# Article

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Keywords:
graph; finite unary algebra; partial algebra; subalgebras; subalgebra lattices
Summary:
We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs $\langle \mathbf{L}_{1},\mathbf{L}_{2}\rangle$ of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to $\mathbf{L}_{1}$ and $\mathbf{L}_{2}$, respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary and sufficient conditions are found for quadruples $\langle \mathbf{L}_{1},\mathbf{L}_{2}, \mathbf{L}_{3},\mathbf{L}_{4}\rangle$ of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to $\mathbf{L}_{1},\mathbf{L}_{2}, \mathbf{L}_{3},\mathbf{L}_{4}$, respectively.
References:
[1] W. Bartol: Weak subalgebra lattices. Comment. Math. Univ. Carolin. 31 (1990), 405–410. MR 1078473 | Zbl 0711.08007
[2] 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.
[3] C. Berge: Graphs and Hypergraphs. North-Holland, Amsterdam, 1973. MR 0357172 | Zbl 0254.05101
[4] P. Burmeister: A Model Theoretic Oriented Approach to Partial Algebras. Math. Research Band 32, Akademie Verlag, Berlin, 1986. MR 0854861 | Zbl 0598.08004
[5] P. Crawley, R. P. Dilworth: Algebraic Theory of Lattices. Prentice Hall Inc., Englewood Cliffs, NJ, 1973.
[6] B. Jónsson: Topics in Universal Algebra. Lecture Notes in Mathemathics 250, Springer, Berlin, 1972. MR 0345895
[7] 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
[8] K. Pióro: On subalgebra lattices of a finite unary algebra, part I. Math. Bohem. 126 (2001), 161–170. MR 1826479
[9] K. Pióro: On a strong property of the weak subalgebra lattice. Algebra Univers. 40 (1998), 477–495. MR 1681837
[10] H. E. Robbins: A theorem on graphs with an application to a problem of traffic. Am. Math. Monthly 46 (1939), 281–283. DOI 10.2307/2303897 | MR 1524589 | Zbl 0021.35703

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