Spectral algebras for strong equational classes of partial algebras

As it is well known, the generalization of universal-algebraic notions to partial algebras yields different kinds of homomorphisms, subalgebras, validity of equations etc. Three types of equations: existential, weak and strong, have been investigated by many algebraists with interesting results on algebraic characterization of equationally definable classes of partial algebras. One of the most significant papers in this area of research is the work by H. Andréka and I. Németi [1] where the category-theoretical point of view is applied to the problem of HSP-characterization. In effect, the authors have given an algebraic description of equational classes definable by many different kinds of equations and generalized equations (i.e., equational implications). On the other hand, they have introduced tools which turned out to be very useful for searching “good” characterizations of partial varieties, e.g. the universal solution, a free object in some sense. However, the results given in [1] do not include the strong equation case, which is our subject of research. For an example see Section 3. There are other interesting results concerning partial varieties. P. Burmeister [4] characterized existential varieties i.e. classes of partial algebras definable by