Algebraic classification of physical structures with zero.~I

I. A. Firdman

Sibirskij žurnal industrialʹnoj matematiki, Tome 8 (2005) no. 4, pp. 131-148.

Résumé

An algebraic version is considered of the axiomatization of physical structures. An arbitrary set $R$ with a distinguished element $O$ (zero) is taken as a set of measurements. Under an additional condition, understood to be an analog of the requirement that a physical structure is one-metric, the structure of a topological skew field with zero $O$ is introduced on $R$; and on the object sets $\mathcal M$ and $\mathcal N$, the structure of finite-dimensional vector spaces over the skew field is introduced. This leads to a complete classification of the corresponding physical structures. The classification theorem can be considered also as a variant of the axiomatics connected with a bilinear form on a pair of finite-dimensional vector spaces over a skew field; i.e., the variant which uses, as axioms, only the combinatorial properties of a bilinear form as a map $\mathcal M\times\mathcal N\to R$ (i.e., without the axioms of addition and multiplication).

Bibliographie

[1] Kulakov Yu. I., Elementy teorii fizicheskikh struktur, izd. NGU, Novosibirsk, 1968

[2] Kulakov Yu. I., “Ob odnom printsipe, lezhaschem v osnovanii klassicheskoi fiziki”, Dokl. AN SSSR, 193:1 (1970), 72–75 | MR

[3] Kulakov Yu. I., “Geometriya prostranstv postoyannoi krivizny kak chastnyi sluchai teorii fizicheskikh struktur”, Dokl. AN SSSR, 193:5 (1970), 985–987 | MR | Zbl

[4] Kulakov Yu. I., “O novom vide simmetrii, lezhaschem v osnovanii fizicheskikh teorii fenomenologicheskogo tipa”, Dokl. AN SSSR, 201:3 (1971), 570–572

[5] Mikhailichenko G. G., “Dvumernye geometrii”, Dokl. AN SSSR, 260:4 (1981), 803–805 | MR

[6] Lev V. Kh., “Trekhmernye geometrii v teorii fizicheskikh struktur”, Metodologicheskie i tekhnologicheskie problemy informatsionno-logicheskikh sistem, Vychislitelnye sistemy, 125, izd. In-ta matematiki SO AN SSSR, Novosibirsk, 1988, 90–103 | MR

[7] Mikhailichenko G. G., “O gruppovoi i fenomenologicheskoi simmetriyakh v geometrii”, Dokl. AN SSSR, 269:2 (1983), 284–288 | MR

[8] Mikhailichenko G. G., “K voprosu o simmetrii rasstoyaniya v geometrii”, Izv. vuzov. Matematika, 1994, no. 3, 21–23 | MR | Zbl

[9] Mikhailichenko G. G., “Prosteishie polimetricheskie geometrii”, Dokl. RAN, 348:1 (1996), 22–24 | MR | Zbl

[10] Mikhailichenko G. G., “Reshenie funktsionalnykh uravnenii v teorii fizicheskikh struktur”, Dokl. AN SSSR, 206:5 (1972), 1056–1058

[11] Mikhailichenko G. G., “Fenomenologicheskaya i gruppovaya simmetrii v geometrii dvukh mnozhestv (teorii fizicheskikh struktur)”, Dokl. AN SSSR, 24:1 (1985), 39–41 | MR

[12] Mikhailichenko G. G., Matematicheskii apparat teorii fizicheskikh struktur, izd. GAGU, Gorno-Altaisk, 1997

[13] Vladimirov Yu. S., Relyatsionnaya teoriya prostranstva-vremeni i vzaimodeistvii. Ch. 1: Teoriya sistem otnoshenii, Izd-vo MGU, M., 1996

[14] Vladimirov Yu. S., Relyatsionnaya teoriya prostranstva-vremeni i vzaimodeistvii. Ch. 2: Teoriya fizicheskikh vzaimodeistvii, Izd-vo MGU, M., 1998

[15] Mikhailichenko G. G., “Dvumetricheskie fizicheskie struktury ranga $(n+1,2)$”, Sib. mat. zhurn., 34:3 (1993), 132–143 | MR | Zbl

[16] Litvintsev A. A., “Kompleksnaya fizicheskaya struktura ranga $(2,2)$”, v kn.: Mikhailichenko G. G., Matematicheskii apparat teorii fizicheskikh struktur, izd. GAGU, Gorno-Altaisk, 1997, 133–144

[17] Litvintsev A. A., “Kompleksnaya fizicheskaya struktura ranga $(3,2)$”, Matematika, Materialy 35 Mezhdunar. stud. konf. “Student i nauchno-tekhnicheskii progress” (Novosibirsk 22–24 apr. 1997 g.), izd. NGU, Novosibirsk, 1997, 62–63

[18] Ionin V. K., “Abstraktnye gruppy kak fizicheskie struktury”, Sistemologiya i metodologicheskie problemy informatsionno-logicheskikh sistem, Vychislitelnye sistemy, 135, Novosibirsk, 1990, 40–43 | MR | Zbl

[19] Ionin V. K., “K opredeleniyu fizicheskikh struktur”, Tr. In-ta matematiki, 21, 1992, 42–51 | MR | Zbl

[20] Borodin A. N., “Gruda i gruppa kak fizicheskaya struktura”, v kn.: Mikhailichenko G. G., Gruppovaya simmetriya fizicheskikh struktur, izd. Barnaul. gos. ped. un-ta, Barnaul, 2003, 195–203

[21] Simonov A. A., “Fizicheskaya struktura ranga $(3,2)$ na abstraktnykh mnozhestvakh”, Matematika, Materialy 35 Mezhdunar. nauch. stud. konf. “Student i nauchno-tekhnicheskii progress” (Novosibirsk, 22–24 apr. 1997 g.), izd. NGU, Novosibirsk, 1997, 100–101

[22] Simonov A. A., “Obobschennoe matrichnoe umnozhenie kak ekvivalentnoe predstavlenie teorii fizicheskikh struktur”, v kn.: Kulakov Yu. I., Teoriya fizicheskikh struktur, M., 2004, 675–707

[23] Burbaki N., Algebra. Algebraicheskie struktury. Lineinaya i polilineinaya algebra, Fizmatgiz, M., 1962

[24] Burbaki N., Algebra. Moduli, koltsa, formy, Fizmatgiz, M., 1966 | MR