Logical matrix: Difference between revisions

Revision as of 18:03, 3 November 2022

Revision as of 18:05, 3 November 2022

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==Lattice==

Let ''n'' and ''m'' be given and let ''U'' denote the set of all logical ''m'' × ''n'' matrices. Then ''U'' has a [[partial order]] given by

:<math>m\forall A,B \in U, \quad A \subsetleq nB \quad \text{when} \quad \forall i,j \quad m_A_{ij} = 1 \implies n_B_{ij} = 1 .</math>

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