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On some non-obvious connections between graphs and unary partial algebras. (English) Zbl 1046.08002

For an undirected graph two types of subgraphs are defined, namely a weak subgraph and a relative subgraph. For a digraph, moreover, a strong subdigraph and a dually strong subdigraph are defined. Further, partial unary algebras are studied. To any such algebra \(A\) its graph \(G^*(A)\) and its digraph \(G(A)\) are assigned in a natural way. Four types of subalgebras (analogous to the types of subdigraphs) are defined, namely weak subalgebras, relative subalgebras, strong subalgebras and initial segments. Lattices of such subgraphs and subalgebras are studied and interconnections between isomorphisms of those lattices and isomorphisms of corresponding algebras and their graphs and digraphs are investigated.

MSC:

08A55 Partial algebras
08A60 Unary algebras
05C20 Directed graphs (digraphs), tournaments
05C75 Structural characterization of families of graphs
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References:

[1] Bartol, W.: Weak subalgebra lattices. Comment. Math. Univ. Carolin. 31 (1990), 405-410. · Zbl 0711.08007
[2] Bartol, W.: Weak subalgebra lattices of monounary partial algebras. Comment. Math. Univ. Carolin. 31 (1990), 411-414. · Zbl 0711.08007
[3] 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.
[4] Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam 1973. · Zbl 0483.05029
[5] Birkhoff, G., Frink, O.: Representation of lattices by sets. Trans. Amer. Math. Soc. 64 (1948), 299-316. · Zbl 0032.00504 · doi:10.2307/1990504
[6] Burmeister, P.: A Model Theoretic Oriented Approach to Partial Algebras. Math. Research Band 32, Akademie Verlag, Berlin, 1986. · Zbl 0598.08004
[7] Evans, T., Ganter, B.: Varieties with modular subalgebra lattices. Bull. Austral. Math. Soc. 28 (1983), 247-254. · Zbl 0545.08010 · doi:10.1017/S0004972700020918
[8] Grätzer, G.: Universal Algebra, second edition. Springer-Verlag, New York 1979. · Zbl 0412.08001
[9] Grätzer, G.: General Lattice Theory. Akademie-Verlag, Berlin 1978. · Zbl 0436.06001
[10] Grzeszczuk, P., Puczyłowski, E. R.: On Goldie and dual Goldie dimensions. Journal of Pure and Applied Algebra 31 (1984) 47-54). · Zbl 0528.16010 · doi:10.1016/0022-4049(84)90075-6
[11] Grzeszczuk, P., Puczyłowski, E. R.: On infinite Goldie dimension of modular lattices and modules. J. Pure Appl. Algebra 35 (1985), 151-155. · Zbl 0562.16014 · doi:10.1016/0022-4049(85)90037-4
[12] Jónsson, B.: Topics in Universal Algebra. Lecture Notes in Mathemathics 250, Springer-Verlag, 1972. · Zbl 0225.08001
[13] McKenzie, R. N., McNulty, G. F., Taylor, W. F.: Algebras, Lattices, Varieties, vol. I. Wadsworth and Brooks/Cole Advanced Books and Software, Monterey, 1987. · Zbl 0611.08001
[14] Pióro, K.: Uniqueness of a unary partial algebra graph characterization by the weak subalgebra lattice, part I. in preparation. · Zbl 0986.08004
[15] Pióro, K.: Uniqueness of a unary partial algebra graph characterization by the weak subalgebra lattice, part II. in preparation. · Zbl 0986.08004
[16] Pióro, K.: Uniqueness of a unary partial algebra graph characterization by the weak subalgebra lattice, part III. in preparation. · Zbl 0986.08004
[17] Pióro, K.: Uniqueness of a unary partial algebra graph characterization by the weak subalgebra lattice, part IV. in preparation. · Zbl 0986.08004
[18] Sachs, D.: The lattice of subalgebras of a Boolean algebra. Canad. J. Math. 14 (1962), 451-460. · Zbl 0105.25204 · doi:10.4153/CJM-1962-035-1
[19] Shapiro, J.: Finite equational bases for subalgebra distributive varieties. Algebra Universalis 24 (1987), 36-40. · Zbl 0644.08003 · doi:10.1007/BF01188381
[20] Shapiro, J.: Finite algebras with abelian properties. Algebra Universalis 25 (1988), 334-364. · Zbl 0654.08001 · doi:10.1007/BF01229981
[21] Ore, O.: Theory of Graphs. AMS Colloq. Publ. vol. XXXVIII., 1962. · Zbl 0105.35401
[22] Tutte, W. T.: Graph Theory. Encyclopedia of Mathematics And Its Applications, Addison Wesley Publ. Co., 1984. · Zbl 0554.05001
[23] Wilson, R. J.: Introduction to Graph Theory, second edition. Longman Group Limited, London, 1979.
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