2. Analysis: From the start, we shall make use of the analysis that was proposed by Friedrich Solmsen3, that takes Aristotle´s use of the concepts of “acting” (activity) and “suffering” (passivity) in order that he (Aristotle) would be able to select the qualities whose suitable combination constituted the nature of the sublunary elements (fire, air, water and earth). In effect, according to Solmsen, Aristotle was convinced that, in order to give account of the coming to be and passing away of natural beings, it demanded that the ultimate substratum (protomatter) could be actuated by the contrary qualities (hot-cold and moist-dry) that inform it and mold it into the elements4. For this sake, Aristotle chooses one pair of qualities, hot and cold, as active, and other, moist and dry, as passive.  These acting and suffering of the qualitites within the substratum cause the mutual transformation of the elements, so that they are constitutive factors of these latter5.  Furthermore,  Aristotle “bears in mind the two fundamental processes, intermutation and mixture, for which the constituents [qualities] must be able to account”.6  But these operations, intemutation and mixture of the elements, have also been supported by Saint Thomas in his opuscule, De Mixtione Elementorum, according to which he affirms that “we should consider that the active and passive qualities of the elements are contrary to each other and susceptible of plus and minus”.7  Furthermore, “the substantial forms are not given from the outside, but they are extracted from the potency of matter by means of a proper transmutation”8.
Based upon several texts on Saint Thomas about the elemental combination and intermutation, we can add some relevant points: firstly, that the qualities reside into the essence of matter; secondly, that there are operations among the elements through their mixing (combining) and intermutation (self-transformation), resulting from the active and passive mixing of qualities within matter. Let us them present an algebraic model that help us to understanding what may be going on within the essence of protomatter, from an epistemological point of view.
Let us name  the hot quality,  the cold quality,  the dry quality and  the moist quality. The  pair   is active, and the pair  passive, according to Aristotle´s own proposal. Well,  let us suppose we can associate a number to some quality  that entitles us to say that it (the specified quality) varies according to plus and minus, as claimed by Saint Thomas. So, let us represent such intensity by , where  is the symbol to represent any quality and  is the kind of quality (passive, for instance) to which a certain intensity   is associated.  We can also suppose that some ammount of intensity can enable the passage from one edge to the other; if, for instance,  is positive, then we get dry, and if  is negative, we get moist.  In effect, the pair dry-moist might then be represented by one single element   in so far as the range on which  varies enables a full definition of the quality and its intensity. The same applies to the pair hot-cold, which might be represented by the single element .  The reason why the intensity  is represented by an upper index is that it represents activity, as the lower index in the previous element represents passivity.  Well, if one considers a generic quality where one has acitivity and passivity altogether in one element, then we may represent this general fact by .  In so far as one allows the existence of upper and lower indexes to represent the simultaneous existence of activity and passivity in one element, then the fundamental pure active quality might be expressed by , as well as the fundamental pure passive quality might be expressed by . 
What then have we achieved so far? A kind of representation for the fundamental active-passive character of matter, expressed by the basic qualities residing within the substractum. The fundamental pair of constitutive components shall work as a kind of “basis”  from which the other primitive components can be spanned by appropriate operations, once they have been defined accordingly. The primitive components of the basis are called primitive idempotents. In the next section of this series of communications we will develop a quite simple but effective model of the substractum, and we will show that this substractum is not an absolute pure potentiality, but it is  “pregnant”  with the fundamental elements that allow us to properly call it the ens in potentia.