This paper discusses a generic decomposition model that represents an arbit
rary n-ary relation as a disjunctive or conjunctive combination of a number
of n-ary component relations of a prespecified type. An important subclass
of order-preserving decompositions is defined and its properties are deriv
ed. The generic model is shown to subsume various known models as special c
ases, including the models of Boolean factor analysis, hierarchical classes
analysis, and disjunctive/conjunctive nonmetric factor analysis Moreover,
it also subsumes a broad range of new models as exemplified with a novel mo
del for multidimensional parallelogram analysis and novel three-way extensi
ons of nonmetric factor analysis. (C) 1999 Academic Press.