On ranks, Green classes, and the theory of determinants of Boolean matrices
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V. B. Poplavskii
Abstract
We consider the groupoid of all possible matrices over an arbitrary Boolean algebra with partial operation of matrix product. On this groupoid, we define the equivalence classes analogous to the Green classes H, C, R, D, J for semigroups. We introduce the notion of the minor rank of a Boolean matrix. We show that the column, row, factorisation and minor ranks are invariants for the J -class of this groupoid, and the minor ranks do not exceed the column, row, factorisation and permanent ranks.
The key result of this work explains the role of the Boolean determinant. We show that in some J -class there exists a square n × n matrix with nonzero determinant if and only if the column, row, factorisation and minor ranks of any matrix of this class are equal to each other and equal to n. All n × n matrices of this J -class have equal determinants, while the determinants of the square matrices of greater size are equal to zero.
© de Gruyter 2008
Articles in the same Issue
- On stability of a vector combinatorial problem with MINMIN criteria
- On the asymptotic behaviour of the probability of existence of equivalent tuples with nontrivial structure in a random sequence
- Characteristics of random systems of linear equations over a finite field
- On the realisation of Boolean functions by informational graphs
- Estimates of the number of occurrences of vectors on cycles of linear recurring sequences over a finite field
- Finite generability of some groups of recursive permutations
- Independent systems of generators and the Hopf property for unary algebras
- Estimates of the complexity of approximation of continuous functions in some classes of determinate functions with delay
- On ranks, Green classes, and the theory of determinants of Boolean matrices
Articles in the same Issue
- On stability of a vector combinatorial problem with MINMIN criteria
- On the asymptotic behaviour of the probability of existence of equivalent tuples with nontrivial structure in a random sequence
- Characteristics of random systems of linear equations over a finite field
- On the realisation of Boolean functions by informational graphs
- Estimates of the number of occurrences of vectors on cycles of linear recurring sequences over a finite field
- Finite generability of some groups of recursive permutations
- Independent systems of generators and the Hopf property for unary algebras
- Estimates of the complexity of approximation of continuous functions in some classes of determinate functions with delay
- On ranks, Green classes, and the theory of determinants of Boolean matrices
