# Article

Full entry | PDF   (0.4 MB)
Summary:
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 $\mathbf L$.
References:
[Bar1] Bartol, W.: Weak subalgebra lattices. Comment. Math. Univ. Carolin. 31 (1990), 405–410. MR 1078473 | Zbl 0711.08007
[Bar2] Bartol, W.: Weak subalgebra lattices of monounary partial algebras. Comment. Math. Univ. Carolin. 31 (1990), 411–414. MR 1078474 | Zbl 0711.08007
[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 0357172 | Zbl 0483.05029
[BiFr] Birkhoff, G., Frink, O.: Representation of lattices by sets. Trans. Amer. Math. Soc. 64 (1948), 299–316. DOI 10.1090/S0002-9947-1948-0027263-2 | MR 0027263
[Bur] Burmeister, P.: A Model Theoretic Oriented Approach to Partial Algebras. Math. Research Band 32, Akademie Verlag, Berlin, 1986. MR 0854861 | Zbl 0598.08004
[EvGa] Evans, T., Ganter, B.: Varieties with modular subalgebra lattices. Bull. Austral. Math. Soc. 28 (1983), 247–254. DOI 10.1017/S0004972700020918 | MR 0729011
[Grä1] Grätzer, G.: Universal Algebra, second edition. Springer-Verlag, New York 1979. MR 0538623
[Grä2] Grätzer, G.: General Lattice Theory. Akademie-Verlag, Berlin 1978. MR 0504338
[GP1] Grzeszczuk, P., Puczyłowski, E. R.: On Goldie and dual Goldie dimensions. Journal of Pure and Applied Algebra 31 (1984) 47–54). DOI 10.1016/0022-4049(84)90075-6 | MR 0738204
[GP2] Grzeszczuk, P., Puczyłowski, E. R.: On infinite Goldie dimension of modular lattices and modules. J. Pure Appl. Algebra 35 (1985), 151–155. DOI 10.1016/0022-4049(85)90037-4 | MR 0775467
[Jón] Jónsson, B.: Topics in Universal Algebra. Lecture Notes in Mathemathics 250, Springer-Verlag, 1972. MR 0345895
[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 0883644
[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 10.4153/CJM-1962-035-1 | MR 0137666 | Zbl 0105.25204
[Sha1] Shapiro, J.: Finite equational bases for subalgebra distributive varieties. Algebra Universalis 24 (1987), 36–40. DOI 10.1007/BF01188381 | MR 0921528 | Zbl 0644.08003
[Sha2] Shapiro, J.: Finite algebras with abelian properties. Algebra Universalis 25 (1988), 334–364. DOI 10.1007/BF01229981 | MR 0969156 | Zbl 0654.08001
[Ore] Ore, O.: Theory of Graphs. AMS Colloq. Publ. vol. XXXVIII., 1962. MR 0150753 | Zbl 0105.35401
[Tut] Tutte, W. T.: Graph Theory. Encyclopedia of Mathematics And Its Applications, Addison Wesley Publ. Co., 1984. MR 0746795 | Zbl 0554.05001
[Wil] Wilson, R. J.: Introduction to Graph Theory, second edition. Longman Group Limited, London, 1979. MR 0539146

Partner of