Labeled reaction matrices, a histocompatibility model

Abstract

A labeled reaction (or recognition) matrix is a triple ( A, ø, B) where A and B are zero-one (0–1) matrices and φ is a certain relation between the rows and columns of A and B. In the application in this paper B defines the antigens and antibodies that play a role in some histocompatibility experiment by listing specificities. A represents data observed from testing cells against sera in the experiment. Antigens or antibodies (not necessarily monospecific) whose action in the experiment can be isolated are called monic. ( A, φ, B) is called monic if all antigens and antibodies are monic and B is reduced. A partial order ⩽ is put on the collection of 0–1 matrices and it is shown that if ( A, φ, B) is any labeled reaction matrix, then A ω ⩽ B where A ω is the reduction of A. An algorithm for obtaining A ω (that gives a labeling of A) is provided. If ( A, φ, B) is monic, then A ω and B are identical (up to a permutation of rows and columns) and the labeling of cells and sera is essentially unique.