Geodesic
Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure
Mathematica Bohemica, Tome 120 (1995) no. 2, pp. 209-217
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $\Cal A =(A,F,R)$ be an algebraic structure of type $\tau$ and $\Sigma$ a set of open formulas of the first order language $L(\tau)$. The set $C_\Sigma(\Cal A)$ of all subsets of $A$ closed under $\Sigma$ forms the so called lattice of $\Sigma$-closed subsets of $\Cal A$. We prove various sufficient conditions under which the lattice $C_\Sigma(\Cal A)$ is modular or distributive.
Let $\Cal A =(A,F,R)$ be an algebraic structure of type $\tau$ and $\Sigma$ a set of open formulas of the first order language $L(\tau)$. The set $C_\Sigma(\Cal A)$ of all subsets of $A$ closed under $\Sigma$ forms the so called lattice of $\Sigma$-closed subsets of $\Cal A$. We prove various sufficient conditions under which the lattice $C_\Sigma(\Cal A)$ is modular or distributive.
DOI : 10.21136/MB.1995.126220
Classification : 03C05, 04A05, 06A23, 06B10, 06C05, 06D05, 08A05
Keywords: closure system; distributive lattice; lattices of $\Sigma-closed subsets; modular lattice; algebraic structures; $\Sigma$-closed subset; convex subset 
@article{10_21136_MB_1995_126220,
     author = {Chajda, Ivan and Emanovsk\'y, Petr},
     title = {Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure},
     journal = {Mathematica Bohemica},
     pages = {209--217},
     year = {1995},
     volume = {120},
     number = {2},
     doi = {10.21136/MB.1995.126220},
     mrnumber = {1357603},
     zbl = {0833.08002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.1995.126220/}
}
TY  - JOUR
AU  - Chajda, Ivan
AU  - Emanovský, Petr
TI  - Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure
JO  - Mathematica Bohemica
PY  - 1995
SP  - 209
EP  - 217
VL  - 120
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.1995.126220/
DO  - 10.21136/MB.1995.126220
LA  - en
ID  - 10_21136_MB_1995_126220
ER  - 
%0 Journal Article
%A Chajda, Ivan
%A Emanovský, Petr
%T Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure
%J Mathematica Bohemica
%D 1995
%P 209-217
%V 120
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.1995.126220/
%R 10.21136/MB.1995.126220
%G en
%F 10_21136_MB_1995_126220
Chajda, Ivan; Emanovský, Petr. Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure. Mathematica Bohemica, Tome 120 (1995) no. 2, pp. 209-217. doi: 10.21136/MB.1995.126220

[1] Chajda I.: A note on varieties with distributive subalgebra lattices. Acta Univ. Palack. Olomouc, Fac. Rer. Natur., Matematica 31 (1992), 25-28. | MR | Zbl

[2] Chajda I., Emanovský P.: $\Sigma$-isomorphic algebraic structures. Mathem. Bohemica 120 (1995), 71-81. | MR | Zbl

[3] Emanovský P.: Convex isomorphic ordered sets. Mathem. Bohemica 118 (1993), 29-35. | MR

[4] Evans T., Ganter B.: Varieties with modular subalgebra lattices. Bull. Austral. Math. Soc. 28 (1993), 247-254. | DOI | MR

[5] Jakubíková-Studenovská D.: Convex subsets of partial monounary algebras. Czech. Math. J. 38 (1988), no. 113, 655-672. | MR

[6] Marmazajev V.I.: The lattice of convex sublattices of a lattice. Mezvužovskij naučnyj sbornik 6. Saratov, 1986, pp. 50-58. (In Russian.) | MR

Cité par Sources :