PyHasse, a Software Package for Applicational Studies of Partial Orderings
293
Sets equipped with the order relation (1) are called partially ordered sets (abbr.
posets), denoted as (X, ≤). The graphical presentation of posets can be done by
Hasse diagrams and is extremely useful, as long the graphs are not too complex.
The simple Eq. (1) has many facets, for example the notion of conflicts, or of
co-monotony, or of separability etc. This fact requires algorithms, which are (as
mentioned above) most often pretty simple, but awkward to perform manually.
Correspondingly PyHasse includes
(a) graph-theoretical,
(b) order-theoretical,
(c) combinatorial and
(d) statistical aspects.
Due to the school of F. Wille (see Ganter and Wille 1996) one also can
associate
(e) artificial intelligence with partial order theory, see Ganter and Wille 1996 and
a more general approach, based on the concept of t-norms, (Bruggemann et al.
2011; Kerber 2017a, b; Bruggemann and Kerber 2018).
An example of (a) is given by disjoint subsets X1, X2 of X, for which is true:
x ∈ X1, y ∈ X2 ⇒ x
y
( 2 )
Subsets with the property (2) are called separated subsets. Usually objects
belonging to different separated subsets are not connected in directed graphs (for
details, see Bruggemann and Voigt 2011). The identification of such separated
subsets is very useful in the sense of exploring the role of indicators and their values.
An example of (b) and (c) is the construction of a linear order, based on the linear
extensions of a poset (X, ≤). Linear extensions of a poset are linear orders which
preserve the order found in the poset (X, ≤). As an example, consider the object set
X = {a, b, c} for which Eq. 1 is analyzed with the following results: a < b, a < c.
Nothing is said about the relation between b and c, because for b and c the Eq. 1
does not hold, then the sequences (a < b < c) and (a < c < b) represent the original
poset whose order relations are preserved. The set of linear extensions can be further
analyzed leading to many useful concepts (for details see Bruggemann and Carlsen
2011).
An example for (d) is once again given by the set of linear extensions, which
can be evaluated by statistical tools; another example is the combination of cluster
analysis with partial order, as shown in (Bruggemann and Carlsen 2014). Other
examples are concerned with the problem of data noise (Bruggemann and Carlsen
2016) and those attempting to combine statistical proximity with a proximity
concept, based on partial order theory (Bruggemann et al. 2014a).
293
Sets equipped with the order relation (1) are called partially ordered sets (abbr.
posets), denoted as (X, ≤). The graphical presentation of posets can be done by
Hasse diagrams and is extremely useful, as long the graphs are not too complex.
The simple Eq. (1) has many facets, for example the notion of conflicts, or of
co-monotony, or of separability etc. This fact requires algorithms, which are (as
mentioned above) most often pretty simple, but awkward to perform manually.
Correspondingly PyHasse includes
(a) graph-theoretical,
(b) order-theoretical,
(c) combinatorial and
(d) statistical aspects.
Due to the school of F. Wille (see Ganter and Wille 1996) one also can
associate
(e) artificial intelligence with partial order theory, see Ganter and Wille 1996 and
a more general approach, based on the concept of t-norms, (Bruggemann et al.
2011; Kerber 2017a, b; Bruggemann and Kerber 2018).
An example of (a) is given by disjoint subsets X1, X2 of X, for which is true:
x ∈ X1, y ∈ X2 ⇒ x
y
( 2 )
Subsets with the property (2) are called separated subsets. Usually objects
belonging to different separated subsets are not connected in directed graphs (for
details, see Bruggemann and Voigt 2011). The identification of such separated
subsets is very useful in the sense of exploring the role of indicators and their values.
An example of (b) and (c) is the construction of a linear order, based on the linear
extensions of a poset (X, ≤). Linear extensions of a poset are linear orders which
preserve the order found in the poset (X, ≤). As an example, consider the object set
X = {a, b, c} for which Eq. 1 is analyzed with the following results: a < b, a < c.
Nothing is said about the relation between b and c, because for b and c the Eq. 1
does not hold, then the sequences (a < b < c) and (a < c < b) represent the original
poset whose order relations are preserved. The set of linear extensions can be further
analyzed leading to many useful concepts (for details see Bruggemann and Carlsen
2011).
An example for (d) is once again given by the set of linear extensions, which
can be evaluated by statistical tools; another example is the combination of cluster
analysis with partial order, as shown in (Bruggemann and Carlsen 2014). Other
examples are concerned with the problem of data noise (Bruggemann and Carlsen
2016) and those attempting to combine statistical proximity with a proximity
concept, based on partial order theory (Bruggemann et al. 2014a).