Geodesic
On linear operators strongly preserving invariants of Boolean matrices
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 169-186
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Let $\mathbb {B}_{k}$ be the general Boolean algebra and $T$ a linear operator on $M_{m,n}(\mathbb {B}_{k})$. If for any $A$ in $M_{m,n}(\mathbb {B}_{k})$ ($ M_{n}(\mathbb {B}_{k})$, respectively), $A$ is regular (invertible, respectively) if and only if $T(A)$ is regular (invertible, respectively), then $T$ is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over $\mathbb {B}_{k}$. Meanwhile, noting that a general Boolean algebra $\mathbb {B}_{k}$ is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertible, respectively) matrices over $\mathbb {B}_{k}$ from another point of view.
Let $\mathbb {B}_{k}$ be the general Boolean algebra and $T$ a linear operator on $M_{m,n}(\mathbb {B}_{k})$. If for any $A$ in $M_{m,n}(\mathbb {B}_{k})$ ($ M_{n}(\mathbb {B}_{k})$, respectively), $A$ is regular (invertible, respectively) if and only if $T(A)$ is regular (invertible, respectively), then $T$ is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over $\mathbb {B}_{k}$. Meanwhile, noting that a general Boolean algebra $\mathbb {B}_{k}$ is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertible, respectively) matrices over $\mathbb {B}_{k}$ from another point of view.
DOI : 10.1007/s10587-012-0004-y
Classification : 06E05, 15A04, 15A09, 15A86, 15B34, 16Y60
Keywords: linear operator; invariant; regular matrix; invertible matrix; general Boolean algebra 
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Chen, Yizhi; Zhao, Xianzhong. On linear operators strongly preserving invariants of Boolean matrices. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 169-186. doi: 10.1007/s10587-012-0004-y

[1] Bapat, R. B.: Structure of a nonnegative regular matrix and its generalized inverses. Linear Algebra Appl. 268 (1998), 31-39. | MR | Zbl

[2] Beasley, L. B., Guterman, A. E., Lee, S.-G., Song, S.-Z.: Linear transformations preserving the Grassmannian over $M_{n}(Z_{+})$. Linear Algebra Appl. 393 (2004), 39-46. | MR

[3] Beasley, L. B., Guterman, A. E.: The characterization of operators preserving primitivity for matrix $k$-tuples. Linear Algebra Appl. 430 (2009), 1762-1777. | MR | Zbl

[4] Beasley, L. B., Lee, S. G.: Linear operations strongly preserving $r$-potent matrices over semirings. Linear Algebra Appl. 162-164 (1992), 589-599. | MR

[5] Beasley, L. B., Pullman, N. J.: Boolean rank preserving operators and Boolean rank-1 spaces. Linear Algebra Appl. 59 (1984), 55-77. | DOI | MR | Zbl

[6] Beasley, L. B., Pullman, N. J.: Operators that preserve semiring matrix functions. Linear Algebra Appl. 99 (1988), 199-216. | DOI | MR | Zbl

[7] Beasley, L. B., Pullman, N. J.: Fuzzy rank-preserving operators. Linear Algebra Appl. 73 (1986), 197-211. | DOI | MR | Zbl

[8] Beasley, L. B., Pullman, N. J.: Linear operators strongly preserving idempotent matrices over semirings. Linear Algebra Appl. 160 (1992), 217-229. | MR | Zbl

[9] Dénes, J.: Transformations and transformation semigroups I. Seminar Report. Magyar Tud. Akad., Mat. Fiz. Tud. Oszt. Közl. 19 (1969), 247-269 Hungarian. | MR

[10] Golan, J. S.: Semirings and Their Applications. Kluwer Dordrecht (1999). | MR | Zbl

[11] Kim, K. H.: Boolean Matrix Theory and Applications. Pure Appl. Math., Vol. 70 Marcel Dekker New York (1982). | MR | Zbl

[12] Kang, K.-T., Song, S.-Z., Jun, Y.-B.: Linear operators that strongly preserve regularity of fuzzy matrices. Math. Commun. 15 (2010), 243-254. | MR | Zbl

[13] Kirkland, S., Pullman, N. J.: Linear operators preserving invariants of non-binary Boolean matrices. Linear Multilinear Algebra 33 (1993), 295-300. | DOI | MR

[14] Li, H. H., Tan, Y. J., Tang, J. M.: Linear operators that strongly preserve invertible matrices over antinegative semirings. J. Univ. Sci. Technol. China 37 (2007), 238-242. | MR | Zbl

[15] Luce, R. D.: A note on Boolean matrix theory. Proc. Am. Math. Soc. 3 (1952), 382-388. | DOI | MR | Zbl

[16] Orel, M.: Nonbijective idempotents preservers over semirings. J. Korean Math. Soc. 47 (2010), 805-818. | DOI | MR | Zbl

[17] Plemmons, R. J.: Generalized inverses of Boolean relation matrices. SIAM J. Appl. Math. 20 (1971), 426-433. | DOI | MR | Zbl

[18] Pshenitsyna, O. A.: Maps preserving invertibility of matrices over semirings. Russ. Math. Surv. 64 (2009), 162-164. | DOI | MR | Zbl

[19] Rao, P. S. S. N. V. P., Rao, K. P. S. B.: On generalized inverses of Boolean matrices. Linear Algebra Appl. 11 (1975), 135-153. | DOI | MR | Zbl

[20] Rutherford, D. E.: Inverses of Boolean matrices. Proc. Glasg. Math. Assoc. 6 (1963), 49-53. | DOI | MR | Zbl

[21] Song, S.-Z., Beasley, L. B., Cheon, G. S., Jun, Y.-B.: Rank and perimeter preservers of Boolean rank-1 matrices. J. Korean Math. Soc. 41 (2004), 397-406. | DOI | MR | Zbl

[22] Song, S.-Z., Kang, K.-T., Jun, Y.-B.: Linear preservers of Boolean nilpotent matrices. J. Korean Math. Soc. 43 (2006), 539-552. | DOI | MR | Zbl

[23] Song, S.-Z., Kang, K.-T., Beasley, L. B., Sze, N.-S.: Regular matrices and their strong preservers over semirings. Linear Algebra Appl. 429 (2008), 209-223. | MR | Zbl

[24] Song, S.-Z., Kang, K.-T., Beasley, L. B.: Idempotent matrix preservers over Boolean algebras. J. Korean Math. Soc. 44 (2007), 169-178. | DOI | MR | Zbl

[25] Song, S.-Z., Kang, K.-T., Kang, M.-H.: Boolean regular matrices and their strongly preservers. Bull. Korean Math. Soc. 46 (2009), 373-385. | DOI | MR | Zbl

[26] Song, S.-Z., Lee, S.-G.: Column ranks and their preservers of general Boolean matrices. J. Korean Math. Soc. 32 (1995), 531-540. | MR | Zbl

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