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2. Linear maps that strongly preserve regular matrices over the Boolean algebra
- Creator:
- Kang, Kyung-Tae and Song, Seok-Zun
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Boolean algebra, regular matrix, and $(U,V)$-operator
- Language:
- English
- Description:
- The set of all $m\times n$ Boolean matrices is denoted by ${\mathbb M}_{m,n}$. We call a matrix $A\in {\mathbb M}_{m,n}$ regular if there is a matrix $G\in {\mathbb M}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${\mathbb M}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $\min \{m,n\}\le 2$, then all operators on ${\mathbb M}_{m,n}$ strongly preserve regular matrices, and if $\min \{m,n\}\ge 3$, then an operator $T$ on ${\mathbb M}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T(X)=UXV$ for all $X\in {\mathbb M}_{m,n}$, or $m=n$ and $T(X)=UX^TV$ for all $X\in {\mathbb M}_{n}$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Perimeter preserver of matrices over semifields
- Creator:
- Song, Seok-Zun, Kang, Kyung-Tae, and Jun, Young-Bae
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- linear operator, rank, dominate, perimeter, and $(U,V)$-operator
- Language:
- English
- Description:
- For a rank-$1$ matrix $A= {\bold a \bold b}^t$, we define the perimeter of $A$ as the number of nonzero entries in both $\bold a$ and $\bold b$. 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public