| Title:
|
Modularity and distributivity of the lattice of $\Sigma$-closed subsets of an algebraic structure (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Emanovský, Petr |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
120 |
| Issue:
|
2 |
| Year:
|
1995 |
| Pages:
|
209-217 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
closure system |
| Keyword:
|
distributive lattice |
| Keyword:
|
lattices of $\Sigma-closed subsets |
| Keyword:
|
modular lattice |
| Keyword:
|
algebraic structures |
| Keyword:
|
$\Sigma$-closed subset |
| Keyword:
|
convex subset |
| MSC:
|
03C05 |
| MSC:
|
04A05 |
| MSC:
|
06A23 |
| MSC:
|
06B10 |
| MSC:
|
06C05 |
| MSC:
|
06D05 |
| MSC:
|
08A05 |
| idZBL:
|
Zbl 0833.08002 |
| idMR:
|
MR1357603 |
| DOI:
|
10.21136/MB.1995.126220 |
| . |
| Date available:
|
2009-09-24T21:10:36Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126220 |
| . |
| Reference:
|
[1] Chajda I.: A note on varieties with distributive subalgebra lattices.Acta Univ. Palack. Olomouc, Fac. Rer. Natur., Matematica 31 (1992), 25-28. Zbl 0777.08001, MR 1212602 |
| Reference:
|
[2] Chajda I., Emanovský P.: $\Sigma$-isomorphic algebraic structures.Mathem. Bohemica 120 (1995), 71-81. Zbl 0833.08001, MR 1336947 |
| Reference:
|
[3] Emanovský P.: Convex isomorphic ordered sets.Mathem. Bohemica 118 (1993), 29-35. MR 1213830 |
| Reference:
|
[4] Evans T., Ganter B.: Varieties with modular subalgebra lattices.Bull. Austral. Math. Soc. 28 (1993), 247-254. MR 0729011, 10.1017/S0004972700020918 |
| Reference:
|
[5] Jakubíková-Studenovská D.: Convex subsets of partial monounary algebras.Czech. Math. J. 38 (1988), no. 113, 655-672. MR 0962909 |
| Reference:
|
[6] Marmazajev V.I.: The lattice of convex sublattices of a lattice.Mezvužovskij naučnyj sbornik 6. Saratov, 1986, pp. 50-58. (In Russian.) MR 0957970 |
| . |
