viii
Indicators and Partial Orders – An Introduction
series of preliminary indicators a multi-indicator system (MIS). The information
within a certain MIS is often important within a holistic point of view (see
for instance Maggino and Zumbo 2012). The aggregation, independent of which
method is applied, must be more or less considered as an averaging. Thus, it seems
to be appropriate to evaluate the MIS as an interim aspect by mathematical methods,
which are able to analyse multiple indicators with respect to the objective under
which the MIS was constructed. The mathematical method of partial order theory is
very helpful in this aspect, and, therefore, indicators and partial order are closely
interrelated when an evaluation by ranking is wanted. Clearly, the partial order
methodology is not the only possibility for studying an MIS (see, e.g., (Brans and
Vincke 1985; Figueira et al. 2005; Colorni et al. 2001; Munda 2008; Munda and
Nardo 2009; Roy 1972; Roy and Vanderpooten 1996; Maggino 2017)).
Here, however, the interplay of MIS and partial order is the main topic.
Partial Order Methodology
When one takes a closer look at the mathematics of partial ordering, it is closely,
although not exclusively related to the regime of indicators. Partial ordering is a
theory of binary relations and is as such especially well-suited for those indicators,
which are ordinal in nature. The reason is that partial order is mathematically deeply
intertwined with
• Graph theory
• Combinatorics
• Algebra
but not with numerical evaluation in the field of real numbers, see, for instance,
(Trotter 1992). In the following, the three items are described in more detail.
Graph Theory
One of the most important visualization techniques of partial orders is the Hasse
diagram. The Hasse diagram is a transitively reduced, acyclic digraph. This
characterization may be enough for mathematicians, but not for scientists interested
in applications. Thus, a few more details are given here.
Partial order is a binary relation among elements x i and x j of a set X which can
be interpreted as ‘better than’, e.g., x i > x j . This relation obeys three axioms:
• Reflexivity, i.e., an element can be compared with itself.
• Antisymmetry, i.e., if an element x is ‘better’ than an element y, then y cannot be
better than x, unless x and y are identical.
Indicators and Partial Orders – An Introduction
series of preliminary indicators a multi-indicator system (MIS). The information
within a certain MIS is often important within a holistic point of view (see
for instance Maggino and Zumbo 2012). The aggregation, independent of which
method is applied, must be more or less considered as an averaging. Thus, it seems
to be appropriate to evaluate the MIS as an interim aspect by mathematical methods,
which are able to analyse multiple indicators with respect to the objective under
which the MIS was constructed. The mathematical method of partial order theory is
very helpful in this aspect, and, therefore, indicators and partial order are closely
interrelated when an evaluation by ranking is wanted. Clearly, the partial order
methodology is not the only possibility for studying an MIS (see, e.g., (Brans and
Vincke 1985; Figueira et al. 2005; Colorni et al. 2001; Munda 2008; Munda and
Nardo 2009; Roy 1972; Roy and Vanderpooten 1996; Maggino 2017)).
Here, however, the interplay of MIS and partial order is the main topic.
Partial Order Methodology
When one takes a closer look at the mathematics of partial ordering, it is closely,
although not exclusively related to the regime of indicators. Partial ordering is a
theory of binary relations and is as such especially well-suited for those indicators,
which are ordinal in nature. The reason is that partial order is mathematically deeply
intertwined with
• Graph theory
• Combinatorics
• Algebra
but not with numerical evaluation in the field of real numbers, see, for instance,
(Trotter 1992). In the following, the three items are described in more detail.
Graph Theory
One of the most important visualization techniques of partial orders is the Hasse
diagram. The Hasse diagram is a transitively reduced, acyclic digraph. This
characterization may be enough for mathematicians, but not for scientists interested
in applications. Thus, a few more details are given here.
Partial order is a binary relation among elements x i and x j of a set X which can
be interpreted as ‘better than’, e.g., x i > x j . This relation obeys three axioms:
• Reflexivity, i.e., an element can be compared with itself.
• Antisymmetry, i.e., if an element x is ‘better’ than an element y, then y cannot be
better than x, unless x and y are identical.