Logical matrix: Difference between revisions

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In fact, ''U'' forms a [[Boolean algebra]] with the operations [[and (logic)|and]] & [[or (logic)|or]] between two matrices applied component-wise. The complement of a logical matrix is obtained by swapping all zeros and ones for their opposite.

Every logical matrix {{nowrap|1=a = ( a _{i j} )}} has ana '''transpose''' {{nowrap|1=a^T = ( a _{j i} ).}} Suppose ''a'' is a logical matrix with no columns or rows identically zero. Then the matrix product, using Boolean arithmetic, a<supmath>a^{\operatorname{T}}a</supmath> a contains the ''m'' × ''m'' [[identity matrix]], and the product a a<supmath>aa^{\operatorname{T}}</supmath> contains the ''n'' × ''n'' identity.

As a mathematical structure, the Boolean algebra ''U'' forms a [[lattice (order)|lattice]] ordered by [[inclusion (logic)|inclusion]]; additionally it is a '''multiplicative lattice''' due to matrix multiplication.

Every logical matrix in ''U'' corresponds to a binary relation. These listed operations on ''U'', and ordering, correspond to a [[algebraic logic#Calculus of relations|calculus of relations]], where the matrix multiplication represents [[composition of relations]].<ref>[[Irving Copilowish]] (December 1948). "Matrix development of the calculus of relations", [[Journal of Symbolic Logic]] 13(4): 193–203 [https://www.jstor.org/stable/2267134?seq=1#page_scan_tab_contents Jstor link]</ref>