mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Boolean matrix is a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

with entries from a

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

. When the

two-element Boolean algebra In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose ''underlying set'' (or universe or ''carrier'') ''B'' is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that ''B ...

is used, the Boolean matrix is called a

logical matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix (mathematics), matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. ...

. (In some contexts, particularly

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

, 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: :

(AB)_ = \bigcup_^n (A_ \cap B_ ) .

According to one author, "Matrices over an arbitrary Boolean algebra β satisfy most of the properties over β₀ = . The reason is that any Boolean algebra is a sub-Boolean algebra of

\beta_0^S

for some set ''S'', and we have an isomorphism from ''n'' × ''n'' matrices over

\beta_0^S \ \text \ \beta_n^S .

"Ki Hang Kim (1982) ''Boolean Matrix Theory and Applications'', page 249, Appendix: Matrices over arbitrary Boolean Algebras,

Marcel Dekker Marcel Dekker was a journal and encyclopedia publishing company with editorial boards found in New York City. Dekker encyclopedias are now published by CRC Press, part of the Taylor and Francis publishing group. History Initially a textbook p ...

References

* R. Duncan Luce (1952) "A Note on Boolean Matrices",

Proceedings of the American Mathematical Society ''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ' ...

3: 382–8
Jstor link
*

Jacques Riguet Jacques Riguet (1921 to October 20, 2013) was a French mathematician known for his contributions to algebraic logic and category theory. According to Gunther Schmidt and Thomas Ströhlein, "Alfred Tarski and Jacques Riguet founded the modern calcul ...

(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

References

Further reading