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Orthogonality and complementation in the lattice of subspaces of a finite vector space

Ivan Chajda, Helmut Länger (2022)

Mathematica Bohemica

We investigate the lattice 𝐋 ( 𝐕 ) of subspaces of an m -dimensional vector space 𝐕 over a finite field GF ( q ) with a prime power q = p n together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice 𝐋 ( 𝐕 ) satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when 𝐋 ( 𝐕 ) is orthomodular. For...

Pairs of k -step reachability and m -step observability matrices

Augusto Ferrante, Harald K. Wimmer (2013)

Special Matrices

Let V and W be matrices of size n × pk and qm × n, respectively. A necessary and sufficient condition is given for the existence of a triple (A,B,C) such that V a k-step reachability matrix of (A,B) andW an m-step observability matrix of (A,C).

Partial choice functions for families of finite sets

Eric J. Hall, Saharon Shelah (2013)

Fundamenta Mathematicae

Let m ≥ 2 be an integer. We show that ZF + “Every countable set of m-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector...

Perimeter preserver of matrices over semifields

Seok-Zun Song, Kyung-Tae Kang, Young Bae Jun (2006)

Czechoslovak Mathematical Journal

For a rank- 1 matrix A = 𝐚 𝐛 t , we define the perimeter of A as the number of nonzero entries in both 𝐚 and 𝐛 . We characterize the linear operators which preserve the rank and perimeter of rank- 1 matrices over semifields. That is, a linear operator T preserves the rank and perimeter of rank- 1 matrices over semifields if and only if it has the form T ( A ) = U A V , or T ( A ) = U A t V with some invertible matrices U and V.

Possible isolation number of a matrix over nonnegative integers

LeRoy B. Beasley, Young Bae Jun, Seok-Zun Song (2018)

Czechoslovak Mathematical Journal

Let ℤ + be the semiring of all nonnegative integers and A an m × n matrix over ℤ + . The rank of A is the smallest k such that A can be factored as an m × k matrix times a k × n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A . For A with isolation number k , we investigate the possible values of the rank of A ...

Ranks of permutative matrices

Xiaonan Hu, Charles R. Johnson, Caroline E. Davis, Yimeng Zhang (2016)

Special Matrices

A new type of matrix, termed permutative, is defined and motivated herein. The focus is upon identifying circumstances under which square permutative matrices are rank deficient. Two distinct ways, along with variants upon them are given. These are a special kind of grouping of rows and a type of partition in which the blocks are again permutative. Other, results are given, along with some questions and conjectures.

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