x
Indicators and Partial Orders – An Introduction
Table 1 The three indicators
Indicator
Short
Description
Direction
sdg7_warm
sdg7_w
Population unable to keep homes adequately warm
(%)
Low better
sdg7_eurenew sdg7_e
Share of renewable energy in gross final energy
consumption (%)
High better
sdg7_co2twh
sdg7_co2 CO 2 emissions from fuel combustion per electricity
output (MtCO2/TWh)
Low better
context, but not order theoretically). These three objects are members of a socalled antichain.
(3) The group {2010, 2012} has no order theoretical connections with {2013}. The
identification of the reasons in terms of indicator values is one main task in the
applications of Hasse diagrams.
Combinatorics
Combinatorics comes mainly into play when directed graphs of the order relations
are extended to form graphs with more connections maintaining the already given
ones. This enrichment process can be continued until a complete order is obtained.
However, when the Hasse diagram has elements of X that are not in an order
relation, then the enrichment process delivers a set of complete orders, i.e., the set of
linear extensions. However, the generation of linear extensions from a given Hasse
diagram is computationally extremely difficult. Here combinatorics helps to find
algorithms or even to find closed formulas. These, e.g., play an important role in
an approximation, known as a local partial order model. An example would be the
Hasse diagram (Fig. 1). It is possible to extend the graph to a linear order, where the
sequence of years follows its natural order.
Algebra
It seems to be plausible to try to understand empirical partial orders as being
composed of simpler graph structures. Any two partially ordered sets (posets) can
be combined by following strict composition rules. These composition rules, such as
addition, multiplication and disjoint union, have only little to do with the operations
known for numbers. Nevertheless, this kind of composition is an important guideline
to understand empirical posets. A remarkably richer algebra is obtained, when the
order relations obeys additional requirements. The crucial concept is the uniqueness.
Within an empirical poset, two elements of a set X can be in order relation to
several others. However, when the additional requirement is uniqueness, then any
Indicators and Partial Orders – An Introduction
Table 1 The three indicators
Indicator
Short
Description
Direction
sdg7_warm
sdg7_w
Population unable to keep homes adequately warm
(%)
Low better
sdg7_eurenew sdg7_e
Share of renewable energy in gross final energy
consumption (%)
High better
sdg7_co2twh
sdg7_co2 CO 2 emissions from fuel combustion per electricity
output (MtCO2/TWh)
Low better
context, but not order theoretically). These three objects are members of a socalled antichain.
(3) The group {2010, 2012} has no order theoretical connections with {2013}. The
identification of the reasons in terms of indicator values is one main task in the
applications of Hasse diagrams.
Combinatorics
Combinatorics comes mainly into play when directed graphs of the order relations
are extended to form graphs with more connections maintaining the already given
ones. This enrichment process can be continued until a complete order is obtained.
However, when the Hasse diagram has elements of X that are not in an order
relation, then the enrichment process delivers a set of complete orders, i.e., the set of
linear extensions. However, the generation of linear extensions from a given Hasse
diagram is computationally extremely difficult. Here combinatorics helps to find
algorithms or even to find closed formulas. These, e.g., play an important role in
an approximation, known as a local partial order model. An example would be the
Hasse diagram (Fig. 1). It is possible to extend the graph to a linear order, where the
sequence of years follows its natural order.
Algebra
It seems to be plausible to try to understand empirical partial orders as being
composed of simpler graph structures. Any two partially ordered sets (posets) can
be combined by following strict composition rules. These composition rules, such as
addition, multiplication and disjoint union, have only little to do with the operations
known for numbers. Nevertheless, this kind of composition is an important guideline
to understand empirical posets. A remarkably richer algebra is obtained, when the
order relations obeys additional requirements. The crucial concept is the uniqueness.
Within an empirical poset, two elements of a set X can be in order relation to
several others. However, when the additional requirement is uniqueness, then any