


EXTENDING CONCEPT LATTICES BEYOND ATTRIBUTES TO RELATIONS 

To that end we are going to generalize the notions of context and of lattice concepts such as they are defined by R. Wille [4] as follows: A context is a triple (G,M,1) where G is a set of objects, M is a set of attributes, and I is a binary relation between G and M indicating by that the object has the attribute . There is a natural Galois connection between G and M (which we indicate with "prime" marks) defined by for ,and for . A concept of the context (G, M,1) is a pair (A, B) with , A' = B, and B' = A. The set L(G, M, I) of the concepts of is partially ordered by the definition if and only if . is a complete lattice called the latticeconcept of (G, M, I). Wille's concept lattices are described in his article in this volume. From our standpoint that will mean considering each attribute as representing all objects to which it is related. DEFINITION 13. Let J and K be two posets and and the corresponding categories of phanerons. and are said to be connected if, for at least there exists at least such that, if ' is a phenomenological morphism between two objects and of D then there exists a phenomenological morphism between two objects of D' and two phenomenological morphisms and such that we have the commutative diagram: 
The morphisms and are called connections between and and the set of these connections is called the "connecting set" between and
DEFINITION 14:. The triple where and are categories of phanerons and a connecting set between them is called a "representation context." We call it a representation context because this notion captures the substitution of a part of a phaneron of any type by a part of a phaneron of another type. If and are categories of phanerons of which all the objects are diagrams reduced to one element, then the connecting set is a binary relation between two sets which are named respectively objects and attributes by Wille and our representation context is in accordance with his definition of a context. The notion of representation context seems interesting as a theoretical framework for various formalizations. For instance, if is a part of a language in which objects are names and morphisms are predicates we can to talk of "linguistic context of representation"; if is a set of dots and lines with rules of combination we can to talk of "graphic context"; likewise for "pictorial context," "musical context," etc. In the case of Touretzky's semantic net (see Figure 5), is the set of all particular elephants, particular cars and particular motorcycles provided with the complex of real relations existing in the world between them (and present to Touretzky's mind). These relations are monadic, dyadic and also triadic since Touretzky accepts that the elephants are thinking animals. J is the underlying partial order on this set (for instance Clyde < Royal Elephant because there exists a phenomenological morphism of which the source is Clyde into any one of the Royal Elephants). Between Elephants and Motorcycles there is no phenomenological morphism. is Touretzky's net and the connecting set is constituted by the set of phenomenological morphisms which, for instance, maps every elephant onto the pair constituted by a dot and the word "Elephant," and maps also every phenomenological morphism between Royal Elephants and Elephants onto the arrow linking the corresponding dots. It maps every individual preference of every elephant relatively to cars and motorcycles onto the triple of the words (elephant, car, motorcycle) linked as in the network. This representation combines graphical and linguistic representation contexts. DEFINITION 15. If is a representation context and if , every such that there exists a connection of connecting D and D' is a sign of D in this representation context. (This definition agrees with Peirce's meaning of a "sign".) Now, for all sets A of objects of , we define the subset A' of objects of which are signs of all objects of A and, for all subsets B of we define the subsets B' of all objects of of which the objects of B are signs. The maps and form a Galois connection between the power sets of and . DEFINITION 16. All pairs such that A = B' and B=A' are called semiotics of the representation context. Thus, a concept in Wille's sense is a semiotic in which every attribute is a sign of a set of objects and the intention of a concept is the set of attributes related to the concept. Similarly every object is signified by a set of attributes, and the extension is the set of objects related to the concept. By analogy we say that A is the extension of the semiotic (A, B) and B its intention. DEFINITION 17. Let and be two semiotics of the same representation context, we define an order of relationship for the set of semiotics of this representation context by: if and only if and The set of the semiotics of the representation context is partially ordered by this relation , and we will say that the semiotic is better than the semiotic because it represents more objects using less signs. The notion of the representation context can be adapted in order to respond to particular situations. The most interesting cases seem to be when the ordered set is a lattice or better, when it is a complete lattice with a top, because in this latter case there exists a universal semiotic which represents everything. Semantic networks can also be interpreted with the help of representation contexts, being constituted by a unique object (a labelled graph or hypergraph present to the mind of the reader) connected with a category of phanerons of which it must express some knowledge. For instance, in Winograd's network (Figure 1), is a state of things present, any day, to Winograd's mind concerning supposed physical entities (Kazuo, Fido) which are instances of subconcepts (person, dog) of a superconcept (animal), facts concerning these entities (to eat, to own, to be an ingredient) and laws governing these facts. The connecting set contains two classes of elements: the first (graphical convention) is a functor which maps the physical entities and their related concepts onto the nodes of the graph on the one hand, and maps the relations between entities, subconcepts and superconcepts onto the arrows of the graph, on the other; the second (linguistic convention) maps the same elements of onto the labels with the implicit rule according to which the labels of the nodes occupy the corresponding place markers of the predicates labels on the arrows which join them. The definition of our representation contexts raises the same questions as are raised by Wille's lattice concepts: How do we determine the semiotics of empirical representation contexts? How do we describe the order graph on the set of the semiotics of a representation context in order to optimize the choice of the semiotics in particular examples (for instance, choice of notations or graphical conventions, conception of a signalling system, the representation of a problem with a view to its computerisation, etc.)? However, taking in account the possibility of the foliation of the set of the phenomenological morphisms we can reduce the general problems of the representation to a few underlying problems which are their corresponding expression in the different "sheets" which are associated with them. Thus it will be possible to categorize notational differences which depend on phenomenological differences in each problem. In this respect we should remind ourselves of D. J. Israel's remark: the kind of things and the qualities of things belong to different phenomenological categories. Therefore we can associate with every category of phanerons the categories of phanerons obtained by decomposing it and then assembling according to their category the decomposition of their diagrams and their morphisms as previously indicated. This being done for the and (since they are simply examples of diagrams) of a representation context we now decompose the morphisms of : we will finally obtain in a natural way a "foliation" of the original representation context into a poset of elementary representationcontexts. These latter are of six types (arities) according to the previous results. This property is naturally transmitted to the "semiotics" of the elementary representation contexts. Thus, the questions of the determination,categorization and the description of these semiotics come down to the study of the same questions in the cases of elementary representation contexts which are of one of the six following types:
where the index numbers of the letters a and 0 indicate the type (arity) of the relational structures and those of the letter the types of elementary morphisms. These elementary representation contexts are naturally ordered so as to form a lattice. This order corresponds to the hierarchy of the "cenopythagorean categories." We obtain the lattice represented in Figure 6. Moreover, we have similar lattices with the semiotics of every elementary representation context,. Going back to semantic networks regarded as particular representation contexts we see that the introduction of the phenomenological aspect leads to a foliated conception (meaning that we can distinguish between different hierarchized levels which can be compared with geological strata) through the resulting distribution of the nodes and relations possibly reduced to their elementary forms between the different levels corresponding to the classes in Figure 6. 
Figure 6. The lattice of the different possible species of "representation context" depending on the fundamental categories (1, 2 or 3) of the entities on either side of our (Galois) "representation relations." R. Wille's lattice concepts are of type [2, 1] because they consist of sets of dyads objects described using monad attributes. There are five other representation possibilities, depending on the "arity" of the entities. can be the designated and the designata, and is the representation relation itself. 
As an example we will deal with Winograd's network,
(Figure 1) based on our analytical method and we would suggest a
few additions in order to ensure phenomenological harmony and consistency.
Our proposal is the diagram, Figure 7, in which the levels of the
elementary representation contexts are the planes and the relations
are dotted lines. 

We consider that the representation contexts are determined and connected by means of social processes and agreements which govern all inter individual communications. This latter only actualizes the social consent about representation of things by other things. However, this is a process which changes at the same time as the real world changes. Many discussions and conflicts of a semantic order come from disagreements and changes which occur during this process. In other words, we communicate always within a framework imposed by society, but we can create new connections and/or new categories of phanerons which will or will not be incorporated in our culture. Nevertheless, most of the time, the social consent is sound and we communicate without difficulty. We find in the writings of Peirce the notion of "Commens" or "Cominterpretant," an entity which is required so that communication can take place, which is very close to this conception. In order to take these remarks into account, we can undertake our formal approach, which generalises the notion of the "fusion" of individual contexts proposed by Wille [22], to the fusion of representation contexts. In these formalisms, the phenomenological reduction can effectively contribute to the simplification of the description of the phenomena. 