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unary algebra; partial algebra; subalgebra lattice; directed graph; finite unary algebra
One of the main aims of the present and the next part [15] is to show that the theory of graphs (its language and results) can be very useful in algebraic investigations. We characterize, in terms of isomorphisms of some digraphs, all pairs $\langle \mathbf{A},\mathbf{L}\rangle $, where $\mathbf{A}$ is a finite unary algebra and $L$ a finite lattice such that the subalgebra lattice of $\mathbf{A}$ is isomorphic to $\mathbf{L}$. Moreover, we find necessary and sufficient conditions for two arbitrary finite unary algebras to have isomorphic subalgebra lattices. We solve these two problems in the more general case of partial unary algebras. In the next part [15] we will use these results to describe connections between various kinds of lattices of (partial) subalgebras of a finite unary algebra.
[1] M. Barr, C. Wells: Category Theory for Computing Science. Series in Computer Science, Prentice Hall International, London, 1990. MR 1094561
[2] W. Bartol: Weak subalgebra lattices. Comment. Math. Univ. Carolin. 31 (1990), 405–410. MR 1078473 | Zbl 0711.08007
[3] W. Bartol, F. Rosselló, L. Rudak: Lectures on Algebras, Equations and Partiality. Rosselló F. (ed.), Technical report B-006, Univ. Illes Balears, Dept. Ciencies Mat. Inf., 1992.
[4] C. Berge: Graphs and Hypergraphs. North-Holland, Amsterdam, 1973. MR 0357172 | Zbl 0254.05101
[5] P. Burmeister: A Model Theoretic Oriented Approach to Partial Algebras. Math. Research Band 32, Akademie Verlag, Berlin, 1986. MR 0854861 | Zbl 0598.08004
[6] T. Evans, B. Ganter: Varieties with modular subalgebra lattices. Bull. Austral. Math. Soc. 28 (1983), 247–254. DOI 10.1017/S0004972700020918 | MR 0729011
[7] P. Grzeszczuk, E. R. Puczyłowski: On Goldie and dual Goldie dimensions. J. Pure Appl. Algebra 31 (1984), 47–54. DOI 10.1016/0022-4049(84)90075-6 | MR 0738204
[8] P. Grzeszczuk, E. R. Puczyłowski: 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
[9] J. Johnson, R. L. Seifer: A survey of multi-unary algebras. Mimeographed seminar notes, U.C. Berkeley, 1967.
[10] B. Jónsson: Topics in Universal Algebra. Lecture Notes in Mathemathics 250, Springer-Verlag, 1972. MR 0345895
[11] E. W. Kiss, M. A. Valeriote: Abelian algebras and the Hamiltonian property. J. Pure Appl. Algebra 87 (1993), 37–49. DOI 10.1016/0022-4049(93)90067-4 | MR 1222175
[12] E. Lukács, P. P. Pálfy: Modularity of the subgroup lattice of a direct square. Arch. Math. 46 (1986), 18–19. DOI 10.1007/BF01197131 | MR 0829806
[13] P. P. Pálfy: Modular subalgebra lattices. Algebra Universalis 27 (1990), 220–229. DOI 10.1007/BF01182454 | MR 1037863
[14] K. Pióro: On some non-obvious connections between graphs and partial unary algebras. Czechoslovak Math. J. 50 (2000), 295–320. DOI 10.1023/A:1022418818272 | MR 1761388
[15] K. Pióro: On subalgebra lattices of a finite unary algebra, Part II. Math. Bohem. 126 (2001), 171–181. MR 1826479
[16] D. Sachs: The lattice of subalgebras of a Boolean algebra. Canad. J. Math. 14 (1962), 451–460. DOI 10.4153/CJM-1962-035-1 | MR 0137666
[17] J. Shapiro: Finite equational bases for subalgebra distributive varieties. Algebra Universalis 24 (1987), 36–40. DOI 10.1007/BF01188381 | MR 0921528
[18] J. Shapiro: Finite algebras with abelian properties. Algebra Universalis 25 (1988), 334–364. DOI 10.1007/BF01229981 | MR 0969156
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