are two copies of a common ancestor gene. To obtain an unambiguous definition of the set of genes, at some autosomal locus, any gene is defined as an ordered pair of zygotes: the zygote who carries the gene, and the parent who transmitted it. The natural ordered structure on the set of zygotes yields an ordered structure upon the set of genes. Any event of the mendelian segregation splits down the set of genes into non-overlaping classes of identical genes: when considered as an ordered sub-set of genes, each class is shown to have the algebraic properties of a tree. Given a sub-set ?? of genes, a family of exclusive events ensuring identity between all genes of ?? is identified as a family of genic trees with some property. This relationship between segregational events and genic trees is extended to the case where two sub-sets ?? and ??′ of genes are considered together. As a consequence, a general method is obtained to compute either identity coefficients involving any number of genes splitted into one or two identity classes, or the fifteen coefficients defined among four genes, whichever the relationships between zygotes and genes might be. Using this approach to deal with the allelic structure in a set of genes carried by related zygotes is suggested. |
