Let \(\nu_ 1\) and \(\nu_ 2\) be two enumerations of a set S and consider the total (respectively, partial) functions from S to S. If all such \(\nu_ 1\)-computable functions are \(\nu_ 2\)-computable and if there is a recursive function that maps each \(\nu_ 1\)-index of a \(\nu_ 1\)- computable function to a \(\nu_ 2\)-index of that function, then \(\nu_ 1\) is called strongly reducible to \(\nu_ 2\) with respect to total (respectively, partial) morphisms. Properties of these reducibilities are examined, both in general and for specific enumerated sets such as the set of unary p.r. functions. A second area covered is that of enumerated partial algebras, which are compared and contrasted with enumerated algebras.