Geodesic
Complex algebras of subalgebras
Algebra i logika, Tome 47 (2008) no. 6, pp. 655-686.

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

Let $\mathcal V$ be a variety of algebras. We specify a condition (the so-called generalized entropic property), which is equivalent to the fact that for every algebra $\mathbf A\in\mathcal V$, the set of all subalgebras of $\mathbf A$ is a subuniverse of the complex algebra of the subalgebras of $\mathbf A$. The relationship between the generalized entropic property and the entropic law is investigated. Also, for varieties with the generalized entropic property, we consider identities that are satisfied by complex algebras of subalgebras.
Keywords: complex algebra, entropic law, mediality, linear identity.
Mots-clés : complex algebra of subalgebras, mode 
@article{AL_2008_47_6_a0,
     author = {K. V. Adaricheva and A. Pilitowska and D. Stanovsk\'y},
     title = {Complex algebras of subalgebras},
     journal = {Algebra i logika},
     pages = {655--686},
     publisher = {mathdoc},
     volume = {47},
     number = {6},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2008_47_6_a0/}
}
TY  - JOUR
AU  - K. V. Adaricheva
AU  - A. Pilitowska
AU  - D. Stanovský
TI  - Complex algebras of subalgebras
JO  - Algebra i logika
PY  - 2008
SP  - 655
EP  - 686
VL  - 47
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2008_47_6_a0/
LA  - ru
ID  - AL_2008_47_6_a0
ER  - 
%0 Journal Article
%A K. V. Adaricheva
%A A. Pilitowska
%A D. Stanovský
%T Complex algebras of subalgebras
%J Algebra i logika
%D 2008
%P 655-686
%V 47
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2008_47_6_a0/
%G ru
%F AL_2008_47_6_a0
K. V. Adaricheva; A. Pilitowska; D. Stanovský. Complex algebras of subalgebras. Algebra i logika, Tome 47 (2008) no. 6, pp. 655-686. http://geodesic.mathdoc.fr/item/AL_2008_47_6_a0/

[1] G. Grätzer, H. Lakser, “Identities for globals (complex algebras) of algebras”, Colloq. Math., 56:1 (1988), 19–29 | MR | Zbl

[2] G. Grätzer, S. Whitney, “Infinitary varieties of structures closed under the formation of complex structures”, Colloq. Math., 48 (1984), 1–5 | MR | Zbl

[3] A. Shafaat, “On varieties closed under the construction of power algebras”, Bull. Aust. Math. Soc., 11 (1974), 213–218 | DOI | MR | Zbl

[4] C. Brink, “Power structures”, Algebra Univers., 30:2 (1993), 177–216 | DOI | MR | Zbl

[5] I. Bošnjak, R. Madarász, “On power structures”, Algebra Discrete Math., 2003, no. 2, 14–35 | MR

[6] A. Romanowska, J. D. H. Smith, Modal theory. An algebraic approach to order, geometry, and convexity, Res. Expo. Math., 9, Heldermann Verlag, Berlin, 1985 | MR | Zbl

[7] A. Romanowska, J. D. H. Smith, “Subalgebra systems of idempotent entropic algebras”, J. Algebra, 120:2 (1989), 247–262 | DOI | MR | Zbl

[8] A. Romanowska, J. D. H. Smith, “On the structure of the subalgebra systems of idempotent entropic algebras”, J. Algebra, 120:2 (1989), 263–283 | DOI | MR | Zbl

[9] A. Pilitowska, Modes of submodes, PhD thesis, Warsaw Univ. Tech., 1996

[10] A. Pilitowska, “Identities for classes of algebras closed under the complex structures”, Discuss. Math. Algebra Stoch. Methods, 18:1 (1998), 85–109 | MR | Zbl

[11] A. Pilitowska, “Enrichments of affine spaces and algebras of subalgebras”, Discuss. Math. Algebra Stoch. Methods, 19:1 (1999), 207–225 | MR | Zbl

[12] J. Ježek, T. Kepka, “Medial groupoids”, Rozpr. Cesk. Akad. Ved, Rada Mat. Priz. Ved, 93:2 (1983), 93 pp. | MR

[13] A. Romanowska, J. D. H. Smith, Modes, World Scientific, Singapore, 2002 | MR | Zbl

[14] T. Evans, “Properties of algebras almost equivalent to identities”, J. Lond. Math. Soc., 37 (1962), 53–59 | DOI | MR | Zbl

[15] J. Dudek, “Small idempotent clones. I”, Czech. Math. J., 48:1 (1998), 105–118 | DOI | MR | Zbl

[16] P. Ðapić, J. Ježek, P. Marković, R. McKenzie, D. Stanovský, “$*$-Linear equational theories of groupoids”, Algebra Univers., 56:3–4 (2007), 357–397 | MR

[17] J. Ježek, R. McKenzie, “The variety generated by equivalence algebras”, Algebra Univers., 45:2–3 (2001), 211–220 | MR

[18] S. Oates-Williams, “Graphs and universal algebras”, Combinatorial mathematics VIII, Proc. 8th Aust. Conf. (Geelong/Aust., 1980), Lect. Notes Math., 884, Springer, Berlin–New York, 1981, 351–354 | MR

[19] W. W. McCune, Otter: An Automated Deduction System, . http:// www-unix.mcs.anl.gov/AR/otter/