On the Subalgebra Lattice of Unary Algebras

We characterize pairs L, A, where Lis a lattice and Ais a unary partial algebra, such that the strong subalgebra lattice S_s(A) is isomorphic to L. Moreover, we find necessary and sufficient conditions for arbitrary unary partial algebras to have isomorphic strong subalgebra lattices. Observe, that for a total algebra A, the lattice S_s(A) is the usual well-known subalgebra lattice. Thus in particular we solve these two problems for total unary algebras and their lattices of (also total) subalgebras.For this purpose we apply some non-obvious connections between unary partial algebras and graphs from 9]. More precisely, we first characterize the pairs L, G, where Lis a lattice and Ga directed graph, such that the strong subdigraph lattice of Gis isomorphic to L. Next, we find a characterization of arbitrary digraphs with isomorphic strong subalgebra lattices. From these results we easily get solutions of our algebraic problems.