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A few notes on subalgebra lattices, part 1

Pióro K.

Demonstratio Mathematica

2000

Vol. 33, nr 4

695-706

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.