A few notes on subalgebra lattices, part 1

• Autorzy: Pióro K.
• Języki publikacji: EN

Abstrakty

EN

First, we apply results proved in [Pió1] and some results of graph theory to formulate and prove a necessary condition for partial (and thus also total) unary algebras to have isomorphic (strong) subalgebra lattices. Although this condition is not sufficient for arbitrary partial unary algebras, we can form, having this fact, a lot of new partial unary algebras with the same subalgebra lattices. Moreover, we use this result to characterize arbitrary two partial (thus in particular also total) monounary algebras with isomorphic (strong) subalgebra lattices. Having this result we can also describe all pairs (A, L), where A is a partial monounary algebra and L a lattice, such that the subalgebra lattice of A is isomorphic to L. In the next part [Pió2] we apply the results of this paper to characterize connections between weak and strong subalgebra lattices of partial (thus also total) monounary algebras.

Słowa kluczowe

EN

subalgebra; lattices

PL

podalgebra; kraty

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2000
• Tom: Vol. 33, nr 4
• Strony: 695--706
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Pióro K.

• Institute of Mathematics Warsaw University ul. Banacha 2 02-097 Warsaw, Poland,

Bibliografia

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• [BRR] W. Bartol, F. Rossello, L. Rudak, Lectures on Algebras, Equations and Partiality, Technical report B-006, Univ. Illes Balears, Dept. Ciencies Mat. Inf., ed. Rossello F., 1992.
• [Ber] C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam 1973.
• [Bur] P. Burmeister, A Model Theoretic Oriented Approach To Partial Algebras, Math. Research Band 32, Akademie Verlag, Berlin, 1986.
• [CrDi] P. Crawley, R. P. Dilworth, Algebraic Theory of Lattices, Prentice Hall Inc., Englewood Cliffs, NJ, 1973.
• [JoSe] J. Johnson, R. L. Seifer, A survey of multi-unary algebras, Mimeographed seminar notes, U.C. Berkeley, 1967.
• [Jon] B. Jonsson, Topics in Universal Algebra, Lecture Notes in Mathemathics 250, Springer-Verlag, 1972.
• [LuPa] E. Lukacs, P. P. Palfy, Modularity of the subgroup lattice of a direct square, Arch. Math. 46 (1986), 18-19.
• [Piol] K. Pióro, On some non-obvious connections between graphs and partial unary algebras - to appear in Czechoslovak Math. J. (2000).
• [Pio2] K. Pióro, A few notes on subalgebra lattices, part II- to appear in Demonstratio Math. 34(2001).
• [Ore] O. Ore, Theory of Graphs, AMS Colloq. Publ. v. 38 (1962).
• [Sach] D. Sachs, The lattice of subalgebras of a Boolean algebra, Canad. J. Math. 14(1962), 451-460.