Uranium Trappers, a Partial Order Study
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I(x), the set of incomparable with x: I(x) = {y ∈ X : x y} I(x). When for the objects
x, y it is valid that q(x) q(y) and q(x) q(y), then x and y are incomparable
(xy).
C(x),the set of comparable with x that is given by: C(x) – {y ∈ X : x y or y x}.
When for the microorganisms x, y it is valid that q(x) q(y) or q(x) q(y), then
x and y are comparable (x⊥y).
The objects located at the top of the HD that have no predecessors in a poset are
called maximal objects and which have no successors are called minimal objects
(Bruggemann and Patil 2011). In PyHasse software, minimal objects are located
near the bottom of the drawing plane.
A special case arises when a HD contains two or more pieces called components.
A component of a poset (X, ) is a local poset (C(X i ), ) of (X, ) (Patil and Taillie
2004).
For y ∈ X,given a poset (X, ), if X i ⊆ X, then C(X i ) = {y : y ⊥ x, x ∈ X i }.
Components partition the poset into disjoint subsets such that, if x is an arbitrary
member of one component and y is an arbitrary member of a different component,
then x y (Bruggemann and Patil 2011).
A poset (X, ) is called a weak order if is transitive and meets linearity. Hence,
the difference between weak and total order is that the former is not antisymmetric
but the latter is Quintero et al. (Quintero et al. 2018).
Likewise, a ranking of X is a two-step procedure where (1) a weak order is found
for X and (2) an ordinal (rank) is assigned to each object of X (Quintero et al.
2018). To assess , the attributes need to be rightly oriented in such a way that,
for example, high values indicate similar ranking aims (common monotonicity)
(Bruggemann and Patil 2011). Further conventions to draw a Hasse diagram are
found in (Bruggemann and Patil 2011; Bruggemann and Halfon 1999).
2.2.2 Ranking Methodologies
Applying HDT on a set under study, a linear order or ranking of objects may not
be directly found; one way to overcome this issue is to determine the so-called
average heights from the concept of linear extensions (De Loof et al. 2011): A linear
order derived from a poset, preserving all its order relations is a linear extension
(Bruggemann and Patil 2011); the sequence of objects due to a linear extension is
described by their values of height; the object at the bottom of a linear extension has
height = 1, the next, height = 2, and so on (Bruggemann and Patil 2011).
The number of linear extensions suggests how complex is the poset and how
many pairs of objects are incomparable. The calculation of average heights is
of interest because, once estimated, a weak order (tied ranks are not excluded)
can be derived (Bruggemann and Annoni 2014). Therefore, average heights are
often called average ranks (rkavs). But, the direct calculation of these average
heights by counting the heights of objects in each linear extension is most often
computationally intractable (Brightwell and Winkler 1991); the reason is because
191
I(x), the set of incomparable with x: I(x) = {y ∈ X : x y} I(x). When for the objects
x, y it is valid that q(x) q(y) and q(x) q(y), then x and y are incomparable
(xy).
C(x),the set of comparable with x that is given by: C(x) – {y ∈ X : x y or y x}.
When for the microorganisms x, y it is valid that q(x) q(y) or q(x) q(y), then
x and y are comparable (x⊥y).
The objects located at the top of the HD that have no predecessors in a poset are
called maximal objects and which have no successors are called minimal objects
(Bruggemann and Patil 2011). In PyHasse software, minimal objects are located
near the bottom of the drawing plane.
A special case arises when a HD contains two or more pieces called components.
A component of a poset (X, ) is a local poset (C(X i ), ) of (X, ) (Patil and Taillie
2004).
For y ∈ X,given a poset (X, ), if X i ⊆ X, then C(X i ) = {y : y ⊥ x, x ∈ X i }.
Components partition the poset into disjoint subsets such that, if x is an arbitrary
member of one component and y is an arbitrary member of a different component,
then x y (Bruggemann and Patil 2011).
A poset (X, ) is called a weak order if is transitive and meets linearity. Hence,
the difference between weak and total order is that the former is not antisymmetric
but the latter is Quintero et al. (Quintero et al. 2018).
Likewise, a ranking of X is a two-step procedure where (1) a weak order is found
for X and (2) an ordinal (rank) is assigned to each object of X (Quintero et al.
2018). To assess , the attributes need to be rightly oriented in such a way that,
for example, high values indicate similar ranking aims (common monotonicity)
(Bruggemann and Patil 2011). Further conventions to draw a Hasse diagram are
found in (Bruggemann and Patil 2011; Bruggemann and Halfon 1999).
2.2.2 Ranking Methodologies
Applying HDT on a set under study, a linear order or ranking of objects may not
be directly found; one way to overcome this issue is to determine the so-called
average heights from the concept of linear extensions (De Loof et al. 2011): A linear
order derived from a poset, preserving all its order relations is a linear extension
(Bruggemann and Patil 2011); the sequence of objects due to a linear extension is
described by their values of height; the object at the bottom of a linear extension has
height = 1, the next, height = 2, and so on (Bruggemann and Patil 2011).
The number of linear extensions suggests how complex is the poset and how
many pairs of objects are incomparable. The calculation of average heights is
of interest because, once estimated, a weak order (tied ranks are not excluded)
can be derived (Bruggemann and Annoni 2014). Therefore, average heights are
often called average ranks (rkavs). But, the direct calculation of these average
heights by counting the heights of objects in each linear extension is most often
computationally intractable (Brightwell and Winkler 1991); the reason is because