Geodesic
On cofactor expansion of determinants of Boolean matrices
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 199-223 
V. B. Poplavskii. On cofactor expansion of determinants of Boolean matrices. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 199-223. http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a9/
@article{FPM_2007_13_4_a9,
     author = {V. B. Poplavskii},
     title = {On cofactor expansion of determinants of {Boolean} matrices},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {199--223},
     year = {2007},
     volume = {13},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a9/}
}
TY  - JOUR
AU  - V. B. Poplavskii
TI  - On cofactor expansion of determinants of Boolean matrices
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2007
SP  - 199
EP  - 223
VL  - 13
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a9/
LA  - ru
ID  - FPM_2007_13_4_a9
ER  - 
%0 Journal Article
%A V. B. Poplavskii
%T On cofactor expansion of determinants of Boolean matrices
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2007
%P 199-223
%V 13
%N 4
%U http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a9/
%G ru
%F FPM_2007_13_4_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

We give necessary and sufficient conditions of a cofactor expansibility of determinants along a row or column for Boolean square matrices over an arbitrary Boolean algebra. First of all we define a natural decomposition of an arbitrary Boolean matrix by interior, exterior, and determinate parts. The introduced notions allow us to establish the main result of this paper. It is shown that the formulas of the cofactor expansion along a row (column) of determinants of an arbitrary square Boolean matrix hold if and only if the formulas of the cofactor expansion along the corresponding row (column) hold for determinants of its interior part.

[8] Poplavskii V. B., “Obratimye i prisoedinennye bulevy matritsy”, Chebyshevskii sb., 6:1 (2005), 174–181 | MR | Zbl

[9] Skornyakov L. A., “Obratimye matritsy nad distributivnymi strukturami”, Sib. mat. zhurn., 27:2 (1986), 182–185 | MR | Zbl

[10] Chesley D. S., Bevis J. H., “Determinants for matrices over lattices”, Proc. Roy. Soc. Edinburgh Sect. A, 68:2 (1969), 138–144 | MR | Zbl

[11] Golan J. S., Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999 | MR

[12] Poplavski V., “On orientability and degeneration of Boolean binary relation on a finite set”, Mathematical Logic in Asia, Proc. 9th Asian Logic Conf. (Novosibirsk, Russia, 16–19 August 2005), World Scientific, Singapore, 2006, 203–214 | MR | Zbl

[13] Poplin P. L., Hartwig R. E., “Determinantal identities over commutative semirings”, Linear Algebra Appl., 387 (2004), 99–132 | DOI | MR | Zbl

[14] Reutenauer C., Straubing H., “Inversion of matrices over a commutative semiring”, J. Algebra, 88 (1984), 350–360 | DOI | MR | Zbl

[15] Rutherford D. E., “Inverses of Boolean matrices”, Proc. Glasgow Math. Assoc., 6:1 (1963), 49–53 | DOI | MR | Zbl