# Math: pyhasse.fuzzy

1. The relation between two objects x and y is replaced by a fuzzy subsethood, and the degree of subsethood of x fuzzy y, and that of y fuzzy x will be calculated. The idea is that by defuzzification a limit can be introduced whether or not a subsethood is considered as a
2. There may be different measures of the subsethood degree. Kerber and Brüggemann  suggest a generalized approach within the framework of fuzzy set algebra. Here, the measure is taken from ideas of Kosko  which were first time transferred into the fuzzy framework by De Walle et al., [De Walle, 1995]. A matrix, where the subsethood of each object versus each other object, is obtained.
3. The crisp transitivity demand as the essential axiom to declare a relational matrix like the subsethood matrix representing a partial order cannot applied. Instead a fuzzy transitivity requirement is needed: The fuzzy relation R(x,y), R(y,z) fulfills the fuzzy transitivity if equation (4) holds: (4)

Here R (Relation) is specialized for the subsethood measure SH, as obtained in step 2.According to Kosko SH(x,y) is calculated as follows: (5)

If the denominator = 0, then SH(x,y) = 1.
4. By an iterative procedure the matrix SH is modified, until the requirement (4) is fulfilled. In PyHasse the matrix method, [De Baets, DeMeyer, 2003] is applied. It results a matrix , which obeys (3) and w from which one can speak of a representation of a fuzzy partial order.
5. A defuzzification is needed, to obtain relations, which can be drawn as Hasse diagram. A crisp matrix is obtained, following equation (6). (6)

The parameter (acut in PyHasse) serves as a threshold. The matrix may still contain equivalence relations. After extracting them, a description of a poset by the resulting matrix, is obtained.

## Examples

See Bruggemann et al., 2011, Altschuh et al, 2015 and Bruggemann, Patil, 2011.