Abstract

Some varieties that are extensions of relational algebras with two constants that play the role of projections are studied. The classes have as a subvariety the abstract fork algebra equivalent variety involving projections. They are obtained by weakening some laws valid in AFA. Some applications of the varieties in the literature and in the specification of abstract data types are exhibited. For each of the classes obtained, an answer is given to the question: 'Is the relational reduct of the class representable?'. For the subvarieties formed with the models that have a representable relational reduct, a representation theorem is proved. For them the finitization problem is studied. Next the varieties presented are compared by means of the inclusion order. For each class the problem of characterizing finite models is considered. Simple models in the varieties are studied. Finally the existence of equivalent classes with a binary operation like fork is studied