Geodesic
On steps of solubility of lattices and degrees of idempotency of prevarieties of lattices
Sbornik. Mathematics, Tome 24 (1974) no. 3, pp. 435-449 
V. B. Lender. On steps of solubility of lattices and degrees of idempotency of prevarieties of lattices. Sbornik. Mathematics, Tome 24 (1974) no. 3, pp. 435-449. http://geodesic.mathdoc.fr/item/SM_1974_24_3_a5/
@article{SM_1974_24_3_a5,
     author = {V. B. Lender},
     title = {On steps of solubility of lattices and degrees of idempotency of prevarieties of lattices},
     journal = {Sbornik. Mathematics},
     pages = {435--449},
     year = {1974},
     volume = {24},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1974_24_3_a5/}
}
TY  - JOUR
AU  - V. B. Lender
TI  - On steps of solubility of lattices and degrees of idempotency of prevarieties of lattices
JO  - Sbornik. Mathematics
PY  - 1974
SP  - 435
EP  - 449
VL  - 24
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1974_24_3_a5/
LA  - en
ID  - SM_1974_24_3_a5
ER  - 
%0 Journal Article
%A V. B. Lender
%T On steps of solubility of lattices and degrees of idempotency of prevarieties of lattices
%J Sbornik. Mathematics
%D 1974
%P 435-449
%V 24
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1974_24_3_a5/
%G en
%F SM_1974_24_3_a5

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

Let $\mathfrak R$ be a sub-prevariety of a fixed prevariety (residually closed class) $\mathfrak U$. The smallest ordinal number $\gamma$ such that an algebra $A\in \mathfrak U$ is $\gamma$-step $\mathfrak R$-soluble is called the step of $\mathfrak R$-solubility of $A$. The smallest ordinal number $\eta\ne1$ such that there exists an $\eta$-step $\mathfrak R$-soluble algebra $A\in\mathfrak U$ is called the degree of idempotency of $\mathfrak R$ relative to $\mathfrak U$. In the paper $\mathfrak U$ is taken to be the class of all lattices, and all ordinal numbers that can be degrees of idempotency of prevarieties of lattices are found. Further, a description is given, depending on the degree of idempotency of a prevariety $\mathfrak R$, of the ordinal numbers that can be steps of $\mathfrak R$-solubility of suitable lattices. Bibliography: 11 titles.

[1] L. N. Shevrin, L. M. Martynov, “O dostizhimykh klassakh algebr”, Sib. matem. zh., XII:6 (1971), 1363–1381

[2] A. I. Maltsev, “Ob umnozhenii klassov algebraicheskikh sistem”, Sib. matem. zh., VIII:2 (1967), 346–365

[3] A. I. Maltsev, “Obobschenno nilpotentnye algebry i ikh prisoedinennye gruppy”, Matem. sb., 25 (67) (1949), 347–366

[4] S. M. Vovsi, “O beskonechnykh proizvedeniyakh klassov grupp”, Sib. matem. zh., XIII:2 (1972), 272–285 | MR

[5] L. M. Martynov, “O dostizhimykh klassakh grupp i polugrupp”, Matem. sb., 90 (132) (1973), 235–245 | MR | Zbl

[6] L. M. Martynov, “O razreshimykh koltsakh”, Matem. zapiski Uralsk. un-ta, 8:3 (1972), 82–93 | MR

[7] L. N. Shevrin, V. B. Lender, “O dostizhimykh klassakh struktur”, Izv. VUZ'ov, Matematika, 1972, no. 12, 111–115 | MR | Zbl

[8] V. B. Lender, “Ob idempotentnykh predmnogoobraziyakh struktur”, XII Vsesoyuzn. algebraicheskii kollokvium, Rezyume soobsch. i dokl., Sverdlovsk, 1973, 283

[9] G. Birkgof, Teoriya struktur, IL, Moskva, 1952

[10] K. Kuratovskii, A. Mostovskii, Teoriya mnozhestv, izd-vo «Mir», Moskva, 1970 | MR

[11] V. I. Igoshin, “Kharakterizuemye klassy algebraicheskikh sistem”, XII Vsesoyuzn. algebraicheskii kollokvium, Rezyume soobsch. i dokl., Sverdlovsk, 1973, 274