Geodesic
Some properties of the weak subalgebra lattice of a partial algebra of a fixed type
Archivum mathematicum, Tome 38 (2002) no. 2, pp. 81-94 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We investigate, using results from [[p3]], when a given lattice is isomorphic to the weak subalgebra lattice of a partial algebra of a fixed type. First, we reduce this problem to the question when hyperedges of a hypergraph can be directed to a form of directed hypergraph of a fixed type. Secondly, we show that it is enough to consider some special hypergraphs. Finally, translating these results onto the lattice language, we obtain necessary conditions for our algebraic problem, and also, we completely characterize the weak subalgebra lattice for algebras of some types.
We investigate, using results from [[p3]], when a given lattice is isomorphic to the weak subalgebra lattice of a partial algebra of a fixed type. First, we reduce this problem to the question when hyperedges of a hypergraph can be directed to a form of directed hypergraph of a fixed type. Secondly, we show that it is enough to consider some special hypergraphs. Finally, translating these results onto the lattice language, we obtain necessary conditions for our algebraic problem, and also, we completely characterize the weak subalgebra lattice for algebras of some types.
Classification : 05C65, 06B10, 06B15, 08A30, 08A55
Keywords: hypergraph; weak subalgebra; weak subalgebra lattice; partial algebra 
@article{ARM_2002_38_2_a0,
     author = {Pi\'oro, Konrad},
     title = {Some properties of the weak subalgebra lattice of a partial algebra of a fixed type},
     journal = {Archivum mathematicum},
     pages = {81--94},
     year = {2002},
     volume = {38},
     number = {2},
     mrnumber = {1909590},
     zbl = {1069.08002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2002_38_2_a0/}
}
TY  - JOUR
AU  - Pióro, Konrad
TI  - Some properties of the weak subalgebra lattice of a partial algebra of a fixed type
JO  - Archivum mathematicum
PY  - 2002
SP  - 81
EP  - 94
VL  - 38
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ARM_2002_38_2_a0/
LA  - en
ID  - ARM_2002_38_2_a0
ER  - 
%0 Journal Article
%A Pióro, Konrad
%T Some properties of the weak subalgebra lattice of a partial algebra of a fixed type
%J Archivum mathematicum
%D 2002
%P 81-94
%V 38
%N 2
%U http://geodesic.mathdoc.fr/item/ARM_2002_38_2_a0/
%G en
%F ARM_2002_38_2_a0
Pióro, Konrad. Some properties of the weak subalgebra lattice of a partial algebra of a fixed type. Archivum mathematicum, Tome 38 (2002) no. 2, pp. 81-94. http://geodesic.mathdoc.fr/item/ARM_2002_38_2_a0/

[1] Bartol W.: Weak subalgebra lattices. Comment. Math. Univ. Carolinae 31 (1990), 405–410. | MR | Zbl

[2] Bartol W., Rosselló F., Rudak L.: Lectures on Algebras, Equations and Partiality. Technical report B–006, Univ. Illes Balears, Dept. Ciencies Mat. Inf., ed. Rosselló F., 1992.

[3] Berge C.: Graphs and Hypergraphs. North-Holland, Amsterdam 1973. | MR | Zbl

[4] Burmeister P.: A Model Theoretic Oriented Approach To Partial Algebras. Akademie Verlag, Berlin, 1986. | MR | Zbl

[5] Crawley P., Dilworth R. P.: Algebraic Theory of Lattices. Prentice Hall Inc., Englewood Cliffs, NJ, 1973. | Zbl

[6] Grätzer G.: Universal Algebra. second edition, Springer-Verlag, New York 1979. | MR

[7] Jónsson B.: Topics in Universal Algebra. Lecture Notes in Mathemathics 250, Springer-Verlag, 1972. | MR

[8] Pióro K.: The weak subalgebra lattice of a partial algebra of a given finite type. Acta Sci. Math. 65 (1999), 439–460. | MR

[9] Pióro K.: On connections between hypergraphs and algebras. Arch. Math. (Brno) 36 (2000), 45–60. | MR