Group relationships and homomorphisms of Boolean matrix semigroups
Let A be a Boolean n-square matrix. Taking a partition of the set \(\{\) 1,...,n\(\}\) and dividing the matrix into blocks leads to the so called image matrix by replacing each zero block by a single zero and each nonzero block by a single one. Taking the image matrix is always a homomorphism under Boolean addition. The question is when it is a multiplicative homomorphism. Some sufficient condition for this is given. This condition leads to the consideration of nonzero antichains of Boolean vectors. It is obtained as corollary that the maximal nonzero n- square Boolean matrix semigroups are precisely the following: \(S_ C=\{n\)-square Boolean matrix A: (\(\forall v\in C)\) (\(\exists w\in C)\) vA\(\geq w\}\), where C is a nonzero antichain of n-vectors. The maximal invariant under conjugations by a permutation matrix nonzero semigroups of Boolean matrices have the form: \(G_ i=\{A:\) the sum of any i rows (or all rows if \(i\geq n)\) of A has at least i ones (or all ones)\(\}\). The authors believe that their results cover all known conditions for the image to be a homomorphism. All the subject appears by analyzing small group structures in Sociology.
scientific article; zbMATH DE number 3908708
scientific article