Boolean spectral theory

AbstractThe fact that there is an isomorphism between the matrices over the Boolean algebra of subsets of a k-element set and the k-tuples of Boolean binary [i.e. (0, 1)] matrices appears to have deterred the presentation of solutions, if not the consideration of problems, concerning general (i.e. nonbinary) Boolean matrices. We try to point out that there are interesting features of the general case that have been lost from view. We reopen the investigation of eigenvalues and their corresponding eigenspaces for matrices over an arbitrary finite Boolean algebra B, answering several questions raised by the pioneering work of D.E. Rutherford and providing concise proofs of some of his results. We also extend a result of Wedderburn characterizing the bases of the vector space of n-tuples over B to a characterization of the bases of all vector spaces over B. Using this characterization of bases, we present a straightforward way to find a basis for the eigenspace of each eigenvalue