Boolean matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Boolean matrix is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> with entries from a <a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebra</a>. When the <a href="/facts/Two-element_Boolean_algebra/8nIWUpwd">two-element Boolean algebra</a> is used, the Boolean matrix is called a <a href="/facts/Logical_matrix/pS4BnBzj">logical matrix</a>. (In some contexts, particularly <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, 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:

(
        A
        B
        
          )
          
            i
            j
          
        
        =
        
          ⋃
          
            k
            =
            1
          
          
            n
          
        
        (
        
          A
          
            i
            k
          
        
        ∩
        
          B
          
            k
            j
          
        
        )
        .
      
    
    {\displaystyle (AB)_{ij}=\bigcup _{k=1}^{n}(A_{ik}\cap B_{kj}).}

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 
 
 
 
 
 β
 
 0
 
 
 S
 
 
 
 
 {\displaystyle \beta _{0}^{S}}
 
 for some set S, and we have an isomorphism from n × n matrices over 
 
 
 
 
 β
 
 0
 
 
 S
 
 
  
 
 to
 
  
 
 β
 
 n
 
 
 S
 
 
 .
 
 
 {\displaystyle \beta _{0}^{S}\ {\text{to}}\ \beta _{n}^{S}.}
 
"

<ul><li><a href="/facts/R._Duncan_Luce/QSJFYkIM">R. Duncan Luce</a> (1952) "A Note on Boolean Matrices", <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a> 3: 382–8, <a href="https://www.jstor.org/stable/2031888?seq=1#page_scan_tab_contents">Jstor link</a> <a href="/facts/MR_(identifier)/uP137L11">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0050559">0050559</a></li>
<li><a href="/facts/Jacques_Riguet/z8aRuzPk">Jacques Riguet</a> (1954) "Sur l'extension du calcul des relations binaires au calcul des matrices à éléments dans une algèbre de Boole", <a href="/facts/Comptes_Rendus/9yR9XTN5">Comptes Rendus</a> 238: 2382–2385</li></ul>

Boolean matrix open-in-new

Boolean matrix