Group relationships and homomorphisms of Boolean matrix semigroups
type: Item
instance of: scholarly article
title: Group relationships and homomorphisms of Boolean matrix semigroups
zbMATH Open document ID: 0569.20052
DOI: 10.1016/0022-2496(84)90011-7
published in: Journal of Mathematical Psychology
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 91D99
Mathematics Subject Classification ID: 20M20
zbMATH DE Number: 3908708
zbMATH Keywords: Boolean n-square matrix
zbMATH Keywords: image matrix
zbMATH Keywords: homomorphism
zbMATH Keywords: Boolean addition
zbMATH Keywords: antichains of Boolean vectors
zbMATH Keywords: Boolean matrix semigroups
author: Ki Hang Kim
author: Fred Roush
reviewed by: Grigori I. Zhitomirski
MaRDI profile type: Publication
full work available at URL: 0022-2496(84)90011-7
OpenAlex ID: W2069758399
cites work: Q3958592
cites work: The analysis of semigroups of multirelational systems
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description: scientific article; zbMATH DE number 3908708