A Reply to McConaghy

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Abstract
Contrasting accounts of the common structure in the blockmodels for two community elites (Breiger and Pattison, 1978; McConaghy, 1980) are shown to follow from two distinct means of interpreting the semigroup representation of role structure in a single population. The two candidate sets of interpretative features for the semigroup are the homomorphisms which it admits and the equations by which it is defined. These alternative descriptions lead naturally to the respective constructions of thejoint homomorphic image (Boorman and White, 1976)and the common role structure semigroup (Bonacich, 1980; McConaghy, 1980) as representatives of common structure in two populations. It is argued that the use of descriptive features in the form of homomorphic images, in general, and the joint homomorphic image, in particular, defines the potentially more useful approach, primarily because homomorphic images of the semigroup may be given relational referents in the generating blockmodel.