As long as the different networks in the populations being compared may be matched between the two social structures, the blockmodel algebras of the two populations may both be embedded in the congruence lattice of the same free semigroup. Common membership in this lattice provides a natural basis for comparing the algebras with reference to the length of the lattice path connecting the two given algebras via their lattice union. This lattice union, named the 'joint reduction' by B&W, is also a semigroup, so that the comparative information derived from the approach is structural and relational as well as metric. This algebraic strategy furnishes a direct method for comparing or contrasting social structures arising in radically different environments. The social groups compared may be of different sizes, and the blockmodels used to describe them may have different number of blocks. Bonacich objects to basing a comparison of two blockmodel algebras on joint reduction and observes a case in which the B&W joint reduction fails to be 'generator-preserving.' This paper answers these two criticisms. Figures, equations, and about 26 references are included.