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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Lemma**

**1.**

**Proof.**

- $p\le m$ and $q\le n$, and $T\left(U\right)=P\left[\right(U\circ B)\oplus O]Q$ for any $U\in {\mathbb{M}}_{p,q}\left(\mathbb{S}\right)$ or
- $p\le n$ and $q\le m$, and $T\left(U\right)=P[{(U\circ B)}^{t}\oplus O]Q$ for any $U\in {\mathbb{M}}_{p,q}\left(\mathbb{S}\right)$.

## 3. Linear Maps that Preserve TR of Matrices over Anti-Negative Commutative Semirings

- (i)
- preserves TR t if $\tau \left(T\right(U\left)\right)=t$ whenever $\tau \left(U\right)=t$ for all $U\in {\mathbb{M}}_{p,q}\left(\mathbb{S}\right)$;
- (ii)
- doubly (or strongly) preserves TR t if $\tau \left(T\right(U\left)\right)=t$ if and only if $\tau \left(U\right)=t$ for all $U\in {\mathbb{M}}_{p,q}\left(\mathbb{S}\right)$;
- (iii)
- preserves TR if it preserves any TR t with $t\le p$.

**Lemma**

**2.**

**Theorem**

**1.**

**Lemma**

**3.**

**Lemma**

**4.**

**Lemma**

**5.**

**Lemma**

**6.**

**Lemma**

**7.**

**Lemma**

**8.**

**Lemma**

**9.**

**Lemma**

**10.**

**Lemma**

**11.**

**Theorem**

**2.**

- T preserves any two TR t and TR s, with $t<s$ and $(t+1)<p;$
- T doubly preserves any one TR t, with $1\le t\le p;$
- T preserves TR;
- T is the $(P,Q,B)$-block map.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Beasley, L.B.; Pullman, N.J. Term-rank, permanent, and rook-polynomial preservers. Linear Algebra Appl.
**1987**, 90, 33–46. [Google Scholar] [CrossRef][Green Version] - Kang, K.T.; Song, S.Z.; Beasley, L.B. Linear preservers of term ranks of matrices over semirings. Linear Algebra Appl.
**2012**, 436, 1850–1862. [Google Scholar] [CrossRef][Green Version] - Song, S.Z.; Beasley, L.B. Linear transformations that preserve term rank between different matrix spaces. J. Korean Math. Soc.
**2013**, 50, 127–136. [Google Scholar] [CrossRef] - Beasley, L.B. Preservers of term ranks and star cover numbers of symmetric matrices. Electron. J. Linear Algebra
**2016**, 31, 549–564. [Google Scholar] [CrossRef][Green Version] - Beasley, L.B.; Song, S.Z.; Kang, K.T. Preservers of term ranks of symmetric matrices. Linear Algebra Appl.
**2012**, 436, 1727–1738. [Google Scholar] [CrossRef][Green Version] - Beasley, L.B.; Song, S.Z. Linear operators that preserve term ranks of matrices over semirings. Bull. Malays. Math. Sci. Soc.
**2014**, 37, 719–725. [Google Scholar] - Kang, K.T.; Song, S.Z.; Jun, Y.B. Linear operators that strongly preserve regularity of fuzzy matrices. Math. Commun.
**2010**, 15, 243–254. [Google Scholar]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kang, K.T.; Song, S.-Z.; Jun, Y.B. Linear Maps that Preserve Any Two Term Ranks on Matrix Spaces over Anti-Negative Semirings. *Mathematics* **2020**, *8*, 41.
https://doi.org/10.3390/math8010041

**AMA Style**

Kang KT, Song S-Z, Jun YB. Linear Maps that Preserve Any Two Term Ranks on Matrix Spaces over Anti-Negative Semirings. *Mathematics*. 2020; 8(1):41.
https://doi.org/10.3390/math8010041

**Chicago/Turabian Style**

Kang, Kyung Tae, Seok-Zun Song, and Young Bae Jun. 2020. "Linear Maps that Preserve Any Two Term Ranks on Matrix Spaces over Anti-Negative Semirings" *Mathematics* 8, no. 1: 41.
https://doi.org/10.3390/math8010041