Geodesic
Interpretability Types for Regular Varieties of Algebras
Algebra i logika, Tome 43 (2004) no. 2, pp. 229-234.

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

It is proved that for every regular variety $V$ of algebras, an interpretability type $[V]$ in the lattice ${\mathbb L}^{\rm int}$ is primary w.r.t. intersection, and so has at most one covering. Moreover, the sole covering, if any, for $[V]$ is necessarily infinite. For a locally finite regular variety $V$, $[V]$ has no covering. Cyclic varieties of algebras turn out to be particularly interesting among the regular. Each of these is a variety of $n$-groupoids $(A; f)$ defined by an identity $f(x_1,\ldots, x_n)=f(x_{\lambda(1)},\ldots, x_{\lambda(n)})$, where $\lambda$ is an $n$-cycle of degree $n\geqslant 2$. Interpretability types of the cyclic varieties form, in ${\mathbb L}^{\rm int}$, a subsemilattice isomorphic to a semilattice of square-free natural numbers $n\geqslant 2$, under taking $m\vee n=[m,n]$ (l.c.m.). 
@article{AL_2004_43_2_a6,
     author = {D. M. Smirnov},
     title = {Interpretability {Types} for {Regular} {Varieties} of {Algebras}},
     journal = {Algebra i logika},
     pages = {229--234},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2004_43_2_a6/}
}
TY  - JOUR
AU  - D. M. Smirnov
TI  - Interpretability Types for Regular Varieties of Algebras
JO  - Algebra i logika
PY  - 2004
SP  - 229
EP  - 234
VL  - 43
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2004_43_2_a6/
LA  - ru
ID  - AL_2004_43_2_a6
ER  - 
%0 Journal Article
%A D. M. Smirnov
%T Interpretability Types for Regular Varieties of Algebras
%J Algebra i logika
%D 2004
%P 229-234
%V 43
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2004_43_2_a6/
%G ru
%F AL_2004_43_2_a6
D. M. Smirnov. Interpretability Types for Regular Varieties of Algebras. Algebra i logika, Tome 43 (2004) no. 2, pp. 229-234. http://geodesic.mathdoc.fr/item/AL_2004_43_2_a6/

[1] R. McKenzie, S. Swierczkowski, “Non-covering in the interpretability lattice of equational theories”, Algebra Univers., 30:2 (1993), 157–170 | DOI | MR | Zbl

[2] D. M. Smirnov, “O mnogoobraziyakh, opredelimykh podstanovkami”, Algebra i logika, 42:2 (2003), 237–354 | MR

[3] B. Jónsson, E. Nelson, “Relatively free products in regular varieties”, Algebra Univers., 4:1 (1974), 14–19 | DOI | MR | Zbl

[4] B. Jónsson, “Congruence varieties”, Algebra Univers., 10:3 (1980), 355–394 | DOI | MR | Zbl

[5] R. McKenzie, “On the covering relation in the interpretability lattice of equational theories”, Algebra Univers., 30:3 (1993), 399–421 | DOI | MR | Zbl

[6] D. M. Smirnov, “Algoritm postroeniya mnogoobraziya proizvolno zadannoi konechnoi razmernosti”, Algebra i logika, 37:2 (1998), 167–180 | MR | Zbl

[7] D. M. Smirnov, “Mnogoobraziya, opredelimye podstanovkami”, Algebra i logika, 39:1 (2000), 104–118 | MR | Zbl

[8] O. C. Garcia, W. Taylor, The lattice of interpretability types of varieties, Mem. Am. Math. Soc., 50 (305), Am. Math. Soc., Providence, RI, 1984 | MR