I just wanted to add that it is unneccessary, in defining essentially algebraic theories to worry about a hierarchy of types in which the domain of each is defined equationally in terms of simpler types. I realized this because I once tried to show that the theories classified by essentially algebraic theories are essentially weaker than those classified by finite limit sketches. They are not because if you had two types, the domain of each of which was described equationally in terms of the other (or a type whose domain was described equationally in terms of itself), you could add new dummy types at the lowest level of the hierarchy, use them to define the domains of the type(s) in question and then add new unary operations of each of the dmmy types taking values as the other and finally equations that forced those operations to be isomorphisms. Nonetheless, I think that FL sketches are the way to go and the definition of essentially algebraic a kludge. Michael ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++