We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs ⟨L1, L2⟩ of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to L1 and L2, 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 ⟨L1, L2, L3, L4⟩ of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to L1, L2, L3, L4, respectively.
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 A, L, where A is a finite unary algebra and L a finite lattice such that the subalgebra lattice of A is isomorphic to 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.