scholarly journals Linear Operators That Preserve Arctic Ranks of Boolean Matrices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1087
Author(s):  
Kyung-Tae Kang ◽  
Seok-Zun Song
Keyword(s):  
Boolean Matrix ◽  
Linear Operators ◽  
The Arctic ◽  
Matrix Space ◽  
Boolean Rank ◽  
Term Rank

We study some properties of arctic rank of Boolean matrices. We compare the arctic rank with Boolean rank and term rank of a given Boolean matrix. Furthermore, we obtain some characterizations of linear operators that preserve arctic rank on Boolean matrix space.

scholarly journals Linear operators that preserve zero-term rank over fields and rings

2002 ◽  
Vol 341 (1-3) ◽  
pp. 143-149 ◽  
Author(s):  
Leroy B. Beasley ◽  
Seok-Zun Song
Keyword(s):  
Linear Operators ◽  
Term Rank

scholarly journals Linear Maps that Preserve Any Two Term Ranks on Matrix Spaces over Anti-Negative Semirings

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 41
Author(s):  
Kyung Tae Kang ◽  
Seok-Zun Song ◽  
Young Bae Jun
Keyword(s):  
Linear Operators ◽  
Linear Map ◽  
Linear Maps ◽  
Term Rank ◽  
Q Matrix ◽  
Matrix Spaces

There are many characterizations of linear operators from various matrix spaces into themselves which preserve term rank. In this research, we characterize the linear maps which preserve any two term ranks between different matrix spaces over anti-negative semirings, which extends the previous results on characterizations of linear operators from some matrix spaces into themselves. That is, a linear map T from p × q matrix spaces into m × n matrix spaces preserves any two term ranks if and only if T preserves all term ranks if and only if T is a ( P , Q , B )-block map.

scholarly journals The arctic rank of a Boolean matrix

2015 ◽  
Vol 433 ◽  
pp. 168-182 ◽  
Author(s):  
LeRoy B. Beasley ◽  
Alexander E. Guterman ◽  
Yaroslav Shitov
Keyword(s):  
Boolean Matrix ◽  
The Arctic

Preservers of term ranks and star cover numbers of symmetric matrices

2016 ◽  
Vol 31 ◽  
pp. 549-564 ◽  
Author(s):  
LeRoy Beasley
Keyword(s):  
Linear Operators ◽  
Symmetric Matrices ◽  
Minimum Number ◽  
Term Rank

Let $\S$ denote the set of symmetric matrices over some semiring, $\s$. A line of $A\in\S$ is a row or a column of $A$. A star of $A$ is the submatrix of $A$ consisting of a row and the corresponding column of $A$. The term rank of $A$ is the minimum number of lines that contain all the nonzero entries of $A$. The star cover number is the minimum number of stars that contain all the nonzero entries of $A$. This paper investigates linear operators that preserve sets of symmetric matrices of specified term rank and sets of symmetric matrices of specific star cover numbers. Several equivalences to the condition that $T$ preserves the term rank of any matrix are given along with characterizations of a couple of types of linear operators that preserve certain sets of matrices defined by the star cover number that do not preserve all term ranks.

Factor rank preservers of matrices over additively-idempotent multiplicatively-cancellative semirings

2020 ◽  
pp. 2150072
Author(s):  
Sushobhan Maity ◽  
A. K. Bhuniya
Keyword(s):  
Boolean Algebra ◽  
Linear Algebra ◽  
Linear Operators ◽  
Multilinear Algebra ◽  
Boolean Rank ◽  
Max Algebra ◽  
Linear Multilinear Algebra

Here, we characterize the linear operators that preserve factor rank of matrices over additively-idempotent multiplicatively-cancellative semirings. The main results in this paper generalize the corresponding results on the two element Boolean algebra [L. B. Beasley and N. J. Pullman, Boolean-rank-preserving opeartors and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984) 55–77] and on the max algebra [R. B. Bapat, S. Pati and S.-Z. Song, Rank preservers of matrices over max algebra, Linear Multilinear Algebra 48(2) (2000) 149–164]; and hold on max-plus algebra and some other tropical semirings.

scholarly journals Linear operators that preserve Boolean rank of Boolean matrices

2013 ◽  
Vol 63 (2) ◽  
pp. 435-440 ◽  
Author(s):  
LeRoy B. Beasley ◽  
Seok-Zun Song
Keyword(s):  
Linear Operators ◽  
Boolean Rank

scholarly journals Zero-term rank inequalities and their extreme preservers

Filomat ◽  
2014 ◽  
Vol 28 (9) ◽  
pp. 1827-1833
Author(s):  
Seok-Zun Song ◽  
Seong-Hee Heo
Keyword(s):  
Boolean Algebras ◽  
Linear Operators ◽  
Term Rank ◽  
Rank Of A Matrix ◽  
Ordered Pairs ◽  
Extremal Properties

The zero-term rank of a matrix A over a semiring S is the least number of lines (rows or columns) needed to include all the zero entries in A. In this paper, we characterize linear operators that preserve the sets of matrix ordered pairs which satisfy extremal properties with respect to zero-term rank inequalities of matrices over nonbinary Boolean algebras.

The Arctic Climate System

2005 ◽  
Author(s):  
Mark C. Serreze ◽  
Roger G. Barry
Keyword(s):  
Climate System ◽  
Arctic Climate ◽  
The Arctic
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