Hasse diagrams

Bruggemann, R., J. Schwaiger, and R.D. Negele. 1995. Applying Hasse diagram technique for the evaluation of toxicological fish tests. Chemosphere 30 1767-1780. 

Hollert, H., Heise, S., I udcnz, S., Briiggemann, R., Ahlf, W. and Braunbeck, T. (2002b) Application of a Sediment Quality Triad and different statistical approaches (Hasse diagrams and fuzzy logic) for the comparative evaluation of small streams, Ecotoxicology 11, 311-321. 

Partial order ranking (POR) is based on elementary methods of discrete mathematics (e.g., Hasse diagrams) — if A < B and B < C, then A < C in the ranking procedures. POR does not assume linearity or any assumptions about distribution properties such as normality. The disadvantage is that often a preprocessing of data is needed to avoid the effects of stochastic noise. Combining POR with PCA may improve its usefulness. POR can only be applied for interpolation. 

Note that COR is not a graph because here the vertex lying above is mapped onto the vertex lying below. Objects of this type are called Hasse diagrams, but because in our case they possess certain special properties (see below) we proposed for them the name corals". 

Theorem 17 [ 115]. The Hasse diagram corresponding to the mapping f is connected and possesses no cycles. 

Figure 19 The partial order by means of the Hasse diagram for Archimedean solids based on the four complexity indices...

Brtiggemann R, Pudenz S, Carlsen L, Sprensen PB, Thomsen M, Mishra RK. The use of Hasse diagrams as a potential approach for inverse QSAR. SAR QSAR Environ Res 2001 11 473-87. 

Another approach to the description of DNA sequences is based on the partial-ordering given by the Hasse diagrams. The order relationships between the C, T, A, G bases of a DNA sequence are recorded into the -x Hasse matrix [Todeschini, Consonni et al, 2006]. The variables used in building the Hasse matrix are the position of the bases in the sequence and a physico-chemical property of the bases, such as the mass. For example, the same small DNA sequence given above ... 

Figure B7 The Hasse diagram obtained by the sequence in the text. For each element, the number corresponds to its absolute position in the sequence.

From each property that ranks the four bases in a different way, a different Hasse diagram is obtained. Descriptors of the DNA sequence are finally obtained from the absolute values of the Hasse matrix elements and the largest eigenvalue was proposed to analyze similarity/diversity of DNA sequences. 

Hasse matrix H and Hasse diagram calculated from five objects described by two variables. Dominance is defined by the maximum values for both variables. 

Fig. 7. A Hasse diagram depicting the 18 isomers of octane with a partial ordering based on the number of paths of length two (pj) and of length three (ps) for each isomer. The diagram derives from the work of Randic and Wilkins [84]...

The detection of the periodic system of chemical elements was a break through in the theoretical understanding of chemistry. Hefferlin discusses periodicities of chemical elements and small molecules. He shows how general the concept of posets is. Why not explore the properties of small molecules by means of a Hasse diagram Hefferlin shows by the example of Phosphorus oxides how this may be done. 

The basic definitions related to the general topic of ordering are reviewed and exemplified including partial ordering, posets, Hasse diagrams, ma-jorization of structures and comparable / incomparable structures. 

The Basic Terms Examples of Posets, The Hasse Diagram... 

The graph, which describes a particular poset, is called a Hasse diagram after the 20- century German mathematician Helmut Hasse (1898-1979) (Rosen 1991). See also chapter by Halfon p. 385. 

The Hasse diagram can be drawn in different ways maintaining the main information, the order relations. Such Hasse diagrams are isomorphic to each other. 

For this particular case the Hasse diagram is simply a cube, labelled as shown in Fig. 2. 

Fig. 2. The Hasse diagram of S = a, b, c. Each subset is attached to its direct offspring, so that the descendant (less important components) lies in lower levels...

In Fig. 5 we show the posets (Hasse diagrams), which correspond to hydrocarbons containing 3,4 and 5 terminal hexagons. 

Fig. 6. The Hasse diagram of ordering 8 (out of 9) Kekule structures of triphenylene...

Hasse Diagrams and their Relation to Molecular Periodicity... 

Hasse diagrams are applied to molecules and radiation phenomena. Then the relation of these diagrams to periodicity in atoms is noted. The possibility is raised that Hasse diagrams can also be related to the growing body of evidence that periodicity exists in molecules with two, three, and four atoms in binary inorganic molecules and in some organic molecules. 

The definition and partially ordered set theory behind the Hasse diagram are given in the chapter by El-Basil, p. 3 and are not repeated we proceed immediately to discuss some interesting Hasse diagrams. 

Would it be possible to construct a molecule or a decay scheme that is its own Hasse diagram Fig. 1 shows such a hypothetical species. The first number in each box is the period number in which the atom is found the second is the number of valence electrons. The lone pair on carbon is not shown and the left-hand line is actually a double bond. Fig. 2 shows such a possible atomic decay scheme similar radiation pathways occur in molecules and also in nuclei. 

The vertical axis shows the heats of formation (Gurvich 1978, 1979, 1981, 1982) of the molecules from free atoms all of the molecules are in the gas phase (except for P2O4, which is in the crystal phase). The Hasse diagram shown in Fig. 4 represents the upper half of Fig. 3 it is easily seen how the lower half would be appended. 

Reaction diagrams are considered especially for the circumstance of progressive substitution (or addition) on a fixed molecular skeleton, and it is noted that these naturally form Hasse diagrams for a partially ordered set (or poset) of the substituted structures. The possibility that different properties are similarly ordered is a further natural consideration, and is here illustrated for several different properties for (methyl chloro) substituted benzenes. 

---