Geodesic
Separability of the lattice of $\tau$-closed totally $\omega$-composition formations of finite groups
Trudy Instituta matematiki, Tome 31 (2023) no. 2, pp. 44-56 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $\mathfrak{X}$ be a non-empty class of finite groups. A complete lattice $\theta$ of formations is said $\mathfrak{X}$-separable if for every term $\eta(x_1, \ldots , x_n)$ of signature $\{\cap, \vee_{\theta}\}$, $\theta$-formations $\mathfrak{F}_1, \ldots , \mathfrak{F}_n$, and every group $A\in \mathfrak{X}\cap \eta(\mathfrak{F}_1, \ldots , \mathfrak{F}_n)$ are exists $\mathfrak{X}$-groups $A_1\in\mathfrak{F}_1, \ldots , A_n\in\mathfrak{F}_n$ such that $A\in\eta(\theta\mathrm{form}(A_1), \ldots , \theta\mathrm{form}(A_n))$. In particular, if $\mathfrak{X}=\mathfrak{G}$ is the class of all finite groups then the lattice $\theta$ of formations is said $\mathfrak{G}$-separable or, briefly, separable. It is proved that the lattice $c^{\tau}_{\omega_\infty}$ of all $\tau$-closed totally $\omega$-composition formations is $\mathfrak{G}$-separable. 
@article{TIMB_2023_31_2_a5,
     author = {I. P. Los and V. G. Safonov},
     title = {Separability of the lattice of $\tau$-closed totally $\omega$-composition formations of finite groups},
     journal = {Trudy Instituta matematiki},
     pages = {44--56},
     year = {2023},
     volume = {31},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2023_31_2_a5/}
}
TY  - JOUR
AU  - I. P. Los
AU  - V. G. Safonov
TI  - Separability of the lattice of $\tau$-closed totally $\omega$-composition formations of finite groups
JO  - Trudy Instituta matematiki
PY  - 2023
SP  - 44
EP  - 56
VL  - 31
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMB_2023_31_2_a5/
LA  - ru
ID  - TIMB_2023_31_2_a5
ER  - 
%0 Journal Article
%A I. P. Los
%A V. G. Safonov
%T Separability of the lattice of $\tau$-closed totally $\omega$-composition formations of finite groups
%J Trudy Instituta matematiki
%D 2023
%P 44-56
%V 31
%N 2
%U http://geodesic.mathdoc.fr/item/TIMB_2023_31_2_a5/
%G ru
%F TIMB_2023_31_2_a5
I. P. Los; V. G. Safonov. Separability of the lattice of $\tau$-closed totally $\omega$-composition formations of finite groups. Trudy Instituta matematiki, Tome 31 (2023) no. 2, pp. 44-56. http://geodesic.mathdoc.fr/item/TIMB_2023_31_2_a5/

[1] Shemetkov L. A., Skiba A. N., Formatsii algebraicheskikh sistem, Nauka, M., 1989 | MR

[2] Skiba A. N., Algebra formatsii, Belaruskaya navuka, Minsk, 1997 | MR

[3] Skiba A. N., Shemetkov L. A., “Kratno $\mathfrak{L}$-kompozitsionnye formatsii konechnykh grupp”, Ukrainskii mat. zhurn., 52:6 (2000), 783–797 | MR | Zbl

[4] Selkin V. M., Odnoporozhdennye formatsii, GGU im. F. Skoriny, Gomel, 2011

[5] Vorobev N. N., Algebra klassov konechnykh grupp, VGU im. P. M. Masherova, Vitebsk, 2012

[6] Safonov V. G., “$\mathfrak G$-otdelimost reshetki $\tau$-zamknutykh totalno nasyschennykh formatsii”, Algebra i logika, 49:5 (2010), 692–704

[7] Shemetkov L. A., Skiba A. N., Vorob'ev N. N., “On laws of lattices of partially saturated formations”, Asian-European Journal of Mathematics, 2:1 (2009), 155–169 | DOI | MR | Zbl

[8] Safonov V. G., Safonova I. N., “Otdelimost reshetki $\tau$-zamknutykh totalno $\omega$-nasyschennykh formatsii konechnykh grupp”, Problemy fiziki, matematiki i tekhniki, 2017, no. 4(33), 76–83 | Zbl

[9] Tsarev A. A., “Inductive lattices of totally composition formations”, Revista Colombiana de Matematicas, 52:2 (2018), 161–169 | DOI | MR | Zbl

[10] Tsarev A. A., “On the lattice of all totally composition formations of finite groups”, Ricerche di Matematica, 68:2 (2019), 693–698 | DOI | MR | Zbl

[11] Los I. P., Safonov V. G., “Separability of the lattice of $\tau$-closed totally $\omega$-composition formations of finite groups”, The XII International Algebraic Conference in Ukraine dedicated to the 215th anniversary of V. Bunyakovsky (July 02–06, 2019, Vinnytsia, Ukraine), 2019, 64–65

[12] Los I. P., Safonov V. G., “$\tau$-Zamknutye totalno $\omega$-kompozitsionnye formatsii konechnykh grupp s bulevymi podreshetkami”, Materialy Mezhdunar. konf. po algebre, analizu i geometrii, Tr. Mat. tsentra im. N. I. Lobachevskogo, 60, Izd-vo Akademii nauk RT, Kazan, 2021, 92–94

[13] Los I. P., Safonov V. G., “Ob odnoporozhdennykh i ogranichennykh totalno $\omega$-kompozitsionnykh formatsiyakh konechnykh grupp”, Problemy fiziki, matematiki i tekhniki, 2021, no. 4(49), 101–107

[14] Scherbina V. V., “O dvukh zadachakh teorii chastichno totalno kompozitsionnykh formatsii konechnykh grupp”, Prikladnaya matematika i Fizika, 52:1 (2020), 18–32 | DOI

[15] Scherbina V. V., “Chastichno kompozitsionnye formatsii s zadannoi strukturoi. I”, Prikladnaya matematika i Fizika, 53:3 (2021), 171–204 | DOI

[16] Safonov V. G., “Kharakterizatsiya razreshimykh odnoporozhdennykh totalno nasyschennykh formatsii konechnykh grupp”, Sib. mat. zhurn., 48:1 (2007), 185–191 | MR | Zbl

[17] Doerk K., Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin–New York, 1992 | MR

[18] Ballester-Bolinches A., Ezquerro L. M., Classes of Finite Groups, Springer, Dordrecht, 2006 | MR | Zbl

[19] Shemetkov L. A., “Frattini extensions of finite groups and formations”, Comm. Algebra, 25:3 (1997), 955–964 | DOI | MR | Zbl

[20] Shemetkov L. A., “Lokalnye zadaniya formatsii konechnykh grupp”, Fundamentalnaya i prikladnaya matematika, 16:8 (2010), 229–244