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L. Carlsen and K. Abit
the so-called Hasse diagram. Hasse diagrams are unique visualizations of the order
relations due to Eq. 1.
2.2.1 The Hasse Diagram
The Eq. 1 is the basic for the Hasse diagram technique (HDT) (Bruggemann
and Carlsen 2006; Bruggemann and Münzer 1993; Bruggemann and Patil 2011).
Hasse diagrams are visual representation of the partial order. In the Hasse diagram
comparable elements are connected by a sequence of lines (Bruggemann and
Carlsen 2006; Bruggemann and Münzer 1993; Bruggemann and Patil 2011).
By convention, the Hasse diagram originally introduced by Halfon and Reggiani
(1986) is drawn with.
• x ≤ y locating x below y,
• attempting a symmetric presentation as far as possible and
• by an arrangement of elements in levels that are numbered from the bottom and
upwards.
• each element is placed at the highest possible level in the diagram as possible
Two important concepts, in addition to the level structure can directly obtained
by inspecting a Hasse diagram:
• Chains are subsets of X, where each element is mutually comparable to the
others. Those subsets are denoted as ‘completely ordered’: Inspecting a Hasse
diagram, any sequence of lines upwards or (strictly) downwards is a chain
• Antichains are sets, where each element is mutually incomparable with the
others. Hence, levels are subsets of the set of all antichains
• Maximal elements are elements where for a given element x there is no elements
y where x ≤ y
• Minimal elements are elements where for a given element x there is no elements
y where y ≤ x
• If x is at the same time a maximal and a minimal element, then x is called an
isolated element. Isolated elements are always of interest as they must have a
special data structure, which makes them incomparable to any other element of
X.
For a detailed explanation see the work by Bruggemann and Patil (2011).
2.3 The More Elaborate Analyses
In addition to the basic partial ordering tools some more elaborate analyses have
been used including average ranks (Bubley and Dyer 1999; Bruggemann et al. 2004;
De Loof et al. 2006; Bruggemann and Patil 2011, Bruggemann and Carlsen 2011;
Bruggemann and Annoni 2014) and sensitivity analysis (Bruggemann and Patil
L. Carlsen and K. Abit
the so-called Hasse diagram. Hasse diagrams are unique visualizations of the order
relations due to Eq. 1.
2.2.1 The Hasse Diagram
The Eq. 1 is the basic for the Hasse diagram technique (HDT) (Bruggemann
and Carlsen 2006; Bruggemann and Münzer 1993; Bruggemann and Patil 2011).
Hasse diagrams are visual representation of the partial order. In the Hasse diagram
comparable elements are connected by a sequence of lines (Bruggemann and
Carlsen 2006; Bruggemann and Münzer 1993; Bruggemann and Patil 2011).
By convention, the Hasse diagram originally introduced by Halfon and Reggiani
(1986) is drawn with.
• x ≤ y locating x below y,
• attempting a symmetric presentation as far as possible and
• by an arrangement of elements in levels that are numbered from the bottom and
upwards.
• each element is placed at the highest possible level in the diagram as possible
Two important concepts, in addition to the level structure can directly obtained
by inspecting a Hasse diagram:
• Chains are subsets of X, where each element is mutually comparable to the
others. Those subsets are denoted as ‘completely ordered’: Inspecting a Hasse
diagram, any sequence of lines upwards or (strictly) downwards is a chain
• Antichains are sets, where each element is mutually incomparable with the
others. Hence, levels are subsets of the set of all antichains
• Maximal elements are elements where for a given element x there is no elements
y where x ≤ y
• Minimal elements are elements where for a given element x there is no elements
y where y ≤ x
• If x is at the same time a maximal and a minimal element, then x is called an
isolated element. Isolated elements are always of interest as they must have a
special data structure, which makes them incomparable to any other element of
X.
For a detailed explanation see the work by Bruggemann and Patil (2011).
2.3 The More Elaborate Analyses
In addition to the basic partial ordering tools some more elaborate analyses have
been used including average ranks (Bubley and Dyer 1999; Bruggemann et al. 2004;
De Loof et al. 2006; Bruggemann and Patil 2011, Bruggemann and Carlsen 2011;
Bruggemann and Annoni 2014) and sensitivity analysis (Bruggemann and Patil