Indicators and Partial Orders – An Introduction
ix
• Transitivity, i.e., if x is ‘better’ than y, and y is ‘better’ than z, then x is ‘better’
than z. A classical counterexample is the tournament. One may define ‘better
than’ as team x beats team y. However, although team y may beat team z, it cannot
be excluded that team z beats team x, which is a violation of the transitivity. On
the other hand, when the order relation is associated with numerical, ordinal
indicator values, the order relation between two elements is governed by the
numerical relation between the indicator values. Thus, the elements can be
ordered, i.e., fulfilling the axiom of transitivity.
If two elements of a set X have an order relation, one can define two vertices for
the elements and connect them. Because the relation is oriented, the orientation for
the two elements is indicated by an arrow and the relation can be described by our
usual symbol ‘<’. When this recipe is performed for all elements of a set, a directed
graph is obtained. When the order relation is based on only one single indicator,
then a complete – linear – order is developed and each of the two elements of X are
connected by an arrow. When there are three elements x, y, z and it is found x < y and
y < x, then transitivity demands that x < z. Hence, for most applications the arrow
for x < z can be omitted, as this relation follows due to the transitivity. The process
of eliminating arrows is called a transitive reduction. Furthermore, a sequence of
arrows such as x 0 < x 1 , x 1 < x 2 , . . . , x n-1 < x n , but x n < x 0 is obviously not possible
as it would be a violation of the transitivity as the transitive reduction would cause
a cyclic graph. Eventually, the arrows can be replaced by simple lines, when the
orientation is governed by the vertical position in the drawing plane. The resulting
graph is called a Hasse diagram. Hasse diagrams or comparability graphs (graphs
of the order relation, however without an orientation) can be analysed theoretically.
Note that a sequence of lines which can be followed strictly upwards or downwards
may be called an order theoretical connection, the set of objects within an order
theoretical connection is called a chain.
An analysis can, e.g., investigate whether or not subsets of X dominate others,
or whether subsets of X are strikingly not connected or only weakly connected
with other parts of the graph. It is clear that ‘weakly’ needs a definition. Here it
is used in the sense of ‘only few connections’. As an example of a Hasse diagram,
we can look at the development in Germany (2008–2015) in switching to more
sustainable energy according to the UN Sustainable Development Goal No. 7, using
three indicators (Table 1); for details see (Europe Sustainable Development Report
2019).
Instead of observing three line graphs for each indicator, the Hasse diagram
shows at once some essential facts:
(1) All three indicators are not decreasing in their values for the time evolution:
2008-2009-2011-2012-2015. Other time series can be found, where the indicator values are simultaneously non-decreasing. One can see that for these special
set of years, the pattern of indicator values is co-monotone with the time. Such
subsets of objects, mutually comparable are called chains.
(2) 2010, 2011 and 2013 cannot be compared, because of a counter current development of indicator values. They are connected (in a general graph theoretical
ix
• Transitivity, i.e., if x is ‘better’ than y, and y is ‘better’ than z, then x is ‘better’
than z. A classical counterexample is the tournament. One may define ‘better
than’ as team x beats team y. However, although team y may beat team z, it cannot
be excluded that team z beats team x, which is a violation of the transitivity. On
the other hand, when the order relation is associated with numerical, ordinal
indicator values, the order relation between two elements is governed by the
numerical relation between the indicator values. Thus, the elements can be
ordered, i.e., fulfilling the axiom of transitivity.
If two elements of a set X have an order relation, one can define two vertices for
the elements and connect them. Because the relation is oriented, the orientation for
the two elements is indicated by an arrow and the relation can be described by our
usual symbol ‘<’. When this recipe is performed for all elements of a set, a directed
graph is obtained. When the order relation is based on only one single indicator,
then a complete – linear – order is developed and each of the two elements of X are
connected by an arrow. When there are three elements x, y, z and it is found x < y and
y < x, then transitivity demands that x < z. Hence, for most applications the arrow
for x < z can be omitted, as this relation follows due to the transitivity. The process
of eliminating arrows is called a transitive reduction. Furthermore, a sequence of
arrows such as x 0 < x 1 , x 1 < x 2 , . . . , x n-1 < x n , but x n < x 0 is obviously not possible
as it would be a violation of the transitivity as the transitive reduction would cause
a cyclic graph. Eventually, the arrows can be replaced by simple lines, when the
orientation is governed by the vertical position in the drawing plane. The resulting
graph is called a Hasse diagram. Hasse diagrams or comparability graphs (graphs
of the order relation, however without an orientation) can be analysed theoretically.
Note that a sequence of lines which can be followed strictly upwards or downwards
may be called an order theoretical connection, the set of objects within an order
theoretical connection is called a chain.
An analysis can, e.g., investigate whether or not subsets of X dominate others,
or whether subsets of X are strikingly not connected or only weakly connected
with other parts of the graph. It is clear that ‘weakly’ needs a definition. Here it
is used in the sense of ‘only few connections’. As an example of a Hasse diagram,
we can look at the development in Germany (2008–2015) in switching to more
sustainable energy according to the UN Sustainable Development Goal No. 7, using
three indicators (Table 1); for details see (Europe Sustainable Development Report
2019).
Instead of observing three line graphs for each indicator, the Hasse diagram
shows at once some essential facts:
(1) All three indicators are not decreasing in their values for the time evolution:
2008-2009-2011-2012-2015. Other time series can be found, where the indicator values are simultaneously non-decreasing. One can see that for these special
set of years, the pattern of indicator values is co-monotone with the time. Such
subsets of objects, mutually comparable are called chains.
(2) 2010, 2011 and 2013 cannot be compared, because of a counter current development of indicator values. They are connected (in a general graph theoretical