A Study to Generate a Weak Order from a Partially Ordered Set, Taken. . .
67
et al. 1998, 2000; Pavan and Todeschini 2004; Pudenz and Heininger 2006; Quintero
et al. 2018; Restrepo and Bruggemann 2008; Restrepo et al. 2008a, b; Voigt et al.
2004a, b) with reference to the German mathematician Helmut Hasse (1967). Sets
X, equipped with a partial order (and thus in this paper by the Eqs. 1, 2, and 3) are
called partially ordered sets and are conveniently denoted as posets and indicated by
(X, ≤).
Two objects x, y mutually incomparable are denoted as x y. Within a poset (X,
≤) the number of incomparable pairs x y is called U. When the orientation x ≤ y
or x ≥ y is of minor interest then the mere fact of comparability is denoted by x ⊥ y.
Note that partial order methodology can also be applied, by evaluation of the
space of all possible data profiles, when the indicators are discrete (cf. e.g. Fattore
and Maggino 2014, as well as Maggino et al. this book).
2.3 Hasse Diagram
The construction of a Hasse diagram, starting from a set of partial order relations (as
an outcome of Eq. 3) is frequently explained in the literature (see e.g. Bruggemann
and Halfon 1997). For the sake of reader’s convenience, some words about Hasse
diagrams may nevertheless useful here: The basis is the order relation x < y. Usually
the object x will be drawn below object y; both are vertices of a graph and presented
by small circles, with the label of the object in the centre. In case x < y a line is
connecting x with y, called an edge, if the vertices are in a cover relation, i.e if there
is no object z for which is valid: x < z < y. The orientation of the order relation is
just obtained from the vertical position. When two objects are not connected by a
system of oriented edges the two objects are incomparable.
By this construction a Hasse diagram allows a two-fold interpretation:
1. Upwards: The numerical values of the objects are nondecreasing along a system
of edges. This “vertical” oriented analysis allows a ranking of objects of subsets
of X, so-called chains.
2. In contrast to (1) there is also a “horizontal” evaluation. This evaluation has its
focus on not connected objects. Following the construction principles of a Hasse
diagram, the objects of in the same vertical position are mutually incomparably.
A set of mutually incomparable objects is called an antichain.
2.4 Weak Order
When the general policy of decision is to find not only the optimal option but also
alternatives, then ranking is a good starting point, since suboptimal objects can
be easily identified if the optimal object is not suitable (e.g., due to political or
economic reasons). The task is how to get a ranking, which is at least a weak order,
67
et al. 1998, 2000; Pavan and Todeschini 2004; Pudenz and Heininger 2006; Quintero
et al. 2018; Restrepo and Bruggemann 2008; Restrepo et al. 2008a, b; Voigt et al.
2004a, b) with reference to the German mathematician Helmut Hasse (1967). Sets
X, equipped with a partial order (and thus in this paper by the Eqs. 1, 2, and 3) are
called partially ordered sets and are conveniently denoted as posets and indicated by
(X, ≤).
Two objects x, y mutually incomparable are denoted as x y. Within a poset (X,
≤) the number of incomparable pairs x y is called U. When the orientation x ≤ y
or x ≥ y is of minor interest then the mere fact of comparability is denoted by x ⊥ y.
Note that partial order methodology can also be applied, by evaluation of the
space of all possible data profiles, when the indicators are discrete (cf. e.g. Fattore
and Maggino 2014, as well as Maggino et al. this book).
2.3 Hasse Diagram
The construction of a Hasse diagram, starting from a set of partial order relations (as
an outcome of Eq. 3) is frequently explained in the literature (see e.g. Bruggemann
and Halfon 1997). For the sake of reader’s convenience, some words about Hasse
diagrams may nevertheless useful here: The basis is the order relation x < y. Usually
the object x will be drawn below object y; both are vertices of a graph and presented
by small circles, with the label of the object in the centre. In case x < y a line is
connecting x with y, called an edge, if the vertices are in a cover relation, i.e if there
is no object z for which is valid: x < z < y. The orientation of the order relation is
just obtained from the vertical position. When two objects are not connected by a
system of oriented edges the two objects are incomparable.
By this construction a Hasse diagram allows a two-fold interpretation:
1. Upwards: The numerical values of the objects are nondecreasing along a system
of edges. This “vertical” oriented analysis allows a ranking of objects of subsets
of X, so-called chains.
2. In contrast to (1) there is also a “horizontal” evaluation. This evaluation has its
focus on not connected objects. Following the construction principles of a Hasse
diagram, the objects of in the same vertical position are mutually incomparably.
A set of mutually incomparable objects is called an antichain.
2.4 Weak Order
When the general policy of decision is to find not only the optimal option but also
alternatives, then ranking is a good starting point, since suboptimal objects can
be easily identified if the optimal object is not suitable (e.g., due to political or
economic reasons). The task is how to get a ranking, which is at least a weak order,