On some unary algebras and their subalgebra lattices

ABSTRACT. We first define lattices, called normal, which are uniquely repre-sented 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. In [10] we have investigated unary algebras and their subalgebra lattices. To this purpose we used connections between partial unary algebras and graphs given in [9]. Moreover, we did not restrict our attention to total algebras only, and we consider the more general case of partial algebras. This approach to unary algebras by partiality, and also this graph-algebraic language turned out to be very useful in such investigations. Recall, we first characterized all the pairs