Perhaps this heritability was limited to those cases in which the heiress transmitted the de facto headship of her house.
heritability her·i·ta·bil·i·ty (hěr'ĭ-tə-bĭl'ĭ-tē)
The quality of being heritable.
The proportion of phenotypic variance attributable to variance in genotypes.
heritable her·i·ta·ble (hěr'ĭ-tə-bəl)
Capable of being passed from one generation to the next; hereditary.
Capable of inheriting or taking by inheritance.