Geodesic
Augmented polynomial matrices and algebraization of switching circuits
Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 85-93.

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

Over rings of polynomials with idempotent variables (over arbitrary fields) there are defined classes of augmented matrices (with one distinguished column) that realize Boolean functions. In the latter classes of augmented matrices (over any fields) there is defined a system of equivalent transformations (preserving realized Boolean functions) that generalizes the known system of elementary transformations (of rows and columns) of usual polynomial matrices. It is proved the completeness of this system for the simplest (binary) case – in the class of augmented matrices over the ring of Zhegalkin polynomials. In particular, there is given a method for reducing of an arbitrary augmented matrix over the ring of Zhegalkin polynomials by means of this system to a uniquely determined one-element form. For the same (binary) case, it is shown that the class of binary incidence matrixes of switching circuits is, in essence, a subclass of the class of augmented matrices over the ring of Zhegalkin polynomials. This reveals the simplest «completely algebraic» extension of the class of switching circuits – one of the basic model classes of mathematical theory of control systems.
Keywords: polynomial with idempotent variables; augmented polynomial matrix; full reverse metamorphosis; algebraization of switching circuits; contact hypergraph. 
@article{BGUMI_2017_3_a8,
     author = {Yu. G. Tarazevich},
     title = {Augmented polynomial matrices and algebraization of switching circuits},
     journal = {Journal of the Belarusian State University. Mathematics and Informatics},
     pages = {85--93},
     publisher = {mathdoc},
     volume = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a8/}
}
TY  - JOUR
AU  - Yu. G. Tarazevich
TI  - Augmented polynomial matrices and algebraization of switching circuits
JO  - Journal of the Belarusian State University. Mathematics and Informatics
PY  - 2017
SP  - 85
EP  - 93
VL  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a8/
LA  - ru
ID  - BGUMI_2017_3_a8
ER  - 
%0 Journal Article
%A Yu. G. Tarazevich
%T Augmented polynomial matrices and algebraization of switching circuits
%J Journal of the Belarusian State University. Mathematics and Informatics
%D 2017
%P 85-93
%V 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a8/
%G ru
%F BGUMI_2017_3_a8
Yu. G. Tarazevich. Augmented polynomial matrices and algebraization of switching circuits. Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 85-93. http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a8/

[1] S. V. Yablonskii, “Elementy matematicheskoi kibernetiki”, Vysshaya shkola, 2007 | Zbl

[2] O. B. Lupanov, “Asimptoticheskie otsenki slozhnosti upravlyayuschikh sistem”, Izdatelstvo MGU, 1984 | Zbl

[3] R. G. Nigmatullin, “Slozhnost bulevykh funktsii”, Nauka, 1991 | MR | Zbl

[4] R. Basaker, T. Saati, “Konechnye grafy i seti”, Nauka, 1974 | MR

[5] V. A. Emelichev, O. I. Melnikov, V. I. Sarvanov, “Lektsii po teorii grafov”, Nauka, 1990 | MR

[6] Yu. G. Tarazevich, “Algebraizatsiya i obobschenie kontaktnykh skhem”, Diskretnaya matematika i ee prilozheniya: materialy XII Mezhdunar. seminara im. akad. OB Lupanova (Moskva). Izdatelstvo mekhaniko-matematicheskogo fakulteta MGU, 2016, 170–172

[7] S. V. Yablonskii, “Vvedenie v diskretnuyu matematiku”, Vysshaya shkola, 2003 | Zbl

[8] A. G. Lunts, “Algebraicheskie metody analiza i sinteza kontaktnykh skhem”, Izv. AN SSSR. Ser. mat, 16 (1952), 405–426

[9] E. B. Vinberg, “Kurs algebry”, Faktorial press, 2001

[10] O. Zarisskii, P. Samyuel, “Kommutativnaya algebra”, Inostrannaya literatura, 1 (1963)

[11] A. I. Maltsev, “Osnovy lineinoi algebry”, Nauka, 1970 | MR