On some unary algebras and their subalgebra lattices

Abstract

We first define lattices, called normal, which are uniquely represented by directed graphs. Secondly, we describe all unary algebras (called normal, too) such that their subalgebra lattices are normal. Next, we characterize pairs (A,L) such that the subalgebra lattice of A is isomorphic to L, where A is a normal unary algebra and L is a normal lattice. Further, we describe pairs of normal unary algebras with isomorphic subalgebra lattices. We use these results in the second part of the paper to find necessary and sufficient conditions for pairs of lattices to be isomorphic to a pair of the weak and strong subalgebra lattices of one normal unary algebra