Geodesic
Reducibility and uniform reducibility of algebraic operations
Sbornik. Mathematics, Tome 24 (1974) no. 3, pp. 373-384 
B. R. Frenkin. Reducibility and uniform reducibility of algebraic operations. Sbornik. Mathematics, Tome 24 (1974) no. 3, pp. 373-384. http://geodesic.mathdoc.fr/item/SM_1974_24_3_a2/
@article{SM_1974_24_3_a2,
     author = {B. R. Frenkin},
     title = {Reducibility and uniform reducibility of algebraic operations},
     journal = {Sbornik. Mathematics},
     pages = {373--384},
     year = {1974},
     volume = {24},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1974_24_3_a2/}
}
TY  - JOUR
AU  - B. R. Frenkin
TI  - Reducibility and uniform reducibility of algebraic operations
JO  - Sbornik. Mathematics
PY  - 1974
SP  - 373
EP  - 384
VL  - 24
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1974_24_3_a2/
LA  - en
ID  - SM_1974_24_3_a2
ER  - 
%0 Journal Article
%A B. R. Frenkin
%T Reducibility and uniform reducibility of algebraic operations
%J Sbornik. Mathematics
%D 1974
%P 373-384
%V 24
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1974_24_3_a2/
%G en
%F SM_1974_24_3_a2

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

This paper is devoted to a study of the conditions under which one algebraic operation can be expressed in terms of others by some arrangement of parentheses. The terminology is mainly that of Frenkin (RZhMat., 1972, 2A235). It is shown that the class of $\sigma$-reducible $n$-groupoids is axiomatizable, but not elementary, and the class of $\tau$-uniformly reducible $n$-groupoids is not axiomatizable; a criterion for $\tau$-uniform reducibility in terms of pseudo-isotopies (a generalization of the concept of isotopy) between $\tau$-reducing operations is obtained. It is shown that a free $n$-groupoid of finite rank is not $\tau$-uniformly reducible, but one of infinite rank is $\tau$-uniformly reducible; as a consequence, any $n$-groupoid is a homomorphic image of one which is $\tau$-uniformly reducible. Some results on algebras with unary operations are also obtained. Bibliography: 7 titles.

[1] B. R. Frenkin, “O privodimosti i svodimosti v nekotorykh klassakh $n$-gruppoidov, I”, Matem. issledovaniya, VI, no. 2, Kishinev, 1971, 122–137 | MR

[2] P. M. Kon, Universalnaya algebra, izd-vo «Mir», Moskva, 1968 | MR

[3] A. I. Maltsev, Algebraicheskie sistemy, izd-vo «Nauka», Moskva, 1970 | MR

[4] B. R. Frenkin, “O privodimosti i svodimosti v nekotorykh klassakh $n$-gruppoidov, II”, Matem. issledovaniya, VII, no. 1, Kishinev, 1972, 150–162 | MR

[5] V. D. Belousov, M. D. Sandik, “$n$-arnye kvazigruppy i lupy”, Sib. matem. zh., 7:1 (1966), 31–54 | MR | Zbl

[6] H. A. Thurston, “The structure of an operation”, J. London Math. Soc., 27:3 (1952), 271–278 | DOI | MR

[7] A. G. Kurosh, Lektsii po obschei algebre, Fizmatgiz, Moskva, 1962 | MR