This paper discusses new algebraic methods to predict the qualitative structure of (P, T) and similar type phase diagrams. Chemical multisystems are considered that have n independent chemical components and n + k phases, with no solution. Graphical representations are developed by use of composition and reaction matrices, and based on equilibrium thermodynamics. The problem is expressed in terms of the optimum of the G function, written in a (linear) programming problem. Two types of duality between the representations are stressed, in the geometric and algebraic (or combinatorial) sense. The construction of phase composition space (chemography), reaction space and chemical potential saturation space is discussed in the general framework. A new type of space, in combinatorial and geometric duality with the chemography is introduced: the « affigraphy ». The thermodynamic meaning of affigraphy originates from adding to the condition that n phase assemblages are obtained at optimum or equilibrium, the condition that the k remaining phases cannot appear; this is expressed by the positivity of the k affinities of the dissociation of these phases. Affigraphy is organized by the affinity vectors of k independent chemical reactions around an hyperinvariant point in a k-dimensional space; the vectors are the column vectors of the chemical reaction matrix. As a « fundamental theorem », it is demonstrated that the intersection of the affigraphy space by a two-dimensional plane gives the structure of the feasible (P, T) diagrams. Several consequences of this result are outlined on simple examples. Aspects of the structure of n + 3, n + 4 and n + 5 systems are discussed; a thermodynamic understanding and generalization of Zen's polyhedra is proposed. A rule for predicting all the possible stability successions on univariant lines for n + k systems is given. Other results on the prediction of stability sequences in multidimensional domains, on the structure of degenerate systems, as well as ways of generalization and new types of diagrams are proposed. The qualitative Schreinemakers'rules for two-dimensional diagrams are seen as consequences of the laws that rule the arrangements of vectors and hyperplanes in the affigraphy. New algorithms to construct phase diagrams are proposed, that may particularly be useful when only a partial set of data on the system is available.