1 Introduction Q-algebras

where 1 denotes the universal relation on the base set. These Q-operations have a geometrical origin, but in [Jónsson 1991] Jónsson discusses them only in a context where he compares the expressive power of various clones of operations on binary relations. For instance, the operation defined in the picture above does not belong to the Tarski clone; that is, Q cannot be defined by using the boolean operations together with the ∗Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, the Netherlands. E-mail: [email protected]. The research of this author is sponsored by the Netherlands Organization for the Advancement of Research (NWO). †Faculty of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, the Netherlands. E-mail: [email protected]. The research of this author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. 1For the sake of a more transparent presentation of our axiom schemas, we deviate slightly from Jónsson’s original approach where the reference points are fixed as k = 0, l = 1. It is not difficult to show that in fact, the two approaches are term-definitionally equivalent.