Pióro, Konrad On some non-obvious connections between graphs and unary partial algebras. (English) Zbl 1046.08002 Czech. Math. J. 50, No. 2, 295-320 (2000). 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. Reviewer: Bohdan Zelinka (Liberec) Cited in 1 ReviewCited in 5 Documents MSC: 08A55 Partial algebras 08A60 Unary algebras 05C20 Directed graphs (digraphs), tournaments 05C75 Structural characterization of families of graphs Keywords:partial unary algebra; isomorphism; lattice of subalgebras PDFBibTeX XMLCite \textit{K. Pióro}, Czech. Math. J. 50, No. 2, 295--320 (2000; Zbl 1046.08002) Full Text: DOI EuDML 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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.