Partially ordered sets: ranking and prediction of substances' properties.
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There are at least two significant applications of partial order theory in chemistry: Ranking methods and substances' properties prediction. In both cases, a set of objects is endowed with a partial order relation e.g. "more polluting than", "can be obtained from", "more reactive than" etc. The couple of set and partial order relation is known in mathematics as a partially ordered set (poset). Ranking methods, such as the Hasse diagram technique, lead to a partial order where several incomparabilities (lack of order) appear between pairs of objects. This phenomenon is quite common in ranking studies, and it often is circumvented by a combination of object features leading to a total order. However, such a combination introduces subjectivities and bias in the ranking process. Here a step-by-step procedure is shown to turn incomparabilities into comparabilities taking into account all the possible bias by a linear combination of features. In such a manner, it is possible to predict how probable it is to obtain a particular total order from a given poset. Similarly, it is possible to calculate the needed bias over certain attributes to obtain a particular total order. An example application is shown where substances are ranked according to their bioconcentration factor and biodegradation potential. Another application of partial order theory to chemistry has to do with the prediction of properties for a set of substances related in a (preferably systematic) chemical fashion. A customary relation is "can be obtained from"; if such a relation is set up for a given molecular structure e.g. benzene, and all its substituted derivatives (say chlorinated ones) are considered, then the set of benzene and its chlorinated derivatives are partially ordered. Taking advantage of the poset generated, different methods can be applied to predict properties of the substances considered in the poset. Such methods include the poset-average, cluster expansion, and splinoid methods. In this paper we discuss each one of these methods, its advantages and disadvantages and we outline its applicability to estimate cooperative free energies of hemoglobins with different degree of oxygenation.