Boolean matrix
Introduction In mathematics, a '''Boolean matrix''' is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a logical matrix. (In some contexts, particularly computer science, the term "Boolean matrix" implies this restriction.) Let ''U'' be a non-trivial Boolean algebra (i.e. with at least two elements). Intersection, union, complementation, and containment of elements is expressed in ''U''. Let ''V'' be the collection of ''n'' × ''n'' matrices that have entries taken from ''U''. Complementation of such a matrix is obtained by complementing each element. The intersection or union of two such matrices is obtained by applying the operation to entries of each pair of elements to obtain the corresponding matrix intersection or union. A matrix is contained in another if each entry of the first is contained in the corresponding entry of the second.The product of two Boolean matrices is expressed as follows:According to one author, "Matrices over an arbitrary Boolean algebra β satisfy most of the properties over β0 = {0, 1}. The reason is that any Boolean algebra is a sub-Boolean algebra of  for some set ''S'', and we have an isomorphism from ''n'' × ''n'' matrices over "
References * R. Duncan Luce (1952) "A Note on Boolean Matrices", Proceedings of the American Mathematical Society '''3''': 382–8,  Jstor link * Jacques Riguet (1954) "Sur l'extension du calcul des relations binaires au calcul des matrices à éléments dans une algèbre de Boole", Comptes Rendus '''238''': 2382–2385
Further reading * Stan Gudder & Frédéric Latrémolière (2009) "Boolean inner-product spaces and Boolean matrices", Linear Algebra and Its Applications '''431''': 274–96 * D.E. Rutherford (1963) "Inverses of Boolean matrices", ''Proceedings of the Glasgow Mathematical Association'' '''6''': 49–63 * T.S. Blythe (1967) "Eigenvectors of Boolean Matrices", Proceedings of the Royal Society of Edinburgh '''67''': 196–204 * Steven Kirkland & Norman J. Pullman (1993) "Linear Operators Preserving Invariants of Non-binary Boolean Matrices", ''Linear and Multilinear Algebra'' '''33''': 295–300  * Kyung-Kae Kang, Seok-Zun Song & Young-Bae Jung (2011) "Linear Preservers of Regular Matrices over General Boolean Algebras", ''Bulletin of the Malaysian Mathematical Sciences Society'', second series, '''34'''(1): 113–25