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We describe optimal decompositions of an n m matrix I into a triangular product I ¼ A / B of an n k matrix A and a k m matrix B. We assume that the matrix entries are elements of a residuated lattice, which leaves binary matrices or matrices which contain numbers from the unit interval [0, 1] as special cases. The entries of I, A, and B represent grades to which objects have attributes, factors apply to objects, and attributes are particular manifestations of factors, respectively. This way,doi:10.1016/j.ijar.2009.05.006 fatcat:ujyrmzpffnd7xo6wopgathtdpi