# Necessary and sufficient conditions for a phenotype system to have a factor-union representation

## Abstract

This paper presents an algorithm for determining whether a phenotype system has a factor-union representation, i.e., whether sets of properties (factors) can be assigned to alleles so that the phenotype of each genotype is determined by the union of the factor sets of the alleles. If such a representation is possible, the algorithm can construct one using p or fewer factors, where p is the number of phenotypes. Refinements to the algorithm are presented that often construct representations using few factors. An example is presented to show that some phenotype systems can be represented only by using almost as many factors as there are phenotypes in the system. Finally, it is shown that the problem of finding factor-union representations for multilocus systems reduces to the problem for 1-locus systems. © 1983.