Geodesic
On some non-obvious connections between graphs and unary partial algebras
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 2, pp. 295-320 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras $\mathbf A$ and $\mathbf B$, their weak subalgebra lattices are isomorphic if and only if their graphs ${\mathbf G}^{\ast }({\mathbf A})$ and ${\mathbf G}^{\ast }({\mathbf B})$ are isomorphic. Secondly, it is shown that for two unary partial algebras $\mathbf A$ and $\mathbf B$ if their digraphs ${\mathbf G}({\mathbf A})$ and ${\mathbf G}({\mathbf B})$ are isomorphic, then their (weak, relative, strong) subalgebra lattices are also isomorphic. Thirdly, we characterize pairs ${\mathbf L},{\mathbf A}>$, where $\mathbf A$ is a unary partial algebra and $\mathbf L$ is a lattice such that the weak subalgebra lattice of $\mathbf A$ is isomorphic to $\mathbf L$.
In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras $\mathbf A$ and $\mathbf B$, their weak subalgebra lattices are isomorphic if and only if their graphs ${\mathbf G}^{\ast }({\mathbf A})$ and ${\mathbf G}^{\ast }({\mathbf B})$ are isomorphic. Secondly, it is shown that for two unary partial algebras $\mathbf A$ and $\mathbf B$ if their digraphs ${\mathbf G}({\mathbf A})$ and ${\mathbf G}({\mathbf B})$ are isomorphic, then their (weak, relative, strong) subalgebra lattices are also isomorphic. Thirdly, we characterize pairs ${\mathbf L},{\mathbf A}>$, where $\mathbf A$ is a unary partial algebra and $\mathbf L$ is a lattice such that the weak subalgebra lattice of $\mathbf A$ is isomorphic to $\mathbf L$.
Classification : 05C20, 05C60, 05C75, 05C99, 08A05, 08A55, 08A60 
@article{CMJ_2000_50_2_a5,
     author = {Pi\'oro, Konrad},
     title = {On some non-obvious connections between graphs and unary partial algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {295--320},
     year = {2000},
     volume = {50},
     number = {2},
     mrnumber = {1761388},
     zbl = {1046.08002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2000_50_2_a5/}
}
TY  - JOUR
AU  - Pióro, Konrad
TI  - On some non-obvious connections between graphs and unary partial algebras
JO  - Czechoslovak Mathematical Journal
PY  - 2000
SP  - 295
EP  - 320
VL  - 50
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2000_50_2_a5/
LA  - en
ID  - CMJ_2000_50_2_a5
ER  - 
%0 Journal Article
%A Pióro, Konrad
%T On some non-obvious connections between graphs and unary partial algebras
%J Czechoslovak Mathematical Journal
%D 2000
%P 295-320
%V 50
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2000_50_2_a5/
%G en
%F CMJ_2000_50_2_a5
Pióro, Konrad. On some non-obvious connections between graphs and unary partial algebras. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 2, pp. 295-320. http://geodesic.mathdoc.fr/item/CMJ_2000_50_2_a5/

[Bar1] Bartol, W.: Weak subalgebra lattices. Comment. Math. Univ. Carolin. 31 (1990), 405–410. | MR | Zbl

[Bar2] Bartol, W.: Weak subalgebra lattices of monounary partial algebras. Comment. Math. Univ. Carolin. 31 (1990), 411–414. | MR | Zbl

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

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

[BiFr] Birkhoff, G., Frink, O.: Representation of lattices by sets. Trans. Amer. Math. Soc. 64 (1948), 299–316. | DOI | MR

[Bur] Burmeister, P.: A Model Theoretic Oriented Approach to Partial Algebras. Math. Research Band 32, Akademie Verlag, Berlin, 1986. | MR | Zbl

[EvGa] Evans, T., Ganter, B.: Varieties with modular subalgebra lattices. Bull. Austral. Math. Soc. 28 (1983), 247–254. | DOI | MR

[Grä1] Grätzer, G.: Universal Algebra, second edition. Springer-Verlag, New York 1979. | MR

[Grä2] Grätzer, G.: General Lattice Theory. Akademie-Verlag, Berlin 1978. | MR

[GP1] Grzeszczuk, P., Puczyłowski, E. R.: On Goldie and dual Goldie dimensions. Journal of Pure and Applied Algebra 31 (1984) 47–54). | DOI | MR

[GP2] Grzeszczuk, P., Puczyłowski, E. R.: On infinite Goldie dimension of modular lattices and modules. J. Pure Appl. Algebra 35 (1985), 151–155. | DOI | MR

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

[MMT] McKenzie, R. N., McNulty, G. F., Taylor, W. F.: Algebras, Lattices, Varieties, vol. I. Wadsworth and Brooks/Cole Advanced Books and Software, Monterey, 1987. | MR

[PióI] Pióro, K.: Uniqueness of a unary partial algebra graph characterization by the weak subalgebra lattice, part I. in preparation.

[PióII] Pióro, K.: Uniqueness of a unary partial algebra graph characterization by the weak subalgebra lattice, part II. in preparation.

[PióIII] Pióro, K.: Uniqueness of a unary partial algebra graph characterization by the weak subalgebra lattice, part III. in preparation.

[PióIV] Pióro, K.: Uniqueness of a unary partial algebra graph characterization by the weak subalgebra lattice, part IV. in preparation.

[Sach] Sachs, D.: The lattice of subalgebras of a Boolean algebra. Canad. J. Math. 14 (1962), 451–460. | DOI | MR | Zbl

[Sha1] Shapiro, J.: Finite equational bases for subalgebra distributive varieties. Algebra Universalis 24 (1987), 36–40. | DOI | MR | Zbl

[Sha2] Shapiro, J.: Finite algebras with abelian properties. Algebra Universalis 25 (1988), 334–364. | DOI | MR | Zbl

[Ore] Ore, O.: Theory of Graphs. AMS Colloq. Publ. vol. XXXVIII., 1962. | MR | Zbl

[Tut] Tutte, W. T.: Graph Theory. Encyclopedia of Mathematics And Its Applications, Addison Wesley Publ. Co., 1984. | MR | Zbl

[Wil] Wilson, R. J.: Introduction to Graph Theory, second edition. Longman Group Limited, London, 1979. | MR